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(2016年7月7日更新) [ 日本語 | English ]

微積分学 (calculus)






有珠山 / サロベツ泥炭採掘跡
1986年, 2006年の有珠山火口原. ワタスゲ・エゾカンゾウ

実数, ℜ (real number)


Def. 絶対値 absolute value, |x| < a (a > 0) ⇔ –a < x < a
Th. 三角不等式 triangle inequality
Def. 式 expression: 数学的対象を表わすために記号を連ねたもの

Def. 項 term ≡ 数・文字の掛算で表す1つの単位
Def. 係数 coefficient ≡ 項中の変数以外の数や文字(Ex. 3, a)

Def. 単項式 monomial: 数字と文字を掛けあわせただけもの
Def. 次数 degree: 文字を何個かけたかという数

→ degf [ディグリー, デグ]: 多項式fに対しdegfはその次数を表す
Ex. 5x3 → (xの次数3次) = 次数3
Ex. 2xy2z4 → (xの次数1次 + yの2次 + zの4次) = 次数7

Def. 多項式(整式) polynomial: 単項式の和で表せる (= 加法乗法のみの式)
Def. 多項式の次数: 多項式に含まれる単項式の次数の最大のもの

Ex. x3 + 4xy2 + 5z4 + 7 → 次数4 (xの次数3、yの次数2、zの次数4) → 4次多項式

Def. 同次式 homogeneous expression: 多項式中の単項式次数が全て同じ
方程式の次数: 方程式の未知数の次数

Ex. 2次方程式 = 未知数の次数が2次

Def. 有理式 rational expression (分数式fractional expression): 2つの多項式の商で与えられる式
索引
Def. 合同式 congruence equation: a, bN, ab (mod m) → amを法 modulus として合同 congruence

Ex. 17 – 5 = 1 × 12 → 17 ≡ 5 (mod 12)

Def. 斉次一次式 homogeneous linear expression: x1, x2, …, xn

f(x1, x2, …, xn) = c1x1 + c2x2 + … + cnxn
Ex. ベクトル x = (x1, x2, …, xn), 0 = (0, 0, …, 0) →

f(x + y) = f(x) + f(y), f(cx) = cf(x), f(Σi=1ncixi)
= Σi=1ncif(xi), f(0) = 0

部分分数分解 partial fraction decomposition

Ax. 相加相乗平均 arithmetic-geometric mean

二項定理 (binomial theorem)


Def. 二項係数 binomial coefficient: 二項展開後した各項の係数

nCk = n!/(k!(n - k)!)

n, k: 添字 suffix, k: 無効添字 dummy suffix → 結果に無関係

パスカルの三角形 Pascal's triangle
算術三角形論 (1654): 元の数学者朱世傑『四元玉鑑』(1303)に既述
二項展開 binomial expansion: (a + b)nを展開すること

0C0 (1)
1C0 (1)___1C1 (1)
2C0 (1)___2C1 (2)___2C2 (1)
3C0 (1)___3C1 (3)___3C2 (3)___3C3 (1)
4C0 (1)___4C1 (4)___4C2 (6)___4C3 (4)___4C4 (1)
5C0 (1)___5C1 (5)___5C2 (10)___5C3 (10)___5C5 (5)___5C5 (1)

n = 0___1
n = 1___a + b
n = 2___a2 + 2ab + b2
n = 3___a3 + 3a2b + 3ab2 + b3
n = 4___a4 + 4a3b + 6a2b2 + 4ab3 + b4
n = 5___a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Th. 二項定理 binomial theorem

ニュートンの二項式 (Newton 1644/65) (a + b)n = Σi=0nnCkankbk

Pr0. 統計的証明
(a + b)n = (a + b)(a + b) … (a + b)
________------------ n個 ------------
an-kbkの係数は、n個の因子(a + b)からbk個選ぶ組み合わせ → nCk //
Pr1. 数学的帰納法: 自然数nに対する命題C(n)
[10 n = 1 → 成立, 20 n = k → 成立, n = k + 1 → 成立]
10 n = 1 自明 trivial
20 n = kのとき成り立つと仮定
(a + b)k + 1 = (a + b)k·(a + b) = Σi=0kkCiakibi·(a + b)

= Σi=0kkCiak+1–ibi + Σi=0kkCiakibi+1

i + 1 := jΣi=0kkCiakibi+1 = Σj=1k+1kCj-1ak+1-jbj = Σi=1k+1kCi-1ak+1–ibi
∴ (a + b)k + 1 = kC0ak+1 + Σi=1k(kCi + kCi-1)ak+1-ibi + kCka0bk+1

= ak+1 + Σi=1kk+1Ciak+1-ibi + a0bk+1

∴ (a + b)n = Σi=0nnCianibi //
Def. 一般二項係数, [a:k] (二項係数の拡張): a

[a:0] = 1
[a:k] = {a(a - 1)(a - 2) … (a - k + 1)}/k! (k = 1, 2, 3, …)

実数の連続性 (continuity of real number)


Def. 切断 cut: (a, 実数) ⊇ A, Aを次の2つの部分集合A1, A2に分ける事

Def. A1 切断の下組, A2 切断の上組

切断条件
  1. A1A2 =
  2. A1 ≠ ∅ (空集合), A2 ≠ ∅
  3. A1A2 = ∅
  4. xA1, yA2x < y
xyで必ず順位が出来る = どのような実数でも必ずA1, A2のどちらかに分ける事が出来る
Def. 順序数: a ∈ Aset (a: constant), xA

xaa = maxマックスA ≡ 最大値 / xaa = minミンA ≡ 最小値

Def. 実数, : a, bAset, aba < c < b, cA → 稠密性をもつ

有理数: density, a < b, a < (a + b)/2 < b
実数: density
整数: not density

A1 ≡ {X| x < a}, A2 ≡ {X| x > a} →

(A1, A2) = not cut. If aA1
a = maxA1 → minA2, not exist, and vice versa.

A1 ≡ {X|xa}, A2 ≡ {X|x > a} → (A1, A2) = cut.

aA1a = maxA1, minA2, not exist →
(A1, A2)は切断cut = 1つの実数を境界とするcutを決定

Th. Delkindの定理: 切断を与えるとその境界として1つの実数が決まる
  1. maxA1もminA2ある → A1 ~ A2間にleap (ex. 整数 integer)
  2. maxA1もminA2もない → A1 ~ A2間にgap
  3. maxA1, minA2の一方のみ存在 → A1 ~ A2間は連続continuous (ex. 実数real number)
2) or 3) 有理数
Ex for 2: A = (A1, A2) cut →

A1 = {x|x ≤ 0, or x > 0 and x2 < 2, xA},
A1 = {x|x > 0 and x2 > 2, xA}
→ √2は有理数ではない = maxA1, minA2存在しない

Pr. A1a, a > 0 → A1a > 0, 2 – a2 > 0, 0 < h < aとおくと

0 < h < (2 - a)/3aなるhが存在
(a + h)2 = a2 + 2ah + h2 = a2 + h(2a + h) < a2 + 3ah < a2 + 2 – a2 = 2
∴ (a + h)2 < 2
a + hA1, A1aa + hA1
∴ maxA1 not exist。minA1も同様にnot exist

循環小数の分数問題
Ex. A := 0.3(•) → 10A = 3.3(•)

A = 1/3 → 3A = 0.9(•) = 1 [実数の連続性より矛盾しない]

数列の極限 (limit of sequences)


Def. 数列(実数列) sequence: ある規則に従い並べた数の列
漸化式 recurrence formula: 数列の隣接項の関係式

an + 2 = an + 1 + an (n = 1, 2, …)

Ex. フィボナッチ数列 (Fibonacci sequence)

素数の数列表現がフィボナッチ数列 Ex. 植物の株の増え方

Def. 部分(数)列 {anp}: 数列{an}の一部分をもとの順序に従って並べたもの
Def. 特性方程式 characteristic equation: 特性解(k)を求める方程式

an+1 = r·an + d (r ≠ 1)
a1 := ka1 = a2 = a3 … = k exist
Def. 平衡点(特殊解) ≡ kk = r·k + d ≡ 特性方程式


ε-δ論法 (ε-δ definition)


limxaf(x) = lDef. ε > 0, δ > 0 → |f(x) – l| < ε__[1 > ε 仮定してよい]
Ex. limx→∞1/(2x + 3) = 0
Pr. ε > 0, δ > 0, x < δ → 1/(2x + 3) < δを探す

x > 0の時を考えると十分 (∵ x → ∞)

1/(2x + 3) < ε → 1 < ε(2x + 3) → 1 – 3ε < 2εx
∴ (1– 3ε)/2ε < x
∴ (1 – 3ε)/2εδ を考えれば良い
limxaf(x) = lε > 0, f(ε) > 0, 0 < |xa| < δ → |f(x) – l| < ε //


Def. 極限値 limit value: {an}, a, ε > 0, nn0N, |ana| < ε

⇒ {an}は極限値を持ちaに収束 convergence ≡ limn→∞an = a
Def. 収束数列(収束列): 収束する数列

Def. 発散 divergence: 収束しない数列: limn→±∞an = ±∞ [±∞への発散]
Pr. ε-δ論法 (これ以外証明方法なし)

limn→∞an = a, ε > 0, n0(ε) ≤ n, |ana| < ε [極限の一意性]
n0n > n0, nNは全てn0の資格がありn0は1つとは限らない

Def. 有界数列 bounded sequence, {an} for n, |an| < M = constant

nに対し数列anの各項の絶対値がMよりも小さい数列

Th. {an} 収束 ⇒ 有界
Pr._limn→∞an := a, ε = 1を選ぶとN = N(1)で nN ⇒ |an - a| < 1 exist

M := max{|a1|, |a2|, |a3|, …, |aN|, |a| + 1} (M, nに無関係な正の定数)

1 ≤ nN ⇒ |an| ≤ M

nNならば |an| = |(an - a) + a| ≤ |an - a| + |a| < 1 + |a| ≤ M //

1. 収束数列 (limn→∞an = a) の部分列 {anp} ⇒

もとの極限値に収束 limnp→∞anp = a

Ex. {an}の部分列{anp} = an1, an2 …, ε > 0, n0n

|ana| < εn0np, |anpa| < ε → lim|an| = a

Prop. a, limn→∞an = aとなるN.S.C.

limn→∞a2n-1 = limn→∞a2n = a (a = ±∞でも成立)

Pr. Case. a, ε > 0, N(ε) ∋ N, nN(ε) ⇒ |an - a| < ε

N1(ε) ≥ (N(ε) + 1)/2となるよう1つ決める

nN1(ε) ⇒ 2n - 1 ≥ 2N1(ε) - 1 ≥ N(ε) ⇒ |a2n-1 - a| < ε
limn→∞a2n-1 = a 成立

同様に nN1(ε) ⇒ 2n ≥ 2N1(ε) ≥ N(ε) ⇒ |a2n - a| < ε

limn→∞a2n = a 成立

If limn→∞a2n-1 = limn→∞a2n = aε > 0, N1(ε) and N2(ε) exist

nN1(ε) ⇒ |a2n-1 - a| < ε, nN2(ε) ⇒ |a2n - a| < ε

N(ε) := max(N1(ε), N2(ε)), nN(ε)
n := 2k - 1 > N(ε) > 2N1(ε) - 1 ⇒ kN1(ε) ⇒ |an - a| = |a2k-1 - a| < ε
n := 2k > N(ε) > 2N2(ε) - 1 ⇒ kN2(ε) ⇒ |an - a| = |a2k - a| < ε
nの偶奇によらず nN(ε) ⇒ |an - a| < ε //

2. 収束しない数列の部分数列は収束することがある

Ex. {an} = {(-1)n} 偶数項を全てとり{anp}とおくと1に収束

3. 有界数列は必ずしも収束しない

Ex. (2参照) limn→∞(-1)nlimn→∞a2n = 1, limn→∞a2n-1 = -1 (発散)

4. 収束数列で極限値も同じ限界を超えない

liman → |an| < M for n → |a| < M

Th. {an}, {bn} 収束数列 ⇒ limn→∞anbn = limn→∞an·limn→∞bn

{an}, {bn} 一方が発散する場合は成り立つとは限らない

Ex. an = n, bn = 1/nlimn→∞anbn = 1, limn→∞an·limn→∞bn = ∞·0
Th. a ∈ , {an} (等比数列), limn→∞an

∞ (a > 1), 1 (a = 1), 0 (-1 < a < 1), 発散 (a ≤ -1)

Q. limn→∞{(-1)n/n}の極限
A. nN ⇒ -1/n ≤ (-1)n/n ≤ 1/n, limn→∞(-1/n) = limn→∞(1/n) = 0

limn→∞{(-1)n/n} = 0

Q. a1 = c, an+1 = 1/2·an + 1の時の{an}の極限
A. k = 1/2·k + 1 (特性方程式) ⇒ k = 2, an+1 - 2 = 1/2·(an - 2) [変形]

∴ {an - 2}は初項c - 2、公比1/2の等比数列 ∴ an - 2 = (c - 2)(1/2)n-1
an = (c - 2)/2n-1 + 2 ⇒ limn→∞an = limn→∞((c - 2)/2n-1 + 2) = 2

Axiom [I] 実数の連続性 (continuity of real numbers)

Ax. 連続の公理 (Dedekind の公理)
実数全体にいかなる切断cutを与えてもmaxA1かminA2のいずれか一方しか存在しない
Def. 上(下)界: xA, x < (>) a = constant ⇒

A: 上(下) above(below)に有界 bounded
a: 一つの上(下)界

Def. 有界: 上にも下にも有界
Def. 上(下)限: A

上限(supermum) supサプ/スープA = 最小上界(上界の最小)
下限(infimum) infインフA = 最大下界(下界の最大)
○ 境界点含まない open, ● 境界点含む closed = max, minがある場合

limit Ex. f(x) < |x2 + y2| = 1

0 ≤ f(x) ≤ 1 → boundedだがf(x) = 1とはならない
ε > 0 に対し1 - ε < f(x)となるxは[-1, 1]で存在 → supf(x) = 1

Def. 上限 supA
  1. Ax, xa (a: 上界の一つ)
  2. ε > 0, cA, aε < c (aよりも小なる上界がない)
Pr. 背理法に基づき(2)の否定を否定

ε > 0, cA, aε caεcaεc: aの上界 … (1)
一方aε < a … (2)
(1)(2)は矛盾 (aが最小上界であるのに対しaεがそれより小さい上界となってしまう)

Th. [II] Weierstraßの定理 (Weierstraß theorem)

= 最大値最小値定理(最大値定理), 極値定理 extreme value theorem
有界閉集合上の連続関数は最大(小)値を持つ ≡

A (A ≠ ∅), supA (infA) ⇒ maxA (minA) exist

Pr. : (B1, B2) cut, B1 = {Aの上界全体}, B2 = {それ以外の実数}

[I]より 1) maxB1 exist, minB2 not exist, or
_____2) maxB1 not exist, minB2 exist.
1)でないことを証明すれば十分
If maxB1 = bbB2
xA, b < x, densityだからb < c < xなるc
xA, c < xbB2, cB2
∴ maxB1 = b < c … 矛盾 //

Ax. アルキメデスの公理 axiom of Archimedes

任意の正の実数a, bに対しb < naとなる自然数n存在

Pr. (背理法) nabが成立と仮定: A := {na| nN} → A, A ≠ ∅

A: 上に有界 → Aの上限存在 (Weierstraßの定理より)
ε := a/2 (ε > 0) → xA, s - ε < xs
x = n0aN (from Def.), y := (n0 + 1)a
y = (n0 + 1)a = n0a + a = x + 2ε > s + ε > s + ε > s
yA, y > ss = supA (矛盾) //

Th. 巾根: a > 0 real number, n: integer ⇒ xn = aなる実数x(巾根)存在
Pr. If A = {x > 0, xn > a} → A ≠ ∅ (→ (a + 1)n > a), aAの下界の1つ

Aは下に有界 → [II] infA = b
bn = a → if bn < a, min{1, (abn)/Σr=1n–1nCrbr} > ε > 0
(b + ε)n = bn + Σr=0n–1nCrbrεnr < bn +

(Σr=0n–1nCrbrεn)ε < bn + abn = a

一方infA = b, Ax, bx, b + ε > x0なるx0A
x0 > 0より(b + ε)n > x0n
a > (b + ε)n > x0nx0n < A … 矛盾(定義xn > a)
bn = a ⇒ 以下同様にbn = a //

Def. 単調増加(減少): {an}, nan ≤ (≥) an+1

a1a2 ≤ … anan+1 ≤ …

Def. 狭義単調増加(減少): {an}, nan < (>) an+1
Def. 単調数列 monotonic sequence ≡ 単調増加(減少)する数列

Ex. {an} = 2n + 3, {bn} = 1/2n, {cn} = (-5)n, {dn} = 3

{an} = 狭義単調増加, {bn} = 狭義単調減少, {cn} ≠ 単調数列
{dn} = 単調増加・単調減少, ≠ 狭義単調増加・狭義単調減少

Th. [III] 上(下)に有界な単調増加(減少)数列はsup(inf)を持つ

Pr. {an}, anM = constant for n, A = {x: x = an, n = 1, 2 …}

Aは上に有界
[II] → supA = αε > 0, αε < an0αなるan0
単調性から n0 < nαε < anα
∴ |αan| < εlimn→∞an = α = supA //

Th. limn→∞(1 + 1/n)n = Σn=1(1 + 1/n)n = a exist (eの存在)
Pr._{an}, an = (1 + 1/n)n, n ≥ 1 (二項定理 →)

an = 1 + n·1/n + n·(n -1)/2!·(1/n)2 + … + (n(n - 1)·…·1)/n!·(1/n)n

= 1 + 1 + (1 - 1/n)·1/2! + … + (1 - 1/n)·…·(1 - (n - 1)/n)·1/n!

< 1 + 1 + (1 - 1/(n + 1))·1/2! + …

+ (1 - 1/(n + 1))·…·(1 - (n - 1)/(n + 1))·1/n!
+ (1 - 1/(n + 1))·…·(1 - n/(n + 1))·1/(n + 1)!

= (1 + 1/(n + 1))n+1 = an+1 ⇒ {an} 単調増加
an ≤ 1 + 1 + 1/2! + … + 1/n!__(二項展開)

≤ 1 + 1 + 1/2 + 1/22 + … 1/2n-1 < 3 上に有界

∴ 単調有界数列は収束 ⇒ limn→∞(1 + 1/n)n = a存在 // ⇒

Def. ea (自然対数の底)
Def. e: ネイピア数 Napier's constant / オイラー数 Euler's number

e = limx→∞(1 + x)1/x ≈ 2.718281828459045… [無理数]

ふな一鉢二鉢一鉢二鉢ひとはちふたはちひとはちふたはち至極惜しいしごくおしい

Th. limx→-∞(1 + 1/x)x = limx→-∞(1 - 1/x)-x = limx→0(1 + x)1/x = e
Q. limx→-∞(1 + 1/2n)nの極限値
A. m := 2n, n → ∞, m → ∞ ⇒ limn→-∞(1 + 1/2n)n =

limm→-∞(1 + 1/m)1/m = limm→-∞{(1 + 1/m)m}1/2 = e1/2 = √e

Q. {an}, a1 = 2, an+1 = 1/4·(an2 + 3) (n = 1, 2, 3, …) ⇒ limn→∞anを求める
A. limn→∞an = 1 収束
Th. {an}, limn→∞an = a ⇒ limn→∞{(a1 + a2 + … + an)/n} = a
Th. a1a2a3 ≤ … ≤ … bnbn-1 ≤ … ≤ b2b1 and limn→∞|anbn| = 0

limn→∞an = limn→∞bn

Pr. {an}: 単調増加、上に有界, {bn}: 単調増加、下に有界

[III]からlimn→∞an = a = supA, limn→∞bn = b = infB
limn→∞|anbn| = 0よりlimn→∞(anbn) = 0 = aba = b //

Th. [IV] Cantorの共通部分定理 (Bachmannの区間縮小法)

(1) [a1, b1] ⊇ [a2, b2] ⊇ … ⊇ [an, bn] … かつ (2) limn→∞(anbn) = 0

⇒ [ai, bi] for iの共通点は1点のみ存在

Pr._(1)はTh. IVの仮定を満たす

即ちa1a2a3 ≤ … ≤ … bnbn-1 ≤ … ≤ b2b1
limm→∞am = a = limn→∞bn = b
i) m = n: an < bn 区間(二次元)であるから等号である事はない
ii) m < n: ambm < bn
iii)m > n: am < bmbn
i), ii), iii)より m, nについて am < bm ∴各bnは{an}の上界。aは極限
n → ∞, abanabbn → [an, bn] ∋ a for n
(2)からε > 0, n0n, bnan < ε
ε > bnanba ≥ 0
a = b (ただ1点を証明 → 背理法)
if aa'なる二点a, a' (a < a') →

ana < a' ≤ bn for n, 0 < a' – a < bnan → 0 < ε < a' – a
ε < |a' – a|

limn→∞|bnan| = 0 に対し矛盾 ∴ uniquely determined //


集積点 accumulate point or cluster point


Def. A: 実数(a)の場合, aAの集積点 ⇔

ε > 0に対し0 < |ax| < εなるaの近傍中のxA中に少なくとも1つ(= 無限に多く)存在

aε < x < a + ε, (aε, a + ε): aの近傍(ε-近傍) neighborhood

V(a, r), U(a, r) or V(a), U(a)

必要条件: 開 open であること
Th. a: Aの集積点 → Aa1, a2, … を選びlimn→∞an = aにすることができる
Pr. ε: = 1

0 < |aa0| < 1なるa0A,
0 < |aa1| < 1/2なるa1A,
0 < |aa2| < 1/22なるa2A,
以下同様 → a0, a1, a2
0 < |aan| < 1/2n なる点が存在
limn→∞(1/2n) = 0 → limn→∞an = a

Th. [V] ワイヤシュトラウス-ボルザノの定理

(Weierstraß-Bolzano (W-B) theorem)

(ボルザノ-ワイヤシュトラウスの定理, Bolzano-Weierstraß theorem)
Th. 点集合Aが有界、無限集合 → Aは少なくとも1つ集積点を持つ

≡ 有界数列は少なくとも1集積値を持つ

Pr. A ⊆ [a, b]において[a1, c], [c, b1]

→ 少なくとも一方にはAの元は無限にある
If [c, b] = [a2, b2] → [a1, b1] ⊇ [a2, b2]

→ [a2, b2]にはAの元は無限にある

以下同様に[a1, b1] ⊇ [a2, b2] ⊇ [a3, b3] ⊇ …
またbnan = 1/2n–1(b1a1) → limn→∞(bnan) = 0
[IV] → supan = infbn = α, [ai, bi] ∋ α for i, ε > 0, εn,

[an, bn] ⊆ (αε, α + ε) = U(α): 集積値 //

Th. 点集合Aが上(下)に有界ではない

→ ±∞を集積点に持つ → 無限集合は少なくとも1つ集積点を持つ

Th. Cauchyの収束定理: 数列anが収束するためのN.S.C. (実数の完備性)

数列 an 収束 ⇔ ε, n0 < p, q, |apaq| < ε → N.C.
しかも、[an, bn], a ∈ [an, bn]にはAの点は無限に存在する

Pr. N.C.: ε > 0, n0 < p → |apα| < ε/2, ε > 0, n0 < q, |aqα| < ε/2,

|apaq| = |apα + αaq| ≤ |apα| + |aqα| < ε
S.C. ε > 0, n0 < n0+1
|an0+1aq| < εan0+1 < aq < an+1 + ε
max{|a1|, |a2|, …, |an0|, |an0+1 + ε|, |an0+1ε|} ≤ M
∴ |an| ≤ M //

Def. 基本列 (Cauchy列) ≡ Cauchyの収束定理を満足する列 (注意: 定義だ!)

Cauchy列 ⇒ 有界 (∵ Cauchyの収束定理)

極限値不明でもCauchy列となることを示せば収束列である

↔ チェザロ (Cesaro E 1859-1906): 基本列以外にも収束することがある

Th. W-Bの定理: 有界な数列は、少なくとも1つの集積点を持つ

→ 集積値がただ1つ ⇒ limn→∞an = a
≡ 有界な数列は、収束する部分列をもつ

Pr. 背理法 If 集積点が2つ以上あるとする(α, α': αα')

1/3|αα'| > p > 0とする
集積点の定理よりU(α) ∋ ap and U(α') ∋ aq, |apaq| > pなる点ap, aqは無数に有る
ここで仮定よりρ = ε (ρ > 0) にとるとn0 < p, qに対して|apaq| < ρ
これに対して|aqaq| < 0 → 矛盾 //

実数の連続性 [I] → 上限下限定理 [II] → 単調数列の極限値 [III]__単調数列
____________________________________
Cauchyの収束定理W-Bの定理 [V] → Cantorの定理 [IV]_____一般数列

方程式と関数 (equations and functions)


Def. 恒等式 identiy: どの値を代入しても両辺が等しい式 Ex. 公式 formula
Def. 方程式 equation: ある特定の値でなければ等号の成立しない式

→ その値 ≡ 根 root または解 solution

Def. 代数方程式 algebraic equation: 未知数の冪powerによる多項式の形で与えられた方程式
方程式の代数的解法: 四則演算と冪根power rootのみで解く方法
1次方程式 equation of first degree (linear equation)

ax + b = 0, a ≠ 0 → x = –b/a

以下は1次方程式としては考慮しなくてよい

a = 0, b = 0 → 不定 indeterminate: 根は無数
a = 0, b ≠ 0 → 不能 inconsistent: 根なし

2次方程式 equation of second degree (quadratic equation)

ax2 + bx + c = 0 (a ≠ 0)

Case. 因数分解 factorization 可能

(xα)(xβ) = 0 → x = α, β

Case. 完全平方式による解の式

x = (–b ± √(b2 – 4ac))/2a
→ 根と係数の関係: α + β = –b/a, αβ = c/a

Def. 根の判別式discriminant, Db2 – 4ac

D > 0 実根 real root, s.s.
D = 0 重根 multiple root (D ≥ 0 実根, s.l.)
D < 0 虚根 imaginary root

Th. 存在定理 exsistence theorem

n次方程式: xn + a1xn-1 + a2xn-2 + … + an-1x + an = 0 ⇒
Πi=1n(xαi) = (xα1)(xα2) × … ×(xαn-1)(xαn) = 0
(複素数範囲内にn個の根を持つ)

Def. 関数(function)の極限
Case xa: ε, 0 ≠ |x - a| < δ(ε), |f(x) - α| ≤ ε

Def. 極限値 ≡ α__表現: limxaf(x) = α or f(x) → α (xa)
f(x)はaαに収束 ⇔ 発散: limxaf(x) = ±∞

Case x → ∞: ε, x > L(ε), |f(x) - α| < ε

Def. 極限値 ≡ α__表現: limx→∞f(x) = α or f(x) → α (x → ∞)
f(x)は∞でαに収束 ⇔ 発散: limx→∞f(x) = ±∞

Th. 関数の極限の性質 (数列の極限の性質の拡張)
xa, f(x), f(y) 収束, limxaf(x) := α, limxag(x) := β

(1) limxa{λf(x) + μg(x)} = λα + μβ__(λ, μ)
(2) limxaf(x)g(x) = αβ
(3) limxa(f(x)/g(x)) = α/β__(β ≠ 0)
(4) limxa|f(x)| = |α|
(a = ±∞でも成立)

Def. 合成関数 composite or compound function, z

z = g(f(x)) ⇐ y = f(x), z = g(y)

Th. 合成関数の極限 U(a), xa, limxaf(x) = b, limybg(y) = c

limxag(f(x)) = c, f(x) ≠ b or g(y) ≠ c at U(a)

Pr. limybg(y) = cε > 0, δ > 0

0 < |yb| < δ → |g(y) – c| < ε → |g(f(x)) – c| < ε … (1)
limxaf(x) = bε > 0, δ > 0
0 < |xa| < δ, f(x) ≠ bのとき → 0 < |f(x) – b| < δ → |g(x) – c| < ε
ここで、0 < |xa| < δ, i.e., U(a)はf(x) = bとのき(1)は成り立たない
g(b) = c, g(b) – c = g(f(x)) – c) = 0 → |g(f(x)) – c| < ε //

Th. 関数のCauchyの収束判定法
limxaf(x)が収束するN.S.C.: ε > 0, δ(ε) > 0,

0 < |x - a| < δ(ε), 0 < |y - a| < δ(ε) ⇒ |f(x) - (f(y)| < ε

Pr. limxaf(x) := α, ε > 0, ε* = ε/2 > 0, δ(ε*) > 0 … (1)

0 < |x - a| < δ(ε*) ⇒ 0 < |f(x) - a| < ε*

if 0 < |x - a| < δ(ε*), 0 < |y - a| < δ(ε*) ⇒

|f(x) - f(y)| ≤ |f(x) - α| + |α - f(y)| ≤ ε* + ε* = ε

ε > 0, δ(ε)

0 < |x - a| < δ(ε), 0 < |y - a| < δ(ε) ⇒ || < |f(x) - f(y)| < ε

{an}, ana, limxa = a, δ(ε) > 0, N(ε) ∈ N

nN(ε) ⇒ |an - a| < δ(ε)

if mN(ε), nN(ε) ⇒ 0 < |am - a| < δ(ε), 0 < |an - a| < δ(ε)

(1)より|f(am) - f(an)| < ε__N.C. //

片側極限 one-sided limit
= 左極限 left-hand limit + 右極限 right-hand limit
Def. 左極限, α: f(x) → (a, a + ε0), ε > 0, a < x < a + δ(ε) (> 0)

⇒ |f(x) - α| < ε__(表記) limxa+0f(x) = α or f(x) → α (xa + 0)
収束しなければ発散 ⇒ limxa+0f(x) = ±∞

Def. 右極限, α: limxa-0f(x) = α or f(x) → α (xa - 0)

収束しなければ発散 ⇒ limxa-0f(x) = ±∞

Th. (極限収束N.S.C.) αN, limxaf(x) = αとなるN.S.C. ⇒

limxa+0f(x) = limxa-0f(x) = α

Pr. limxaf(x) := α, ε > 0, δ(ε) > 0

0 ≠ |x - a| < δ(ε) ⇒ |f(x) - α| < ε, a < x < a + δ(ε) ⇒ |f(x) - α| < ε

limxa+0f(x) = α 成立

a - δ(ε) < x < a ⇒ |f(x) - α| < εlimxa-0f(x) = α 成立
⇔ if limxa+0f(x) = limxa-0f(x) = α, ε > 0, δ1(ε) > 0

a < x < a + δ1(ε) ⇒ |f(x) - α| < ε 成立

ε > 0, δ2(ε) > 0

a - δ2(ε) < x < a ⇒ |f(x) - α| < ε 成立

δ(ε) := min{δ1(ε), δ2(ε)}

0 ≠ |x - a| < δ(ε) ⇒ |f(x) - a| < εlimxaf(x) = α //

[ 初等関数 ]

連続関数 (continuous function)


Def. ε > 0, δ > 0, |xa| < δ, U(a, δ)内で全ての点x

|f(x) – f(a)| < εf(x)はx = aで連続
前提はlimxaf(x)なら0 < |xa| < δaで連続なら|x a| < δでよい

Def. 左(右)方連続 left (right) continuous, f(x) ⇒

f(a – 0) = f(a) or f(a + 0) = f(a)

Th. f(x)が点aで連続となるN.S.C. ⇒ f(x)がaで右連続かつ左連続
Def. xD, f(x) continuous for all xf(x): 連続関数
Th. f(x), g(x) continuous ⇒ f(x) ± g(x), f(xg(x), f(x)/g(x) (g(x) ≠ 0),

|f(x)|, kf(x) (k: constant) 全てcontinuous

Pr. trivial. 仮定よりlimxaf(x) = f(a), limxag(x) = g(a) + Th. 関数の極限の性質 //
Th. f(x) continuous at x = a, f(a) ≠ 0

1) f(a) > 0 → f(a)f(x) > 0 at U(a) [f(a)とf(x)が同符号である事を証明]
2) f(a) < 0 → f(a)f(x) > 0 at U(a)

Pr. 1) If limxaf(x) = f(a) > 0, ε = |f(a)|/2 > 0, δ > 0, |xa| < δ

→ |f(x) – f(a)| < |f(a)|/2, δ exist →
0 < f(a) – |f(a)|/2 < f(x) < f(a) + |f(a)|/2__f(a) > 0, f(x) > 0
2)も同様
f(x): x = aでcontinuous ⇔ εδ(ε) > 0, |xa| < δ → |f(x) – f(a)| < ε

Th. f(x) [a, b] continuous, f(a) < 0 < f(b) or f(a) > 0 > f(b) ⇒

f(c) = 0, c ∈ (a, b)

Pr. f(a) < 0 → f(x) < 0 within U(a), i.e., ax < ξからf(x) < 0 … (1)

(1)に適合するξの上限supξ = cとおく
if f(c) > 0 → U(c) st f(x) > 0 → [cξ, c]でf(x) > 0 (such that: st)

一方supξ = c: supξ 最小上界 → c - δ < ξ < cで(1)を満たす →
[cδ, c]でf(x) < 0 … 矛盾__f(c) < 0

if f(c) < 0 → U(c) st f(x) < 0

(1)からc < ξで(1)を満たす … supξ = cに矛盾__f(c) > 0
continuous

Th. 中間値の定理, IVT: f(x): [a, b] continuous, f(a) ≠ f(b)

f(a) < μ < f(b) or f(a) > μ > f(b) ⇒ f(c) = μ, c ∈ (a, b)

Pr._ φ(x) ≡ f(x) – μ … > φ(x), [a, b] continuous,

φ(a) = f(a) – μ < 0, φ(b) = f(b) – μ > 0
(前定理より) φ(c) = 0, c ∈ (a, b) → φ(c) = f(c) – μ = 0 → f(c) = μ
c ∈ (a, b) //

2分法: 中間値の定理に従い区間[an, bn]幅絞り方程式の解の近似値を計算

欠点: 収束精度は良くない
利点: アルゴリズム理解容易, 方程式定める関数が連続なら必ず収束

Th. 有界定理 f(x): [a, b] continuous ⇒ f(x) = 有界 bounded
Pr. If f(x), supA not exist → g, f(t) > g, t ∈ [a, b] exist

f(xn) > n, xn ∈ [a, b], {xn} bounded → 集積値 x0 ∈ [a, b]

{x'n}: limn→∞x'n = x0, limn→∞f(x'n) = +∞ → limxx0f(x) = f(x0) = +∞ → 矛盾 //

continuous

Th. f(x): [a, b] continuous → bounded → max, minを有する

i.e., [a, b] continuous → 上に有界 →
supf(x) = M, f(c) = Mなるc ∈ [a, b]

Pr. (背理法) If f(c) = Mなるxが存在しない

Mf(x) ≠ 0 → 1/(Mf(x)): [a, b]でcontinuous → 上に有界
0 < 1/(Mf(x)) ≤ Λ
f(x) ≤ M – (1/Λ) (1/Λ > 0)
f(x)はMよりも小さい上界 M – (1/Λ)を持つ
supf(x) = Mに矛盾 ∴ f(c) = M //

Def. 単調関数 x1 < x2

単調増加: f(x1) ≤ f(x2) ⇔ 単調減少: f(x1) ≥ f(x2)

Def. 逆関数 inverse function, f-1(x)

y = f(x) → xがただ1つ定まる (≡ 関数となる) ⇒

x := g(y) → (x, y入替) → y = g(x) ≡ y(x)の逆関数 ≡ f-1(x)

Ex. f(x) = x + 1: y = x + 1 → x = y + 1 → y = x - 1 ∴ f(x) = x - 1
Ex. f(x) = x2 (x > 0): y = x2x = y2y = ±√x ∴ 逆関数なし
Ex. f(x) = x2 (y > 0): y = √xf(x) = √x

⇒ 逆関数は常に存在するとは限らない

Th. 逆関数の存在: y = f(x) [a, b] continuous, continuous

狭義単調増加: f(a) = α, f(b) = β
y = f-1(x) 逆関数 inverse function存在, [α, β]:

一価連続、狭義単調増加

Ainto B, Aontof(a) ⊂ c B
恒等写像

f·f-1 = f-1·f = I, f(a) = Bf-1(A)

Pr. f(x) 狭義単調増加, α < β
(中間値の定理) α < γ < β, f(c) = γとなるc ∈ (a, b)が少なくとも1つ存在
If cc', f(c) = f(c') → 狭義単調増加に矛盾
cはだだ1つ、即ちf-1(x)は[α, β]で一価に確定
If [α, β] ∋ γ1γ2, f–1(γ1) = c1, f–1(γ2) = c2f(c1)= γ1, f(c2) = γ2
If c1 > c2f(c1) > f(c2) → γ1 > γ2, If c1 = c2f(c1) = f(c2) → γ1 = γ2

γ1 < γ2c1 < c2f–1: 狭義単調増加

f–1(γ + k) = c + kf(c + h) = γ + k, f–1(γ) = c

f(c) = γ, f–1(γ + k) – f-1(γ) = h

f(c + h) – f(c) = k → limh→0(f(c + h) – f(c)) = 0 →

limh→0f(c + h) = f(c) → limh→0(f–1(γ + k) – f–1(γ)) = 0

∴ limk→0f–1(γ + k) = f–1(γ) → f–1: continuous //

微分法 (differentiation)


導関数 (derivatives)


Def. 一価関数: y = f(x) [a, b]
Case. x [a, b]の一点cからc + h (h = Δx)まで変化(h ≠ 0) →

y: f(c + h) – f(c) = f(c + Δx) – f(c) = Δyだけ変化 ⇒

Def. 平均変化率 Δyx: Δx, Δyx, yの増分, Δyx = {f(x + h) – f(x)}/h

limΔx→0Δyx = limh→0{f(x + h) – f(x)}/h

= limxc{f(x) – f(c)}/(xc) exist (有限確定値)

Def. 微分係数(微係数, 瞬間変化率), f'(c) ≡ x = cにおけるf(x)

y'x = c [プライム, ダッシュ] = df(x)/dx, d/dx·f(x)

Def. f(x)の導関数 ≡ y', f'(x), dy/dx, d/dx·f(x), df(x)

(f(c + h) – f(c))/hf'(c) := ρ(h) → f(c + h) – f(c)) = h·ρ(h) + h·f'(c), limh→0ρ(h) = 0とも表せる

Th. f(x) diff. at af(x) continuous at a
Pr. if xaf(x) = f(a) + (x - a)·{(f(x) - f(a))/(x - a)}

limxaf(x) = limxa{f(a) + (x - a)·{(f(x) - f(a))/(x - a)} = f(a) + 0·f'(a) = f(a) //

⇔ 対偶 contrapositive: f(x) discontinuous at af(x) not diff. at a

Th. f(x) [a, b] diff. ⇒ f(x) [a, b] continuous (逆は偽)
Ex. y = |x| f(x): [a, b] continuous → not differential at x = 0

limh→0[f(c + h) – f(c)] = limh→0[{f(c + h) – f(c)}/h·h] = f'(c)·0 = 0

Def. 接線 ≡ y = f'(a)(x - a) + f(a) (f(x) diff. at a)
Def. limh→0+0{f(x + h) – f(x)}/h = f'+(c) 右微分係数 ⇒ 収束: 右微分可能

limh→0-0{f(x + h) – f(x)}/h = f'(c) 左微分係数 ⇒ 収束: 左微分可能

Th. f(x)が点cで微分可能(可微分) differentiable (diff.)となるN.S.C.

(1) f(x)が点cで右微分・左微分可能
(2) f'+(c) = f'-(c)

Q. f(x) = |x|はx = 0で微分可能ではないことを示せ
A. limh→0(f(h) - f(0))/h = limh→0(|h|/h)

limh→+0(|h|/h) = limh→0(h/h) = 1 ∴ 右微分可能 f'+(0) = 1
limh→-0(|h|/h) = limh→-0(-h/h) = -1 ∴ 左微分可能 f'-(0) = -1
f'+(0) ≠ f'-(0)__(2)の条件を満たさない //

Th. 和・積・商の導関数 derivatives of sum, product and quotient
Th. 合成関数の微分 differentiation of composite function
y = f(x) diff. at x = a, z = g(y) diff. at y = b

z = g(f(x)) diff. at x = a, z' = g'(bf'(a) → 微分係数
定理の結果はdz/dx = dz/dy·dy/dx, dz/dx = dz/dt·dt/ds·ds/dx (3関数合成)等と表し利用される

Pr. 0 < |ya| < δ, g(y) = g(a), S(δ) = [y| g(y) = g(a)なる全てのy]
1) Case: δS(δ) not exist

0 < |h| < δf(a + h) – f(a) = k

k ≠ 0, f(a + h) = b + k (∵ b = f(a))

{f(a + h) – f(a)}/h = a, さらにh → 0 → k → 0
{g(f(a + h)) – g(f(a))}/h

= {g(b + k) – g(b)}/h = {g(b + k) – g(b)}/k·{f(a + h) – f(a)}/h

limh→0{g(f(a + h) – g(a)}/h = g'(bf'(a)

2) Case: δS(δ) exist: 証明省略可(大部分の関数は(1)である)
対数微分法(logarithmic differentiation)
y = f(x) (f(x) ≠ 0) ⇒ log|y| = log|f(x)|とし右辺が微分しやすい時に使う

y = f(x) ⇒ logy = logf(x) (f(x) > 0 ⇒ 絶対値不用)
(logy)' = y'/y = d/dx·logf(x)

Ex. y = xsinx (x > 0) ⇒ y': logy = logxsinx = sinxlogx

y'/y = cosxlogx + sinx·(1/x)
y' = y(cosxlogx + sinx/x) = xsinx(cosxlogx + sinx/x)

Th. 逆関数の微分(逆関数定理) differentiation of inverse function
x = αの近傍 nbd で連続狭義単調変化, diff. at x = α (f'(α) ≠ 0) ⇒

y = f(x)の逆関数 x = y-1(x) = φ(y) diff. at y = β = f(α), φ(β) = 1/f'(α),
dx/dy = 1/(dy/dx)

Pr. x = f'(y), y = βのnbdで一価連続狭義単調増加 → f·f'(x) = xを示す

f(α + h) – f(α) := k 狭義単調増加からk ≠ 0 → h ≠ 0
[f'(β + k) – f'(β)]/k = [f-1(f-1(α + h) – f-1(f(α))]/(f(α + h) – f(α))

= (α + hα)/(f(a + h) – f(α)) = 1/[(f(α + h) – f(α))/h]
= 1/f'(α) = φ'(β) (h → 0)

[ 初等関数 elementary function ]

初等関数の微分 differentiation of elementary functions
Th. 対数関数・指数関数の微分 diff. of log and exponential functions

(logax)' = 1/x·logae__Case a = e: (logex)' = 1/x
(ax)' = ax·loga_____ Case a = e: (ex)' = ex

Pr. for (logax)' = 1/x·logae

(loga(x + h) – logax)/h = 1/h·loga(1 + h/x) = 1/x·x/h·loga(1 + h/x) = 1/x·loga(1 + h/x)x/h → 1/x·logae (h → 0)

Th. 冪関数の微分 f'(x) = (xα)' = αxα–1
Pr. y = xα = eαlogx

y' = eαlogx·(αlogx)' = eαlogx·(a/π) (= (a/πy)) = αxα-1
y = xα (対数微分法)

logy = αlogx, (1/yy' = α/x, y' = α/x·y = α/x·xα = αxα-1

Ex. (実質公式) y = √x (= x1/2) ⇒ y' = 1/2·x-1/2 = 1/(2√x)
Ex. y = xxy': logy = logxx = xlogxy'/y = x'logx + x(logx)' = logx + 1

y' = y(logx + 1) = xx(logx + 1)

Ex. y = (diff.)nを微分:

logy = diff., logxyn = y[(diff.)n·logx + diff.·(1/x)]

Th. 他の初等関数の微分 differentiation of the following functions

三角関数 trigonometric function
逆三角関数 inverse trigonometric function
双曲線関数 hyperbolic function

Th. x = g(t), y = f(t), diff. at t, x = g(t)に逆関数存在, g'(t) ≠ 0 ⇒

dy/dx = f'(t)/g'(t) = (dy/dt)/(dx/dt)

Pr. t = g-1(x) exist, diff. ∴ y = f(t) = f(g-1(x))

dy/dx = df/dt(g-1(x))·d/dx(g-1(x)) = f'(t)·1/g'(t) = f'(t)/g'(t) //

Q. x = (1 - t2)/(1 + t2), y = 2t/(1 + t2) ⇒ 導関数をtの関数で表せ
A. dx/dt = -4t/(1 + t2)2, dy/dt = (2 - 2t2)/(1 + t2)2

dy/dx = dy/dt·dx/dt = (t2 - 1)/2t

微分可能性の判定
Q. x = 0 diff.を判定。if diff. → f'(0)を求める

(1) f(x) = x|x|__ (2) f(x) = xlog|x| (x = ≠ 0), or 0 (x = 0)

A._ (1) limx→0(f(x) - f(0)/(x - 0) = limx→0(x|x| - 0)/x = limx→0|x| = 0 ∴ diff. f'(0) = 0

(2) limx→0(f(x) - f(0)/(x - 0) = limx→0(xlog|x| - 0)/x)

= limx→0log|x| = -∞ ∴ non. diff.

Q. f(x) = x2 + 1 (x ≤ 1), or (ax + b)/(x + 1) (x > 1)がx = 1でdiff.となるa, b
A. f(x) diff. at x = 1 ⇒ f(x) continuous at x = 1

f(x)はx = 1で左連続、f(1) = 12 + 1 = 2
limx→1+0f(x) = limx→1+0((ax + b)/(x + 1)) = (a + b)/2
x = 1で連続 → f(1) = limx→1+0f(x) ∴ 2 = (a + b)/2 ∴ b = 4 -a
x > 0, f(x) = (ax + 4 - a)/(x + 1)

(f(x) - f(1))/(x - 1) = ((ax + 4 - a)/(x + 1) - 2)/(x - 1)

= ((a - 2)(x - 1))/((x + 1)(x - 1)) = (a - 2)/(x + 1)

x = 1, 右微分係数: f'+(1) = (a - 2)/2
x = 1, 左微分係数: f'-(1) = limx→1-0(f(x) - f(1))/(x - 1)

= limx→1-0((x2 + 1) - 2)/(x - 1) = limx→1-0(x + 1) = 2

f'+(1) = f'-(1) ∴ (a - 2)/2 = 2 ⇒ a = 6, b = -2

Eq. y = f(x)の接線 ⇒ f(x) ≈ f'(a)(x - a) + f(a), x = aのnbd (1次近似)
陰関数f(x, y): ある区間で一価で幾つかの関数(分枝)に分けられればdiff.

y': 両辺をxで微分 f1(x, y) + f2(x, yy' = 0 (f2(x, y) ≠ 0)
y' = -f1(x, y)/f2(x, y)

Ex. x3 – 9xy + y3 = 0を微分 ⇒ 両辺をxで微分 3x2 –9(y + xy') + 3y2y' = 0

y' = (x2 – 3y)/(3xy2) //

高次導関数 (high-order derivatives)


f(x) diff. ⇒ f'(x): (第)1次導関数, f'(x) diff. ⇒ f''(x) = f(2)(x): 2次導関数, …

f(n–1)(x) diff. ⇒ f(n)(x): n次導関数 (n回微分可能)

Def. 高次導関数: n ≥ 2の導関数 y(n), f(n)(x), dny/dxnと表す
Def. Cn級関数 Cn-class function:

I, f(x), n diff., f(n)(x) continuous in ICn級 (C級もある)

Ex. y = x2y' = 2x, y'' = 2, y(n) = 0 (n ≥ 3) ∴ y = x2C級関数
Ax. n次導関数公式 equations of n-order derivatives
逐次微分法 successive differentiation: 高次導関数を求める方法:

n次微係数: f(n)(x), x = cのときのf(n)(c)

Ex. f(x) = xαf(n)(x) = α(α – 1)(α – 2) … (αn + 1)xαn
Ex. y = 1/(x + 1) → y = (x + 1)-1, y' = (-1)(x + 1)-2, y'' = (-1)(-2)(x + 1)-3,

y''' = (-1)(-2)(-3)(x + 1)-4 …, y(n) = (-1)nn!(x + 1)-(n+1)

Th. f(x), g(x): n次導関数を持つ

⇒ (f(x) ± g(x))(n) = f(n)(x) + g(n)(x) ⇒ (c·f(x))(n) = c·f(n)(x)

Law Leibnizの法則 (generalized) Leibniz rule
f(x), g(x), n次導関数持つ ⇒ (f(xg(x))(n) = Σk=0nnCkf(nk)(xg(k)(x)

≡ 積の微分法を高次導関数に拡張

Pr. (数学的帰納法)
n = 1: (f(xg(x))' = f'(xg(x) + f(xg'(x)であるから成立
n = nのとき成立つと仮定しn = n + 1を考える:

(f(xg(x))(n+1) = ((f(xg(x))(n))' = (Σk=0nnCkf(nk)(xg(k)(x))'
= Σk=0nnCkf(nk+1)(xg(k)(x) + Σk=0nnCkf(nk)(xg(k+1)(x)

ここでk - 1はk index

= f(n+1)(xg(x) + Σk=1n(nCk + nCk–1)f(nk+1)(xg(k)(x) + f(xg(n+1)(x)
= n+1C0f(n+1)(xg(0)(x) + Σk=1nn+1Ckf(nk+1)(xg(k)(x)

+ n+1Cn+1f(0)(xg(n+1)(x)

= Σk=0n+1n+1Ckf(nk+1)(xg(k)(x) //

Ex. f(x) = tan-1(x)のf(n)(0)

f'(x) = 1/(1 + x2)
∴ (1 + x2)f'(x) = 1, [(1 + x2)f'(x)](n+1) = 1(n+1) = 0 → [微分方程式]
Leibnizの法則より [(1 + x2)f'(x)](n+1)

= n+1C0(1 + x2f(n+2)(x) + n+1C1(1 + x2)'·f(n+1)(x)

+ n+1C2(1 + x2)''·f(n)(x) = 0

∴ (1 + x2)f(n+2)(x) + (n + 1)·2x·f(n+1)(x) + [n(n + 1)/2]·2·f(n)(x) = 0
x := 0, f(n+2)(0) + n(n + 1)·f(n)(0) = 0
f(n+2)(0) = –n(n + 1)·f(n)(0) [漸化式]
n = 2m (偶数)_____n = 2m – 1 (奇数):
_________________f(2m+1)(0) = –2m(2m – 1)·f(2m–1)(0)
f(0)(0) = f(0) = 0___f'(0) = 1/(1 + 0) = 1 = 1!
f''(0) = 0__________f(3)(0) = -2·1·1 = -2!,
f(4)(0) = 0_________f(5)(0) = -4·3·(-(2)!) = 4!
________________f(7)(0) = -6·5·(4)! = -6!
_________________
f(2m)(0) = 0____f(2m+1)(0) = (-1)m·(2m)! → 0! = 1

平均値の定理 (mean-value theroem)


Th. Rolleの定理: f(x) [a, b] continuous, (a, b) diff., f(a) = f(b)

f'(c) = 0, c ∈ (a, b)

Pr._ i) f(x) ≡ 0 → f'(c) = 0, st. c = (a + b)/2

ii) f(x) > 0 (f(d) > 0, x = d), M > f(d) > 0, f(a) = f(b) = 0

f(c) = M > 0なる点ca, bとは異なる
i.e., a < c < bに対してf(c) = Mf(c + h) for h ≠ 0
f(c + h) - f(c) ≤ 0
h > 0 → (f(c + h) – f(c))/h ≤ 0, h < 0 → (f(c + h) - f(c))/h ≥ 0 limh→0+0{f(c + h) – f(c)}/h = f'+(c) ≤ 0: non positive,
limh→0–0{f(c + h) – f(c)}/h = f'(c) ≥ 0: non positive
f'+(≤0)(c) = f'–(≥0)(c) = f'(c) = 0

iii) f(x) < 0の時は最小値mをとって同様に証明される //

Th'. f(a) = f(b) = k ≠ 0, g(x) ≡ f(x) – k, g(x) [a, b] continuous, (a, b) diff.,

g(a) = g(b) = 0 ⇒ g'(c) = 0, c ∈ (a, b)

Th. 平均値の定理: f(x) [a, b] continuous, (a, b) diff.

f(b) = f(a) + (ba)f'(c), c ∈ (a, b)
変形 (ca)/(ba) = θ (0 < θ < 1)とおいた次の4形は同意

f(b) = f(a) + (ba)f'(a + θ(ba))
f(a + h) = f(a) + h·f'(a + θh)
f(x + h) = f(x) + h·f'(x + θh)
Δy = Δx·f'(x + θ, Δx)

Pr. g(x) := f(x) – f(a) – [(f(b) – f(a))/(ba)]·(xa)

g'(c) = 0 = g(b), g'(c) = 0, c ∈ (a, b),
g'(c) = f'(c) – (f(b) – f(a))/(ba) = 0 //

Th. Cauchyの平均値の定理 f(x), g(x) [a, b] continuous, (a, b) diff. →

(a, b) g'(x) ≠ g(a), or (f(b) – f(a))/(g(b) – g(a)) = f'(ξ)/g'(ξ), ξ ∈ (a, b)
g(b) ≠ g(a) ∵ if g(b) = g(a) → g'(ξ) = 0, ξ ∈ (a, b)
[g(x) = x とおいた平均値の定理の拡張である]

Pr. φ(x) := f(x) – f(a) – k(g(x) – g(a)), φ(b) = 0なるkを決めると

k = (f(b) – f(a))/(g(b) – g(a)) … (1)
φ(x)は[a, b] continuous, (a, b) diff., φ(a) = φ(b) = 0となる
(Rolleの定理) φ'(c) = 0, ξ ∈ (a, b), φ'(x) = f'(x) – kg'(x) = 0
k = f'(ξ)/g'(ξ) … (2) //

Th. f(n–1)(x) [a, b] continuous, (a, b) f(n)(x),

G(x) [a, b] continuous, (a, b) G'(x) ≠ 0

f(b) = f(a) + f'(a)(ba) + (f''(a)/2!)·(ba)2 + …

+ (f(n–1)(a)/(n – 1)!)·(ba)n–1 + Rn

Rn = (f(n)(c)/(n – 1)!)·(bc)n–1·(G(b) – G(a))/G'(c), c ∈ (a, b)

Pr. x := a,

F(x) = f(b)– {f(x) + f'(x)(bx) + (f''(x)/2!)·(bx)2 + …

+ (f(n–1)(x)/(n – 1)!)·(bx)n–1}

F(x) [a, b] continuous, (a, b) diff. (Cauchyの平均値の定理 ⇒)

(F(b) – F(a))/(G(b) – G(a)) = F'(c)/G'(c), c ∈ (a, b)

F(b) = f(b) – f(b) = 0
F(a) = f(b)– {f(a) + f'(a)(ba) + (f''(a)/2!)·(ba)2 + …

+ (f(n–1)(a)/(n – 1)!)·(ba)n–1} = Rn

d(f(k)(x)/k!·(bx)k)/dx = f(k+1)(x)/k!·(bx)kf(k)(x)/(k – 1)!·(bx)k–1,

k = 1, 2, …, n – 1

F'(x) = –f'(x) –Σk=1n–1{f(k+1)(x)/k!·(bx)kf(k)(x)/(k – 1)!}

= –f(n)(x)/(n – 1)!·(bx)n–1

∴ (0 – Rn)/(G(b) – G(a)) = (–f(n)(x)/(n – 1)!·(bx)n–1)/G'(c)
Rn = (f(n)(c)/(n – 1)!)·(bc)n–1·(G(b) – G(a))/G'(c) //

Th. Taylorの定理 Taylor's theorem: f(x) [a, b] continuous, (a, b) diff.

f(b) = f(a) + {(ba)/1!}f'(a) + {(ba)2/2!}f''(a) + …

+{(ba)n - 1/(n – 1)!}f(n-1)(a) + Rn, c ∈ (a, b)

Lagrangeの剰余: Rn = (1/n!)f(n)(c)(ba)n, c ∈ (a, b)
Cauchyの剰余: Rn = f(n)(c)/(n – 1)!·(bc)n–1(ba), c ∈ (a, b)
Pr. φ(x) := (b x)n, φ(b) = 0, φ(a) = (ba)n, φ'(c) = -n(bc)n–1
Lagrange (前定理より)

Rn = (f(n)(c)/(n – 1)!)·(bc)n–1·(0 – (ba)n)/(-n(bc)n–1) =

f(n)(c)/n!·(ba)n

φ(x) := bx, φ(b) = 0, φ(a) = ba, φ'(c) = -1

Cauchy (前定理より)

Rn = (f(n)(c)/(n – 1)!)·(bc)n–1·(0 – b + a)/(-1) =

f(n)(c)/(n – 1)!·(bc)n–1(ba) //

n = 1 ≡ 平均値の定理: (ca)/(ba) = θ, 0 < θ < 1 → c = a + (ba)θ

Lagrange': Rn = (1/n!)f(n)(a + (ba)θ)(ba)n
Cauchy': Rn = f(n)(a + (ba)θ)/(n – 1)!·(ba)n(1 – θ)n–1

a = x, b = x + h

Lagrange'': Rn = f(n)(x + θh)/nhn
Cauchy'': Rn = f(n)(x + θh)/(n – 1)!·(1 – θ)n–1hn
Taylor式において

b = x,
f(x) = f(a) + f'(a)(xa) + f''(a)/2!·(xa)2 + …

+ f(n-1)(a) /(n – 1)!·(xa)n-1 + Rn

f(x)のx = aを中心とするTaylorの展開式

Lagrange''': Rn = f(n)(a + θ(xa))/n!·(xa)n
Caucny''': Rn = f(n)(a + θ(xa))/(n – 1)!·(1 – θ)n–1·(xa)n

[ ⇓ Taylorの定理でa = 0]
Th. Maclaurinの定理 (Maclaurinの展開) Maclaurin's theorem
f(x) = f(0) + f'(0)·x + f''(0)/2!·x2 + … + f(n–1)(0)/(n – 1)!·xn–1 + Rn

Rn = f(n)(θx)/nxn, 0 < θ < 1
Rn = f(n)(θx)/(n – 1)!·(1 – θ)n–1xn, 0 < θ < 1

Roche-Schlömlichの剰余: Taylorの定理の剰余Rnにおいてpを任意の定数

Rn = f(n)(c)/((n – 1)!p)·(bc)np(ba)p, c ∈ (a, b)
Rn = f(n)(a + θ(ba))/((n – 1)!p)·(1 – θ)np·(ba)n, θ ∈ (0, 1)

Ex. f(x) n次多項式 → x = aを中心とするTaylor展開

→ [f(x)をxaの昇べき順に展開したもの]
f(n+1)(x) = 0
f(x) = f(a) + f'(a)(xa) + f''(a)/2!·(xa)2 + …

+ f(n-1)(a)/(n – 1)!·(xa)n-1 + f(n)(a)/n!·(xa)n

微分の応用 (application of differentiation)


Def. ランダウの記号: limx→a(h(x)/g(x)) = 0 ⇒ h(x) = o(g(x)) (xa)
Ex. limx→0(cosx - 1)/x = limx→0(-sinx/1) = 0 ∴ sinx - x = o(x)
Th. 漸近展開 asymptotic expansion: f(x) ∈ I, Cn-class

f(x) = Σk=0nf(k)(0)/kxk + o(xn) (x → 0)

Pr. (MacLaurinの定理) xI, 0 < θ < 1 ⇒ f(x) = Σk=0n-1f(k)(0)/nxn

f(x) = Σk=0nf(k)(0)/kxk + (f(n)(θx) - f(n)(0))/nxn
h(x) := (f(n)(θx) - f(n)(0))/xn, f(x) Cn-class → f(n)(x) continuous ⇒

|θx| < |x| → 0 (x → 0)

limx→0(h(x)/xn) = limx→0(f(n)(θx) - f(n)(0))/n! = (f(n)(0) - f(n)(0))/n! = 0
f(x) = Σk=0n(f(k)(0)/kxk + o(xn) (x → 0) ∵ h(x) = o(xn)

Ax. 漸近展開 (asymptotic expansion elementary function)

関数の変動

Def. 増加/減少 f(x): x = c ± h (h > 0) ⇒

f(ch) < f(c) < f(c + h) 増加 ⇔ f(ch) > f(c) > f(c + h) 減少

Th. 1. f(x), differential at x = c

f'(c) > 0 増加 increase ⇔ f'(c) < 0 減少 decrease_ f'(c) = 0 増減不明

Pr. f(c + h) – f(c) = h{f'(c) + ρ(h)}, h → 0

ρ(h) → 0, 0 < |h| < δ, δ > 0 ⇒ |ρ(h)| < |f'(c)|
f'(c) + ρ(h)はf'(c)と同符号 ⇒ h符号変化 ≡ f(c + h) – f(c)も符号変化

Th. 1'.: f(x) diff at [a, b] continuous, (a, b) diff., f'(x) = 0

f(x) ≡ 定数関数 constant function

Pr. a < x0b, f(x) continuous at [a, x0], diff. in (a, x0)

(f(x0) - f(a))/(x0 - a) = f'(c), a < c < x0 (平均値の定理 ⇒)

f'(c) = 0 ∴ f(x0) - f(a) = 0 ∴ f(x0) = f(a) //

Th. 2. f(x): [a, b] continuous, (a, b) differential

⇔ [f'(x) ≥ 0 increase, f'(x) ≤ 0 decrease]

Pr. (f(x2) – f(x1)/(x2x1)) = f'(c), c ∈ (x1, x2)

f'(x) ≥ 0 → f(x2) ≥ f(x1) ∴ increase
f'(x) ≤ 0 → f(x2) ≤ f(x1) ∴ decrease //

Def. 極値 extreme (value) ≡ 極大値 + 極小値

f(x), x = c ± h (h > 0) → f(x ± h) < f(c) 極大, f(x ± h) > f(c) 極小
Remark. f(x): x = cでdiff., f(x) = cで極値 ⇒ f'(c) = 0

必ずしも逆は成り立たない ex. f(x) = x3

Th. 3. f(x): [a, b] continuous, (a, b) diff., 適当にcをとる.

f'(x) > 0 in (a, c), f'(x) < 0 in (c, b) ⇒ f(x) = cで極大

Pr. 狭義単調増加 a < x1 < cf(x1) < f(c)

狭義単調減少 c < x2 < bf(c) > f(x2)
∴ 極大 at x = c

Th. 4. f(x): x = ccn級, f'(c) = f''(c) = f'''(c) = … = f(n–1)(c) = 0, f(n)(c) ≠ 0

f'(c) = f''(c) = f'''(c) = … = f(n–1)(c) = 0, f(n)(c) ≠ 0

1) n = even: f(n)(c) > 0 → 極大 / f(n)(c) < 0 → 極小
2) n = odd: f(n)(c) > 0 → increase / f(n)(c) < 0 → decrease (極値ではない)
Pr. n = 1 ⇒ Th.3の拡張 = 自明。n ≥ 2の証明のみ必要
f(c + h) = f(c) + (f'(c)/1!)·h

+ (f''(c)/2!)·h2 + … + (f(n–2)(c)/(n – 2)!)·hn–2 + Rn-1

仮定 f'(c) = f''(c) = … = f(n-1)(c) = 0から

f(c + h) – f(c) = [hn–1/(n – 1)!]f(n–1)(c + θh) (0 < θ < 1)
from Lagrange

(Purpose: h > 0, h < 0でも常に左辺 > 0, < 0であることを証明)
f(n)(c) > 0 →Th. 1 f(n–1)(c)はx = cで増加の状態
1) h < 0のとき(n: even): f(n–1)(c + h) < 0, c + h < c + θh < c

f(n–1)(c + h) < f(n–1)(c + θh) < f(n–1)(c), f(n–1)(c + h) < 0, f(n–1)(c) = 0
f(n-1)(c + θh) < 0__f(c + h) – f(c) > 0

2) h > 0のとき(n: even), hn-1 > 0

f(c + h) > 0, c < c + θh < c + h, f(n–1)(c) = 0, f(n–1)(c + h) > 0
f(n–1)(c + θh) > 0__f(c + h) – f(c) > 0

Def. 最大 the maximum (max), 最小 the minimum (min)

f(x), Dx, c: f(c) ≥ f(x) → 最大 ⇔ f(c) ≤ f(x) → 最小

最大・最小: local → c0-class, continuous
極大・極小: global → cn-class

Def. a < x < b at I

(f(x) - f(a))/(x - a) ≤ (f(b) - f(x))/(b - x) ⇒ 凸関数(下に凸)
(f(x) - f(a))/(x - a) < (f(b) - f(x))/(b - x) ⇒ 狭義の凸関数

Th. 凸(凹)関数: f(x) diff. at I [1-3は同等]

1) f(x) 凸(凹)関数 at I
2) f'(x) 増加(減少)関数 at I
3) f(x)は任意の点における接線より下(上)にない

Th. f'(x) (cδ, c)単調増加(減少), f'(x) (c + δ, c)単調減少(増加)

cf(x)の変曲点

Th. f''(x) exist at I, f''(c) = 0 (cI), f'(cδf'(c + δ) < 0 ⇒ c 変曲点
Th. Hermiteの多項式 Hn(x) ≡ (–1)nex2(dnex2/dxn)

n次多項式, n ≥ 2, Hn(x) = 0はn個の互いに異なる実根をもつ
⇒ 各根はHn–1(x) = 0の根によって隔てられる

Pr. Hn(x) = 0の根 ≡ fn(x) = dnex2/dxn = (–1)nex2Hn(x) = 0の根

H1 = 2xx = 0, H2 = 4x2 – 2 → x = ±1/√2
n = 2のとき成立
n = k + 1のときHn+1 = 2xHnH'nn + 1次多項式
Hn = 0の根 c1 < c2 < … < cnとすると

limx→–∞fn(x) = fn(c1) = fn(c2) = … = fn(cn) = limx→–∞fn(x)

Rolleの定理から fn+1(d1) = fn+1(d2) = …

= fn+1(dn+1), d1 < c1 < d2 < c3 < … < cn > dn+1
なるdi (Hn+1 = 0の根)存在 //

Th. Newton法 (Newton method) [a, b], f(a) = 0, f'(a) ≥ 0, (a, b) → f''(x) > 0,

{xn}, x1 = b, xn+1 = xn - f(xn)/f'(xn) (n = 1, 2, 3, …)
⇒ {xn} 狭義単調減少でaに収束

Pr._f'(x), [a, b] 狭義単調増加, f'(a) ≥ 0 → f'(x) > 0 (a < xb) … (1)

f(x), [a, b] 狭義単調増加, f(a) = 0 → f(x) > 0 (a < xb) … (2)

(数学的帰納法) {an}, a < xnb for all nを証明

n = 1, x1 = b → a < x1b成立
n = k, a < xkb成立と仮定, [a, xk] Taylorの定理

f(a) = f(xk) + f'(xk)(a - xk) + f''(c)/2·(a - xk)2

となるa < c < xk exist
xk = xk+1 + f(xk)/f'(xk), f(a) = 0を代入

0 = f(xk) + f'(xk){a - xk+1 - f(xk)/f'(xk)} + f''(c)/2·(a - xk)2

= (a - xk+1)f'(xk) + f''(c)/2·(a - xk)2

xk+1f'(xk) = af'(xk) + f''(c)/2·(a - xk)2
a < xkb, (1)より f'(xk) > 0, f''(c) > 0 →

xk+1 = a + f''(c)/2f'(xk)·(a - xk)2 > a
(1), (2)より xk+1 = xk - f(xk)/f'(xk) < xk < xkb

a xnbn = k + 1で成立 ⇒ Nについてa < xnbが成立
xn+1 = xn - f(xn)/f'(xn) < xn ⇒ {xn} 下に有界な狭義単調減少 = 収束

β := limn→∞xn

a < xnba < βbf(xn) = f'(xn)(xn - xn+1)
f(x), f'(x) continuous → f(β) = f'(β)(β - β) = 0
(2)より β = alimn→∞xn = β = a //

不定形の極限値 limit of indeterminate forms

Th. ロピタルの定理 (l'Hôpital's rule): f(x) → 0, g(x) → 0, xa,

limxa(f(x)/g(x)) = 0/0 = ∞ – ∞ = 00の場合極限値をとる

Case 0/0: f(x), g(x), (ar, a + r) – {a} diff.

limxa±0f(x) = 0, limxa±0g(x) = 0, g'(x) ≠ 0,
limxa±0f'(x)/g'(x) = A (有限 finite or 無限大 infinite) ⇒ limxa±0f(x)/g(x) = A

Pr. define f(a) = g(a) = 0 → f(x), g(x): (ar, a + r) continuous

(ar, a + r) ⊃ [ar, a + r], f(x), g(x) [a, a + h] continuous,
(a, a + h) diff., g'(x) ≠ 0 ⇒ limxa±0f'(x)/g'(x) = A
f(a) = g(a) = 0 ∴ limxa±0f'(a)/g'(a) = A
f(x)/g(x) = f'(c)/g'(c), c ∈ (a, b) (Cauchyの平均値の定理)
limxa±0f(x)/g(x) = A

Cauchy系 f(x), g(x) (R, +∞) diff., limx→∞f'(x)/g'(x) = A

limx→∞f(x)/g(x) = A (∵ x:= 1/tx → ∞ ≡ t → 0)

f(x) = f(1/t) = F(t), g(x) = g(1/t) = G(t) → F(t), G(t) (0, 1/R) continuous
F'(t) = -1/t2·f'(1/t) = -1/t2·f'(x), G'(t) = -1/t2·g'(x), F'(t)/G'(t)

= f'(x)/g'(x) (= f(x)/g(x))

t → 0 ∴ limta±0F'(x)/G'(x) = limx→∞f'(x)/g'(x) = limx→∞f(x)/g(x) = A
Th. Case. ∞/∞: f(x), g(x) (a, a + r) diff.,

limx→∞f(x) = ∞, limx→∞g(x) = ∞, g(x) ≠ 0, g'(x) ≠ 0,
limxa+0f'(x)/g'(x) = A (finite or infinite) ⇒ limxa+0f(x)/g(x) = A

Pr. Case. A infinte

a < x < x' < a + r, [x, x'] continuous, (x, x') diff., g'(x) ≠ 0
→ (f(x') - f(x))/(g(x') - g(x)) = f'(c)/g'(c), x < c < x'
limxa±0f'(x)/g'(x) = Alimca+0f'(c)/g'(c) = A (x, x' → 0, ca + 0)
ε > 0, δ > 0, a < c < a + δ,

|f'(c)/g'(c) – A| = |(f(x') – f(x))/(g(x') – g(x)) – A| < ε/3

∴ |g(x') – g(x)|ε/3 > |f(x') – f(x) – A(g(x') – g(x))| → g(x)で両辺を割る
|1 – g(x')/g(x)|ε > |f(x')/g(x) – f(x)/g(x) – g(x')/g(xA + A|

≥ |f(x')/g(x) – f(x)/g(x) + A| – |A·g(x')/g(x)|
= |–f(x')/g(x) + f(x)/g(x) – A| – |A·g(x')/g(x)| ≥

|f(x)/g(x) – A| – |f(x')/g(x)| – |A·g(x')/g(x)|

∴ |1 – g(x')/g(x)|ε/3 + |f(x')/g(x)| + |A·g(x')/g(x)| > |f(x)/g(x) – A|
ここでx': fixed, xa (他の値は全てreduceされる)

→ |f(x)/g(x) – A| < 3ε, |1 – g(x')/g(x)| < 1 //

Q. k = constant, logx = kxの実数解の個数
A. x > 0, logx/x = k, f(x) := logx/x

f'(x) = ((1/xx - logx·1)/x2 = (1 - logx)/x2
x = ef'(x) = 0 (ロピタルの定理)
limx→∞f(x) = limx→∞(logx/x) = limx→∞((1/x)/1) = limx→∞(1/x) = 0

_x___(0)____e__ _(∞)
f'(x)_____ _+__ 0___-
f(x)__(-∞)__ _1/e___ (0)

解の個数: k > 1/e … 0. k ≤ 0, k = 1/e … 1. 0 < k < 1/e … 2

Def. 無限小 infinitesimal/無限大 infinite: 変数が0に収束[±∞] ⇒ その変数

a = ±∞としても同様

1) limxa(u/v) = 0 ⇒ uvより高位の無限小 [uvより低位の無限大]
2) limxa(u/v) = ±∞ ⇒ uvより低位の無限小 [uvより高位の無限大]
3) limxa(u/v) = kuvと同位の無限小[大] (k = 1 → 同値の無限小[大])
4) limxa(u/vn) = k (k ≠ 0) ⇒ uvに比してn位の無限小[大]

(= 第n位の無限小[大])

Ex. 2つの無限小uvの差が、u, vいずれよりも高位の無限小

uvは同値の無限小(逆も真)

Pr. uv := δ, lim(δ/v) = 0, u = v + δ, u/v = 1 + δ/v ∴ lim(u/v) = 1 //
Th. Duhamelの定理 x = ∞, αi, βi (i = 1, 2, …, n)は全て正で同値の無限小,

limn→∞(α1 + α2 + … + αn) = A (finite) ⇒ limn→∞(β1 + β2 + … + βn) = A

積分法 (integration)


Def. I = [a, b] ⇒

Iの分割: Δ:a = x0 < x1 < x2 < … < xn-1 < xn = b ⇒ δ = {xk}
Δの幅: |Δ| = max{xk - xk-1|k = 1, 2, … n}
Riemann和: S(f; Δ{ξk}) = Σk=1nf(ξk)(xk - xk-1)

Def. 積分可能: f(x), I = [a, b], 有界 ⇒ a exist, {ξk} ⇒ lim|Δ|→0S(f;Δ,{ξk} = α
Def'. 積分可能: a exist, ε > 0, δ(ε) > 0 exist, |Δ| < δ(ε)なるIの分割Δ, {ξk}

⇒ |S(f;Δ,{ξk} - α| < ε

Def. (Riemann)積分: α = abf(x)dx
Th. ダルブーの定理 (Darboux's theorem): I = [a, b] f(x) ⇒ |Δ| → 0

SΔ(f), sΔ(f) 収束 ⇒
lim|Δ|→0SΔ(f) = -abf(x)dx, lim|Δ|→0sΔ(f) = -abf(x)dx

Pr. ε > ε/2 > 0 ⇒ SΔ0(f) < -abf(x)dx + ε/2, Δ0 exist, n: Δ0の分点数 ⇒

f(x) bounded, M := supxIf(x), m := infxIf(x), Δ = {xk},
Δ1: Δの小区間[xk-1, xk]に分点を1増やしたもの ⇒

mmkMkM, 0 < xk - xk-1 ≤ |Δ|

∴ 0 ≤ SΔ(f) - SΔ1 (f) ≤ (Mkmk)(xkxk−1) ≤ (Mm)|Δ|

δ(ε) := ε/(2n(M - m)) > 0, |Δ| < δ(ε), I → Δ分割 ⇒

0 ≤ SΔ(f) - SΔ'(f) ≤ n(M - m)|Δ| < n(M - m)δ(ε) = 2/ε

Δ0 ∪ Δ' → SΔ'(f) ≤ SΔ0(f)

∴ 0 ≤ SΔ(f) - -abf(x)dx

= (SΔ(f) - SΔ'(f)) + (SΔ'(f) - SΔ0(f)) + (SΔ0(f) - -abf(x)dx)

< ε/2 + 0 + ε/2 = ε

lim|Δ|→0sΔ(f) = -abf(x)dxについても同様 //

Def. 振幅, ω(f, I): I = [a, b], f(x) bounded ⇒

ω(f, I) = supxIf(x) - infxIf(x) = supx,yI|f(x) - f(y)| ≥ 0

∴ Δ = {xk} ⇒ ω(f, I): I = supxIf(x) - infxIf(x) = Mk - mk
Σk=1nω(f, Ik)(xk - xk-1) = Σk=1n(Mk - mk) = SΔ(f) - sΔ(f)

過剰和と不足和の差は振幅で分かる

Th. (可積分N.S.C.): I = [a, b], f(x) bounded (1-3は同値) ⇒
(1) f(x) integrable on I
(2) lim|Δ|→0SΔ(f) = lim|Δ|→0sΔ(f), viz. -abf(x)dx = -abf(x)dx
(3) Δ = {xk}, Ik := {xk-1, xk} ⇒ lim|Δ|→0Σk=1nω(f, Ik)(xk - xk-1) = 0

[不定積分基本公式 basic formulae]

不定積分 (indefinite integrals)


Def. 原始関数 primitive function, F(x)
Def. 不定積分 indefinite integrals, f(x)dx = F(x)

f(x) ⊂ I integrable, aI, CF(x) = axf(t)dt + C (xI)

Th. f(x) continuous on IF(x)はf(x)の不定積分 ⇔ F(x)はf(x)の原始関数
Th'. f(x) continuous on I, aId/dxaxf(t)dt = f(x) (xI)
Th. f(x), g(x) indefinite integrals exist, λ, μ

{λf(x) + μg(x)}dx = λf(x)dx + μg(x)}dx

Pr. f(x)dx = axf(t)dt + C1, g(x)dx = axg(t)dt + C2

λf(x)dx + μg(x)}dx = λ(∫axf(t)dt + C1) + μ(∫axg(t)dt + C2)

= ax{λf(x)dx + μg(x)dx} + (C1 + C2) //

Th. f(x) indefinite integrals exist on IF(x) continuous function

Th. 置換積分 integration by substitution

F(x) = f(x)dx, x = g(t): diff. w.r.t.t. → f(x)dx = f{g(t)}·g'(t)dt

x := g(t) → dx = g'(t)dt求めf(x)のxの代りにg(t)をdxの代りにg'(t)dt代入

Pr. F(x) = F(g(t)) ∴ d(F(g(x))/dt = dF/dx·dx/dt = f(xg'(t) = f(g(t))·g'(t) //
Cf. f'(x)/f(x)dx = logf(x), f'(x)/nf(x)n–1dx = nf(x)はよく使う置換積分

Th. 部分積分 integration by parts

(f(xg(x))' = f'(xg(x) + f(xg'(x) ⇒ (f'(xg(x) + f(xg'(x))dx = f(xg(x)

f(x)·g'(x)dx = f(xg(x) – f'(xg(x)dx, g(x) := x

f(x)dx = x·f(x) – x·f'(x)dx

Ex. I = logxdx: f(x) = logx, f'(x) = 1/x

I = xlogxx·(1/x)dx = xlogx1dx = xlogx – x

Ex. In = sinnxdx (n 自然数)

[:= sinn–1xsinxdx, f = sinn–1x (f' = (n – 1)sinn–2xcosx),
g' = sinx (g = –cosx)]
In = –cosxsinn–1x + (n – 1)sinn–2xcos2dx

= –cosxsinn–1x + (n – 1)sinn–2x(1 – sin2x)dx

In = –cosxsinn–1x + (n – 1)In–2 – (n – 1)In
In = 1/n·{–cosxsinn–1x + (n – 1)In–2}
簡約公式: I1, I2等簡単な計算部分を使ってInを求められる

Cf. In = sinnxdx = 1/n·{sinxcosn–1x + (n – 1)In–2},
___In = tannxdx = 1/(n – 1)·(tann–1xIn–2)

有理関数の積分 integration of rational functions

i) A/(xa)dx = A·log|xa| + C
ii) A/(xa)kdx = -A/[(k – 1)/(xa)k-1] + C (k ≠ 1)
iii) (Lx + M)/(x2 + q2)kdx = Lx/(x2 + q2)kdt(1)=Ik + M1/(x2 + q2)kdx(2)=Jk
A. (1) x2 + q2 := t, dt/2 = xdx, Ik = 1/2tkdt

I1 = 1/2·log|x2 + q2|, I2 = –1/2·(x2 + q2),

Ik = –1/2·[(k – 1)(x2 + q2)k–1] (k = 1, 2, 3 …)

__(2) f' = 1, g = 1/(x2 + q2)として部分積分

Jk = x·{1/(x2 + q2)k} + 2k(x2/(x2 + q2)k+1)dt

= x/(x2 + q2)k + 2k[{(x2 + q2) – q2}/(x2 + q2)k+1]dt
= x/(x2 + q2)k + 2k1/(x2 + q2)kdt – 2kq21/(x2 + q2)k+1dt
= x/(x2 + q2)k + 2kJk – 2kq2Jk+1

Jk+1 = 1/q2[x/{2k(x2 + q2)k} + {(2k – 1)/2k}Jk]

k + 1 := k, k = k – 1

Jk = 1/q2[x/{2(k – 1)(x2 + q2)k–1} + {(2k – 3)/2(k – 1)}Jk] [漸化式]

iv) (Lx + M)/[(xp)2 + q2]kdx

= Lx/[(xp)2 + q2]kdx(1)=In + M1/[(xp)2 + q2]kdx(2)=Jn

A.__(1) xp = t, dx = dt, In = L(t + p)/(t2 + q2)kdt → reduce (iii)

(2) (iii-2)とどちらを用いてもよい
J1= (1/(t2 + q2))dt = (1/q)·tan-1(t/q), f'(t) = t/(t2 + q2)k, q(t) = t
{1/(q2tan2θ + q2)k)}qsec2θdθ = 1/q2k–1(1/sec2kθ)sec2θdθ

= 1/q2k–11/sec2k–2θdθ = 1/q2k–1cos2k–2θdθ [漸化式]

Ex. (x3/(x + 1))dx = (x2 - x + 1 - 1/(x + 1))dx = x3/3 - x2/2 + x - log|x + 1|
Ex. (1/(x2 + 4x + 7))dx = 1/((x + 2)2 + 3)dx = (1/√3)·tan-1((x + 2)/√3)
Th. f(x) = a0 + a1x + a2x2 + … + amxm = Σi=0maixi (1, 2, …, m: integer)

f(x) = 0の1つの実根 af(x) = (xa)r·F(x) (F(a) ≠ 0)なるr
f(x) = 0の根が虚根 x = p + qipqiもまた根を持つ

Pr. 共約のとき一方が根ならば他方も必ず根

f(p + qi) = u(p, q) + iv(p, q) = 0 → u(p, q) = 0, v(p, q) = 0
ここでi2 = -1 = (-i)2f(pqi) = u(p, q) – v(p, q) = 0 ∴ pqiも根
f(x) = (xpqi)φ(x) = (xpqi)·(xp + qi)φ(x) …

= [(xp)2 + q2]kΦ(x)

以上の事から[(xp)2 + q2]s, (pa)r, amの形で表わせる //

実根の場合の積分
f(x) = a0 + a1x + … + amxm = (xa)r·F(x), F(x) ≠ 0.
g(x) = b0 + b1x + … + bmxm

A1 := g(x) – A1F(a) = 0, g(x) – A1F(x) = (xag1(x)
g(x)/f(x) = ((xag1(x) + A1F(x))/{(xa)r·F(x)}

= A1/(xa)r + A2/(xa)r–1 + … + Ar/(xa) + g1(x)/F(x)

無理関数の積分 integration of irrational function (虚根の積分)

(1) I = R(x, n√(ax + b))dx

n√(ax + b) := tax + b = tn, x = (tnb)/a, dx = (n/atn–1dt
I = (n/a)R√(tnb)/a, t)tn–1dt

(1)' I = R(x, n√{(ax + b)/(cx + e)}dx

n√{(ax + b)/(cx + e)} ≡ t,
(ax + b)/(cx + e) = tn,
x = (betn)/(ctna),
dx/dt = (bcaentn–1/(atn – c)2
I = n(bcae)R((betn)/(ctna), t)tn–1/(atnc)2dt

(1)'' I = R(x, p√(ax + b)), q√(ax + b))dx

l√(ax + b) = tとして(1)'に還元reduceしp, qL.C.M.をとるとn/pp, n/qq

(2) I = R(x, √(ax2 + bx + c)dx

A) a = 0 → reduce to (1)
B) a > 0: √(ax2 + bx + c) = t – √a·x, ax2 + bx + c = t2 – √a·x + ax2
x = (t2c)/(b + 2√a·t)
∴ √(ax2 + bx + c) = t – (t2c)√a/(b + 2√a·t)
dx = {2(√a·t2 + bt + √a·c)/(b + 2√a·t)2}dt

これをIに代入するとIは有理関数の積分となる

C) a < 0
i) D = b2 – 4ac < 0

ax2 + bx + c < 0 (√(ax2 + bx + c)は虚数) not exist

ii) D = 0 → x ≠ -b/2a, ax2 + bx + c < 0 not exist
iii) D > 0 → 異る2実根 α < x < β

→ √(ax2 + bx + c) = √a(xα)(xβ) := t(xα)

t2 = a(xα)/(xβ),
x = (αt2)/(t2a),
dx/dt = 2a(βα)t/(t2α)

I = R((αt2)/(t2a), t)(t2α)/{2a(βα)t}dt → reduce (1)

(3) I = xm(axn + b)q/pdx (a ≠ 0, b ≠ 0. a, b, p, q: integer)

(axn + b)1/p := t, axn + b = tp
x = {(tpb)/a}1/n, dx = ptp–1/(an·xn–1)dt
I = (p/an)xmn+1tq+p–1dt = (p/an)[{(tpb)/a}1/n]mn+1tq+p–1dt

= (p/an){(tpb)/a}(m+1)/n–1tq+p–1dt

p + q – 1 integer ∴ (m + 1)/n: integer → integrable

Ex. I = x13(1 + x7)3/5dx
A. (m + 1)/n = (13 + 1)/1 = 14: integer, x7 = t, x = t1/7, dx = 1/7t–6/7dt

I = t13/7·(1 + t)3/5·(1/7t–6/7)dt = 1/7t·(1 + t)3/5dt

= 1/7(t + 1)8/5dt – 1/7(t + 1)3/5dt
= 1/7{13/5(x7 + 1)13/5 – 8/5(x7 + 1)8/5}

Ex. Im, n = sinmx·cosnxdxの簡約公式 [漸化式 → 部分積分]

sinm–1x(sinx·cosnx)dx,

f = sinm–1x,
f' = (m – 1)sinm–2xcosx,
g' = sinx·cosnx,
g = –1/(n + 1)cosn+1x

Im, n = sinm–1x{–1/(n + 1)cosn+1x}

(m – 1)sinm–2xcosx{–1/(n + 1)cosn+1x}dx

= –1/(n + 1)[sinm–1xcosn+1x – (m – 1)sinm–2xcosn+2xdx]

→ cosn+2x = cosnx(1 – sin2x)

= 1/(n + 1){–sinm–1xcosn+1x + (m – 1)

(sinm–2xcosnx – sinmxcosnx)dx}

= 1/(n + 1){–sinm–1xcosn+1x + (m – 1)(Im–2, nIm, n)}

Im, n + (m – 1)/(n + 1)Im, n

= 1/(n + 1){–sinm–1xcosn+1x + (m – 1)Im–2, n}

Im, n = 1/(m + n){–sinm–1xcosn+1x + (m – 1)Im–2, n}

以下の簡約公式は同じ
= 1/(m + n){sinm+1xcosn–1x + (n – 1)Im, n–2}
= 1/(n + 1){–sinm+1xcosn+1x + (m + n + 2)Im, n+2}
= 1/(m + 1){sinm+1xcosn+1x + (m + n + 2)Im+2, n}

定積分 (definite integrals)


Th. Heine-Borelの被覆定理
f(x) [a, b] continuous, ε1 = ε/2 > 0 (fixed),

[a, b] ∋ x1, x2, 0 < |xx'| < δ'x, |f(x) – f(x')| < ε1, 1/2δ'x = δx

⇒ (x1δx1, x1 + δx1)∪(x2δx2, x2 + δx2)∪ …

∪(xnδxn, xn + δxn) ⊃ [a, b] → 近傍δxiの和は[a, b]を被覆する

Pr. (背理法) [a, b] ∉ (x1δx1, x1 + δx1)∪(x2δx2, x2 + δx2)∪ …

∪(xnδxn, xn + δxn)を仮定し矛盾を説明

I1 := [a, b] ∉ [a, (a + b)/2], [(a + b)/2, b],
I1を覆わない部分がある方をI2 = [a1, b1]
nが十分に大きい(n → ∞) In = [an–1, bn–1], |In| = (ba)/2n–1,

aa1a2 ≤ … ≤ < b, bb1b2 ≥ … > a

∴ limn→∞an = limn→∞bn = c ∈ [a, b]
In = [an–1, bn–1] ⊂ (cδc, c + δc) … 仮定と矛盾 //
Def. 一様連続 uniformly continuous
ε > 0, δ > 0, [a, b] ∋ x1, x2, |x1 < x2| ⇒ |f(x1) – f(x2)| < ε ≡ 一様連続
Pr. Heine-Borel定理においてminΣi=1nδ = δ, |xx'| < δ → |f(x) – f(x')| < ε

x ∈ (xiδxi, xi + δxi), |xxi| < δxi, |xx'| < δδi
∴ |x' – xi| ≤ |x' – x| + |x' – xi| < 2δxi = δ'xi
∴ |f(x') – f(xi)| < ε/2, |f(x) – f(xi)| < ε/2, |f(x) – f(x')| < ε/2 //

Ex. f(x) = 1/x, I = (0, ∞) ≠ 一様連続
Pr. (背理法) f(x) = 1/xI = (0, ∞)で一様連続と仮定

ε = 1, δ = δ(1) > 0

x, yI, |x - y| < δ ⇒ |1/x - 1/y| < 1

nN, x := 1/n, y = 1/n + δ/2, |x - y| = δ/2 < δ

|1/x - 1/y| = |n - 2n/(2 + δn)| = δn2/(2 + δn) < 1

⇔ limn→∞(δn2/(2 + δn)) = ∞ (矛盾) //

Th. f(x), g(x)はI上で一様連続 ⇒ λ, μ, λf(x) + μg(x)もI上で一様連続
Pr._ Case. λ = μ = 0: trivial

Case (λ, μ) ≠ (0, 0): ε > 0, ε' = ε/(λ + μ) > 0, δ1(ε'), δ2(ε') > 0

x, yI, |x - y| < δ1(ε') ⇒ |f(x) - f(y)| < ε'
x, yI, |x - y| < δ2(ε') ⇒ |g(x) - g(y)| < ε'

δ(ε) := min{δ1(ε'), δ2(ε')}, |x - y| < δ(ε), x, yI

|λf(x) + μg(x) - (λf(y) + μg(y))|

≤ |λ||f(x) - f(y)| + |μ||g(x) - g(y)| < (|λ| + |μ|)ε' = ε //

Th. 最大・最小値の定理 f(x) [a, b] continuous

⇒ maxf(x) = M, minf(x) = m, mf(x) ≤ M

Pr.___┌──┬──┬──┬──┬──┬──┬──┐ Δ分割

a[x0__x1 ………………… xn-2 xn-1_xn], b[xi, xi+1]
Mimi, xi+1xi = δi

SΔ = f(x0)(x1x0) + f(x1)(x2x1) + … + f(xn–1)(xnxn–1)

= Σi=0n–1f(xi)(xi+1xi) = Σi=0n–1f(xi)δi

_________δi
┏━━━━━━━┓
┣━┳━━━━━┫
┗━┻━━━━━┛
x_δi1_____δi2___x'

SΔ = sΔ
1) Δ'分割: Δ分割に新しい分点を付け加える

miδi = miδi1 + miδi2mi1δi1 + mi2δi2 (Mimi)

Miδi1 + Miδi2Mi1δi1 + Mi2δi2 = Miδi

miδiMiδi

2) 任意の2つの分割Δ、Δ'を考えこれらの分点を分点とする分割Δ''を考える

(1)からSΔ'' ≤ SΔ', sΔ ≤ sΔ'' → sΔ ≤ sΔ'' ≤ SΔ'' ≤ SΔ'

3) 有限個の分点を持つあらゆる分割を考える

SΔ, sΔ supsΔ = s, infSΔ = S, sΔ ≤ SΔ' → sΔ ≤ SsS //

3)で特にsΔ ≤ sΔ'' ≤ SΔ'' ≤ SΔ, s = S

abf(x)dx, f(x) [a, b] (定)積分可能 integrable (a: 下端, b: 上端)

Th. f(x) [a, b] continuous ⇒ 一様連続

cf. f(x) = 1/x (0 < x ≤ 1) continuousだが一様連続でない事もある

Pr. (背理法) f(x)はI = [a, b]上で一様連続ではないと仮定し解く
Ax. I = (a, b], f(x) 連続, 右極限 limxa+0f(x)収束 ⇒ f(x) 一様連続
Ax. I = [a, ∞), f(x) 連続, 極限 limx→∞f(x)収束 ⇒ f(x) 一様連続
Th. f(x) 単調 (monotone) on If(x) integrable
Th. f(x) continuous on I = [a, b] ⇒ f(x) integrable (SΔ = sΔを証明)

(cf. [a, b] 不連続 → ルベック積分)

Pr. Δを分割し各部分区間もみなδよりも小さくなるようにすると

sΔ ≤ supsΔ = s, S = infSΔ ≤ SΔ

SsSΔ – sΔ = limn→∞(SΔ – sΔ) = (baε //

Th. f(x), g(x) continuous on I = [a, b], f(x) ≥ g(x) (axb),

f(ξ) > g(ξ), ξI exist ⇒ abf(x)dx > abg(x)dx

Pr. h(x) := f(x) - g(x) → h(x) continuous on I

δ > 0 exist, C (constant) > 0 exist →

|x - ξ| < δh(x) > C

abh(x)dxξ-δξ+δh(x)dxξ-δξ+δCdx ≥ 2 > 0
abh(x)dx = abf(x)dx - abg(x)dx > 0 //

Th. 積分の平均値の定理: f(x) continuous on I = [a, b] ⇒

1/(b -a)abf(x)dx = f(ξ)

Pr. Max = f(x1), min = f(x2) exist → mf(x) ≤ M (axb)

abmdx < abf(x)dx < abMdxm < 1/(b -a)abf(x)dx < M
f(ξ) = m < 1/(b -a)abf(x)dx__(x1 < ξ < x2, a < ξ < b) //

Th. (積) f(x), g(x) integrable on If(x)g(x) integrable
Th. (商) f(x), integrable on I, f(x) ≠ 0, 1/f(x) bounded ⇒ 1/f(x) integrable
定積分の求め方
1) 区分求積(分)法 quadrature by parts: あらゆる分割の極限を整頓

limn→∞SΔ計算 ⇒ 直接abf(x)dx求める

2) 不定積分との関連
Th. 区分求積法: f(x), [a, b] 積分可能 ⇒

abf(x)dx = limn→∞((b - a)/n)Σk=1nf(a + (b - a)/n·k)

= limn→∞((b - a)/n)Σk=0n-1f(a + (b - a)/n·k)

a = 0, b = 1 → 01f(x)dx = limn→∞(1/nk=1nf[k:n] = limn→∞(1/nk=0n-1f[k:n]
Th. 定積分の性質 (f(x), g(x) [a, b] continuous)
1) (線形性) abλf(x)dx = λabf(x)dx (λ: 定数)
1') ab{λf(x) + μg(x)}dx = λabf(x)dx + μabg(x)dx
2) a, b, cの大小がどうであってもabf(x)dx = acf(x)dx + cbf(x)dx
3) (単調性) f(x) ≥ 0, a < b

abf(x)dx ≥ 0, 更にf(x) ≥ g(x) → abf(x)dxabg(x)dx
(f(x) ≡ g(x) 等号成立)

Pr. Δ = {xk}, ξ ∈ [xk-1, xk] → f(ξk) ≤ g(ξk)

S(f;Δ,{ξk}) = Σk=1nf(ξk)(xk - xk-1) ≤ Σk=1ng(ξk)(xk - xk-1) = S(g;Δ,{ξk})
|Δ| → 0 ⇒ abf(x)dx = lim|Δ|→0S(f;Δ,{ξk}) ≤ lim|Δ|→0S(g;Δ,{ξk}) = abg(x)dx //

4) 絶対値の不等式 |abf(x)dx|ab|f(x)|dx (ab)
Pr. –|f(x)| ≤ f(x) ≤ |f(x)|, a < b → –ab|f(x)|dxabf(x)dxab|f(x)|dx //
5) 積分の第一平均値の定理 abf(x)dx = (ba)f(c), a < c < b
Pr. [a, b], maxf(x) = G, minf(x) = gg(ba) ≤ abf(x)dxG(ba)

gabf(x)dx/(ba) = f(c) ≤ G, a < c < b (中間値の定理) //

Th. 微積分学の基本定理 f(x) [a, b] continuous, F(x) = axf(x)dx, a < x < b

dF(x)/dx = f(x) 閉区間であれば必ず原始関数存在
(≡ G'(x) = f(x) → abf(x)dx = G(b) – G(a) = [G(x)]ab)

Pr. F(x + h) – F(x) = ax+hf(x)dxaxf(x)dx = hf(ξ) (x < ξ < x + h)

(一様連続なものがintegrableであることの証明)
h → 0, ξxF(x + h) – F(x) = hf(x)
dF(x)/dx = f(x) ここでG(x) (f(x)の原始関数), G(x) = F(x) + C
Δ分割 a = a0 < a1 < … < an = b

G(ai) – G(ai–1) = G'(xi–1)(aiai–1) = f(xi–1)(aiai–1)

G(b) – G(a) = Σi=0n–1f(xi–1)(aiai–1), δ(Δ) → 0

Σi=0n–1f(xi–1)(aiai–1) = abf(x)dx //

置換積分 integration by substitution

f(x) [a, b] continuous, x = g(u) [a, b] (強意の)単調増加(減少),
g(α) = a, g(β) = b, g'(u) [α, β] continuous
abf(x)dx = αβf(g(u))g'(u)du

部分積分 integration by parts: abf'(x)g(x)dx

= [f(x)g(x)]ababf(x)g'(x)dx, f(x) = x ⇒ [xg(x)]ababxg'(x)dx

Th1. abf(x)dx = baf(a + bx)dx = [F(a + bx)]ba = F(b) – F(a)
Th2. 0af(x)dx = 0a/2(f(x) + f(ax))dx

Ex. 0πsinnxdx = 0π/2(sinnx +sinn(πx))dx = 20π/2sinnxdx

Th3. -aaf(x)dx = 0a(f(x) – f(–x))dx

Ex. -π/2π/2cosnxdx = 0π/2(cosnx – cosn(–x))dx = 20π/2cosnxdx

Th4. abf(cx)dx = 1/ccacbf(x)dx
Th5. f(ax) = f(x) ⇒ 0af(x)dx = 20a/2f(x)dx,

f(ax) = –f(x) ⇒ 0af(x)dx = 0

Th6. f(x) 偶関数 even f. f(-x) = f(x) ⇒ -aaf(x)dx = 20af(x)dx,

__f(x) 奇関数 odd f. f(-x) = –f(x) ⇒ -aaf(x)dx = 0

Pr. Case. f(x) continuous

-aaf(x)dx = 0af(x)dx + -a0f(x)dx
x := -t, x: -a → 0, t: a → 0, dx = -dt
-a0f(x)dx = a0f(t)(-dt) = 0af(-t)dt

0af(t)dt even f., -0af(t)dt odd f.

∴ when f(x) is odd f., -aaf(x)dx = 0af(x)dx + -a0f(x)dx

= 0af(x)dx + 0af(x)dx = 20af(x)dx

∴ when f(x) is even f., -aaf(x)dx = 0af(x)dx + -a0f(x)dx

= 0af(x)dx - 0af(x)dx = 0 //

Ex. Schwarzの不等式 {abf(x)g(x)dx}2abf(x)2dx·abg(x)2dx
Pr. f(x) = 0 for x, or a = b → 両辺とも0。よって成立

f(x) ≠ 0なるxが存在と仮定 → abf(x)2dx > 0,
tの2次式 F(t) = t2abf(x)2dx – 2tabf(x)g(x)dx + abg(x)2dx
F(t) = ab{tf(x) – g(x)}2dx ≥ 0, F(t) = 0の判別式 D ≤ 0 //

特異積分(異常積分、広義積分) improper integral


f(x) [a, b] continuous, (a, b) discontinuous

Def. b: 特異点 singular point (不連続点 discontinuous point)
Cf. 特異性 singularity

特異積分: 特異点を有する関数の積分
Def. 可積分 abf(x)dx = limε→0ab-εf(x)dx
1) 独立について f(x) [a, b] continuous → integrable → abf(x)dx = exist
2) f(x) [a, b] continuous → |f(x)| continuous

→ integrable. i.e., ab|f(x)|dx = exist

3) f(x) [a, b) continuous, x = bでdiscontinuous (定義する必要なし)

limε→0ab-εf(x)dx 有限確定値として存在 := abf(x)dx
f(x) (a, b] continuous

limε→0a-εbf(x)dx 有限確定値として存在 := abf(x)dx

f(x) [a, b] ∋ c continuous

limε→0ac-εf(x)dx + limε→0c+εbf(x)dx = abf(x)dx

4) 無限積分 infinite integral: 積分範囲が∞を含む積分

f(x) [a, +∞) continuous → af(x)dx = limb→∞abf(x)dx,
f(x) (-∞, b] continuous → -∞bf(x)dx = lima→–∞abf(x)dx
f(x) (-∞, +∞) continuous → infinite

特異積分 f(x)dx exist ⇔ |f(x)|dxが存在するとは限らない

(但し逆は真 Ex. |f(x)|dx exist ⇒ f(x)dx exist)

Ax. α, constant ⇒

(1) 1(1/xα)dx 収束 N.S.C. ⇒ α > 1
(2) 01(1/xα)dx 収束 N.S.C. ⇒ α < 1

Pr._α = 1 → 1(1/x)dx = ∞, 01(1/x)dx = ∞ 共に発散

α ≠ 1 → 1(1/xα)dx = limt→∞1t(1/xα)dx = limt→∞[(x1-α/(1 - α)]1t

= limt→∞((t1-α - 1)/(1 - α)) ∴ {-1/(1 - α) (1 - α < 0), ∞ (1 - α > 0)}

α > 1の時に収束

______01(1/xα)dx = limt→+0t1(1/xα)dx = limt→+0[(x1-α/(1 - α)]t1

= limt→+0((1 - t1-α)/(1 - α)) ∴ {∞ (1 - α < 0), 1/(1 - α) (1 - α > 0)}

α < 1の時に収束 //

Th. 特異積分の絶対収束 (比較判定法)
f(x), g(x), (a, b], integrable, |f(x)| ≤ g(x) (a < xb) ⇒

abg(x) convergence ⇒ abf(x) convergence
[a, b), [a, ∞], (-∞, b]も同様

Pr. a < t < b, F(t) = tbf(x)dx, G(t) = tbg(x)dx

limta+0G(t) convergence, ε > 0, δ > 0

a < t1 < a + δ, a < t2 < a + δ ⇒ |G(t1) - G(t2)| < ε

t1, t2 ∈ (a, a + δ), t1t2
|F(t1) - F(t2)| = |∫t1t2f(x)dx|t1t2|f(x)|dxt1t2g(x)dx

= G(t1) - G(t2) < ε

t2t1も同様 ⇒ a < t1 < a + δ, a < t2 < a + δ ⇒ |F(t1) - F(t2)| < ε
右極限 limta+0F(t) 収束 (Cauchyの収束定理より) ∴ abf(x)dx 収束
他の区間も同様に証明 //

5) Th. εの極限値定理 f(x) [a, b] continuous, abf(x)dx exist

cbf(x)dx exist___Ex. abf(x)dx = ∞ → cbf(x)dx = ∞

6) 可積分の十分条件
Th. 1. f(x) [c, b] continuous (b singular point), 0 < α < 1,

limxb(bx)αf(x) exist ⇒ cbf(x)dx integrable

Pr. limε→0abf(x)dx integrable in U(b) = (bε, b)を示す

|f(x)| < M/(bx)α, M constant
bc-ε|f(x)|dx < Mbc-ε{1/(bx)α}dx

= M[-1/(1 – α)·(bx)1–α]abε = M/(1 – α){(ba)1–αε1–α}
< M(ba)1–α/(1 – α) (∵ 0 < α < 1) → 単調増加、上に上界

ε → 0 → |f(x)| > 0 → 積分区間増大 → 単調増加

limε→0cb+εf(x)dx exist, (5) → cbf(x)dx integrable

Th. 2. f(x) [a, ∞) continuous, α < 1, limx→∞xαf(x) exist

af(x)dx integrable (1の拡張)

Pr. 十分大きいxに対しxα|f(x)| < M = constant, |f(x)| < M/xα

ax|f(x)|dx < Max1/xαdx = M[1/(1 –α)x1–α]ax

= M/(α – 1)·(a1–αx1–α), α > 1

→ 上に有界, x → +∞ → 単調増加
ax|f(x)|dx exist → axf(x)dx exist //

Ex. [a, b] ∋ x, b 特異点, f(x) ≥ M/(bx) → abf(x)dx

→ +∞ / [a, ∞), x > b, f(x) ≥ M/xaf(x)dx → ∞

[ ガンマ分布 ]

Def. オイラーのガンマ関数 Euler's Γ-function, Γ(s)

= 0e-xxs–1dx (s > 0)

Ax. Γ関数 ⇒ 収束
Pr. (積分区間を分けて考える)

i) 0 < s < 1, x = 0はf(x) = e-xxs-1の特異点

x1-sf(x) = e-x < 1 (0 < x < ∞) ⇒ 収束

ii) x → ∞, e-xxs+1 → 0 ⇒ x2f(x) = e-xxs+1は(1, ∞)で有界 ⇒ 収束
i), ii)よりガンマ関数は収束 //

Def. オイラーのベータ関数 Euler's β-function, B(p, q)

= 01xp–1(1 – x)q–1dx (p, q > 0) (p, q ≥ 1 → ordinary)

Ax. (p > 0 or q > 0) ⇒ 収束(存在), それ以外のp, qで発散
Pr. B(p ,q) = J1=(0, 1/2] + J2=[1/2, 1)

J1 = 01/2xp-1(1 - x)q-1dx, J2 = 1/21xp-1(1 - x)q-1dx
(0, 1/2] ⇒ (1 - x)q-1 ∈ (1, (1/2)q-1) ⇒ 有界

1 - q < 1 ⇒ 01/2xp-1dx = 01/2(1/(x1-p)dxJ1は収束

[1/2, 1) ⇒ xp-1は1と(1/2)p-1 ⇒ 有界

1 - q < 1 ⇒ 1/21(1 - x)q-1dxJ2は収束 //

Ax. B(p, q) = 20π/2sin2p-1θcos2q-1θdθ (p > 0, q > 0)
Pr. x := sin2θ
Ex. 0π/2sin5θcos3θdθ = 1/2·B(3, 2) = 1/2·(Γ(3)Γ(2))/Γ(5)

= 1/2·(2!1!)/4! = 1/24

Euler functionsの特性
Th. Γ(s + 1) = sΓ(s)
Pr. Γ(s + 1) = 0e-xxsdx = [e-xxs]0 + s0e-xxs-1dx

= s0e-xxs-1dx = sΓ(s) //

Th. Γ(n) = (n - 1)! or Γ(n + 1) = n! (n: 正整数)
Pr. Γ(n) = (n – 1)Γ(n – 1) = (n – 1)(n – 2) … Γ(1)

ここでΓ(1) = 1 ∴ Γ(n) = (n – 1)! //

Th. (Γ関数-β関数関係) B(p, q) = (Γ(p)Γ(q))/Γ(p + q) (p, q 正整数)
Pr. 累次積分からニ重積分への変換と極座標を用いて証明
Γ(p)Γ(q) = 40e-x2x2p-1e-x2dx0e-y2y2q-1dy

= 02e-r2r2(p+q)-1dr0π/22cos2p-1θsin2q-1θdθ = Γ(p + q)B(p, q) //

Th. Γ(1/2) = 20e-x2dx = √π
Pr. B(1/2, 1/2) = (Γ(1/2))2/Γ(1), B(1/2, 1/2) = 20π/2 = π

ここでΓ(1) = 1 ⇒ (Γ(1/2))2 = π
Γ(1/2) = 0e-xx-1/2dx, x := y2dx = 2ydy = 2√(xdy

∴ Γ(1/2) = 20e-y2dy = 20e-x2dx //

積分の応用 (application of integral)


面積 (area), S
Th. (極方程式の面積) 曲線 C, r = f(θ) continuous, (αθβ)

C, θ = α, θ = βで囲まれる部分の面積 S
S = 1/2αβr2 = 1/2αβf(θ)2

Pr. [α, β]のΔ分割: α = θ0 < θ1 < … < θn-1 < θn

r = f(θ), θk-1, θkが囲む扇形で近似した面積 1/2·f(θk)2(θk - θk-1)
∴ リーマン和 ≈ 1/2Σk=1nf(θk)2(θk - θk-1), ||Δ|| → 0

⇒ 1/2αβf(θ)2 //

長さ (length), l
Def. 曲線の長さ: 曲線 C (端点 A, B), 分点 P

P0 = A, P1, P2, … Pn-1, Pn = B
CPk-1からPkまでの微小曲線Ck (k = 1, 2, … k)に分割

CkPk-1Pk---------------Σk=1nPk-1Pk--------------- = P0P1----------- + P1P2----------- + … + Pn-1Pn---------------

max1≤knPk-1Pk--------------- → 0 ⇒ Pk-1Pk--------------- = L (converge) ≡ 求長可能 ⇒ L ≡ 長さ

Q. a > 0, C, x = a(t - sint), y = a(1 - cost) (0 ≥ t ≥ 2π

(1) Cx-axisが囲む部分の面積(S)、(2) Cの長さ(L)を求めよ

A. C ⊂ サイクロイド cycloid

(1) S = 02πaydx = 02π(a(1 - costa(1 - cost))dt = 3πa2
(2) dx/dt = a(1 - cost), dy/dt = asint → 8a

体積・側面積 (volume and lateral area), V and Sl
Th. [a, b], f(x) ≥ 0 continuous → y(x), x = a, y = bが囲む図形

x軸で回る回転体の体積 ⇒ V = πab{f(x)}2dx

Th. [a, b], f(x) ≥ 0 C1-class → y(x), x = a, y = bが囲む図形

x軸で回る回転体の側面積 ⇒ S = 2πabf(x)√(1 + {f'(x)}2)dx

Eq. ウォリスの公式 Wallis's formula
limn→∞{(1/√n)·(2n)!!/(2n - 1)!!} = limn→∞{(1/√n)·(22n(n!)2/(2n)!} = π
Eq. スターリングの公式 Stirling's formula

lims→+∞(Γ(s + 1)/(√(2πs(s+1)/2e-s) = 1

Th. B(p, q) = B(q, p) = B(p + 1, q) + B(p, q + 1) = (p + q)/p·B(p + 1, q)
Pr. y = 1 - x (変数変換)

B(p, q) = 01xp-1(1 - x)q-1dx = -10(1 - y)p-1yq-1dy

= 01(1 - y)p-1yq-1dy = 01xq-1(1 - x)p-1dx = B(q, p) //

[多くのTh.は既述Th.からの拡張]

級数 (series)


Th. 収束N.C.: Σan 収束 ⇒ limn→∞an = 0
Ex. Σn=1(2n3/(n3 + 1)), limn→∞(2/(1 + 1/n3)) = 2 (≠ 0) ∴ 発散
⇔ 逆は真とは限らない: 逆 limn→∞an = 0 ⇒ Σan 収束
Ex. Σn=1an = Σn=1(1/(√n + √(n + 1)))

limn→∞an = limn→∞(1/(√n + √(n + 1))) = 0 (しかし以下の通り発散)
Σn=1an (分母有理化): 1/(√n + √(n + 1)) =

(√(n + 1) - √n)/((n + 1) - n) = √(n + 1) - √n = √(n + 1) - √n

Sn = Σk=1(√(k + 1) - √k) = √(n + 1) - 1
Σn=1an = limn→∞Sn = limn→∞(√(n + 1) - 1) = ∞ ∴ 発散

Th. 収束N.S.C. ε > 0, N < 0, mnN (m, n, N integer)

⇒ |an+1 + an+2 + … + am| < ε an → liman = 0]

Pr. trivial (Cauchyの収束定理)
Def. 正項級数 (positive term series, PTS): Σan, an ≥ 0 for n

Def. 絶対値級数 (absolute series) Σ|an|

Def. 交項級数/交代級数 (alternating series)

Σan 項が交互に正負 Ex. a1a2 + a3 – … (an > 0)

Th. (ライプニッツの定理) {an}, an > 0 単調減少数列, limn→∞an = 0

⇒ 交項級数 Σn=1(-1)n-1an 収束

Pr. Sn := Σk=1nak, a2k - a2k+1 ≥ 0 →

S2n = a1 - a2 + a3 - a4 + a5 - … - a2n-2 + a2n-1 - a2n

= a1 - (a2 - a3) - (a4 - a5) - … - (a2n-2 - a2n-1) - a2na1
{S2n} 上に有界

S2n+2 = S2n + a2n+1 - a2n+2S2n → {S2n} 単調増加 → S := limn→∞S2n

limn→∞S2n+1 = → limn→∞(S2n + a2n+1) = S + 0 = S
∴ {Sn}はSに収束 → Σn=1(-1)n-1an 収束 //

Case 正項級数 sn = an+1 + an+2 + … + anM (constant)
Case 交項級数 a1 > a2 > … > an > … > 0, liman = 0
Th. 正項級数の比較判定法: [1, ∞) f(x) > 0 continuous, 単調増加

Σn=1f(n), 1f(x)dx 同時に収束・発散

Def. 一般調和級数 ≡ Σn=1(1/nα) (α > 0)
Σn=1an 正項級数 ⇒
Th. コーシー・アダマール判定法 Cauchy–Hadamard test: limn→∞nan = r

i) 0 ≤ r < 1 → Σn=1an 収束,__ii) 1 < r ≤ ∞ → Σn=1an 発散
r = 1の時には判定できない

Th. ダランベール判定法 d'Alembert test: limn→∞(an+1/an) = r

i) 0 ≤ r < 1 → Σn=1an 収束,__ii) 1 < r ≤ ∞ → Σn=1an 発散
⊂ コーシー・アダマールの判定法

Ex. Σn=1(1/n!) → an := 1/n!, an > 0, limn→∞(an+1/an) =

limn→∞{(n + 1)n+1/(n + 1)!·n!/nn} = limn→∞(1 + 1/n)n = e (< 1) 収束

Th. ラーベ判定法 Raabe's test: limn→∞n(an+1/an - 1) = r

i) -∞ ≤ r < -1 → Σn=1an 収束, __ii) -1 < rΣn=1an 発散

Ex. Σn=1(1/n2): an := 1/n2n(an+1/an - 1) = n(n2/(n2 + 2n + 1) - 1)

= (-2n2 - n)/(n2 + 2n + 1) = -2 (< -1) ∴ 収束

Th. Cauchyの凝集判定法 a1a2 ≥ … ≥ an ≥ … ≥ 0

Σn=1an, Σk=02ka2k 同時に収束・発散

Pr. sn = a1 + a2 + … + an, tk = a1 + 2a2 + … + 2ka2k, 2kn < 2k+1

sna1 + a2 + (a3 + a4) + … + (a2k–1+1 + … + a2k)

≥ 1/2·a1 + a2 + 2a4 + … + 2k–1a2k = 1/2·tk

sna1 + (a2 + a3) + … + (a2k + … + a2k–1)

≤ 1/2·a1 + a2 + 2a4 + … + 2k–1a2k = tk

1/2·tksntk (2kn < 2k+1)
∴ 同時に有解か有解ではない → 同時に収束・発散 //

Ex. Σ(1/n), Σ(1/ns) (0 < s < 1) 発散, Σ(xn/n!) 収束
Th. Σ|an| = sΣan 絶対収束級数 → 収束

(Σan 収束, Σ|an| 発散 → Σan 条件収束級数)

Pr. ε > 0, 十分大きいm, n,

|an+1 + an+2 + … + am| < |an+1| + |an+2| + … + |am| < ε //

Ex. Σ(sinn/n2): |sinn/n2| ≤ 1/n2, Σ(1/n2) 収束

Σ|sinn/n2| 収束 (比較判定法) ∴ Σ(sinn/n2) 絶対収束

Th. 絶対収束級数は項の順序を変えても和は変わらない, Σan = Σbn
Pr. 絶対収束級数 s := a1 + a2 + … + an + …,

sの項の順序を変えた級数 t := b1 + b2 + … + bn + …
Σ|an| 収束 → |a1| + |a1| + … + |an| = A,
tの項はsの項のいずれかに等しいので、
|b1| + |b2| + … + |bn| ≤ |a1| + |a1| + … + |am| ≤ A (nm) ∴ Σ|bn| 収束
sp = a1 + a2 + … + ap, tq = b1 + b2 + … + bq,

tr = b1 + b2 + … + br (rpq) → |sptq| ≤ |b1| + |b2| + … + |br|

ε > 0, N > q → |bq+1| + |bq+2| + … + |br| < ε ∴ |sptq| < ε //

Th. [Riemann] 条件収束級数 Σan, C (Real)

→ 項の順序を変えてできた級数が和Cを持つようにできる

Pr. 正の項 p1, p2, …, 負の項 q1, q2, …,
Σn=1pn & Σn=1(–qn) → ∞ (発散)
m1: p1 + p2 + … + pm1–1C < p1 + p2 + … + pm1P1
n1: P1 – (q1 + q2 + … qn1–1) ≥ C > P1 – (q1 + q2 + … qn1–1 + qn1) ≡ Q1
m2: Q1 + pm1+1 + … + pm2–1C < Q1 + pm1+1 + … + pm2–1 + pm2P2
n2: P2 – (qn1+1 + … + qn2–1) ≥ C > P2 – (pn1+1 + … + pn2–1 + pn2) ≡ Q2
以下同様にm1, m2, …およびn1, n2, …の2組の数列を得る
an → 0, pn → 0, qn → 0 ∴ {Pn}, {Qn} converge

p1 + p2 + … + pm1
┗━━━━━━━━┛P1
_________- (q1 + q2 + … + qn1-1 + pn1)
┗━━━━━━━━━━━━━━━━┛Q1
__________________+ pm1+1 + … + pm2-1 + pm2
┗━━━━━━━━━━━━━━━━━━━━┛P2
__________________________ - (pn1+1 + … + pn2-1 + pn2) + … + pm2+1 + …
┗━━━━━━━━━━━━━━━━━━━━━━━━━┛Q2Cに収束 //

Th. Σan = A, Σbn = B 絶対収束 ⇒ i, j, Σaibj = AB 絶対収束
[級数の加法は普通の加法と同様に扱える]
Pr. tn (Σ|aibj|の第n項の部分和)

Σi=1p|aiΣj=1q|bj| 収束 (p, q: tnai, bj中最大)
Σi=1p|aiΣj=1q|bj| < A*, and Σi=1p|aiΣj=1q|bj| < B*

→ 正項級数 Σ|aibj|は収束

Σaibjの第n2部分和は(a1 + a2 + … + an)(b1 + b2 + … + bn)と並べ替えられ(a1b1 + a2b1 + a1b2 + a2b2 + … + anbn)とすると
T = Σaibj = limn→∞Σi=1naiΣj=1nbj = AB //

Def. Cauchy積級数, Σn=0cn

Σn=0an, Σn=0bn, cn = Σk=0nakbn-kΣn=0cn

Th. Σn=0an, Σn=0bn 絶対収束 ⇒ Σn=0cn 絶対収束

n=0an)(Σn=0an) = Σn=0k=0akbn-k)

関数列 sequence of functions
Def. {fn}, f, x ∈ I fixed, limn→∞f(x) 収束, f(x) = limn→∞fn(x) (xI) ⇒

f ≡ {fn}の極限関数, {fn}はfに各点収束, I ≡ 収束域

Ex. I = [0, 1], fn(x) = xnf(x) = limn→∞xn = 0 (0 ≤ x < 1), or 1 (x = 1)

limx→1-0{limn→∞fn(x)} ≠ limn→∞{limx→1-0fn(x)}
連続関数の各点収束極限関数は連続とは限らない

⇒ 極限と積分は交換できないことがある
Def. (一様収束) {fn}, f, ε > 0, N(ε) exist, xI, nN(ε),

|fn(x) - f(x)| < ε ⇒ {fn}はfに一様収束 ≡ limn→∞supxI|fn(x) - f(x)| = 0

Th. {fn} continuous on Ifに一様収束 ⇒ f continuous on I
Pr. ε > 0, N(ε/3) at ε/3 > 0, xI, nN(ε/3) ⇒ |fn(x) - f(x)| < ε/3

Nε := N(ε/3) → |fNε(x) - f(x)| < ε/3 → fNε continuous on x = a

δ1(ε/3) > 0 exist → |x - a| < δ1(ε/3) ⇒ |fNε(x) - fNε(a)| < ε/3

δ(ε) := δ1(ε/3), |x - a| < δ(ε), xI

|f(x) - f(a)| ≤ |f(x) - fNε(x)| + |fNε(x) - fNε(a)| + |fNε(a) - f(a)|

< ε/3 + ε/3 + ε/3 = ε

f continuous on x = aaIf continuous function on I //

Th. 極限(一様収束) ⇔ 定積分: I = [a, b], {fn} fに一様収束 ⇒

limn→∞abfn(x)dx = abf(x)dx

Th. {fn} on I, C1-class → fに各点収束, {f'n} → gに一様収束 ⇒

f on I, C1-class, f' = gd/dx(limn→∞fn(x)) = limn→∞f'n(x)

Def. 広義一様収束 ≡ {fn}はfI上で一様収束しない, JI, J上で一様収束
Def1. {fn}, Sn(x) := Σk=1nfk(x), {sn} 各点収束, I 収束域 ⇒

関数項級数 ≡ Σk=1nfn(x) := limn→∞Sn(x) (xI)

Def2. Tn(x) := Σk=1nfk(x) → {Tn} 各点収束 ⇒ Σk=1は絶対収束
Def3. {fn}, 部分和 Sn(x) = Σk=1nfk(x) → {Sn} I上で一様収束 ⇒

Σn=1fn ≡ 一様収束 on I

Th. ワイエルシュトラウスのM判定法 Weierstrass M-test
{fn} → |fn(x)| ≤ Mn (xI), Σn=1Mn 収束, {Mn} exist

Σn=1fn I上で絶対収束かつ一様収束

Pr. Σn=1|fn(x)| 収束 (比較判定法) ∴ Σn=1fn 絶対収束

n=1fn(x) - Σn=1mfn(x)| = n=m+1fn(x)|

Σn=m+1|fn(x)| ≤ Σn=m+1Mn

Σn=1Mn 収束 → supxIn=1fn(x) - Σn=1mfn(x)|

Σn=m+1Mn → 0 (m → ∞) ∴ Σn=1fn 一様収束 //

Def. 優級数 ≡ Σn=1Mn

→ ワイエルシュトラウスのM判定法 ≡ 優級数判定法

整級数(べき級数) (power series)


Def. 整級数 Σn=0anxn = a0 + a1x + a2x2 + … + anxn + …

(ai = 1 → |x| < 1, 収束値 = 1/(1 – x)) [収束 → 値はxの関数]

Th. 収束半径(r): Σanxn 整級数,

x = x0 (≠ 0)で収束, |x| < |r| ⇒ 絶対収束
x = x1で発散, |x| < |r| ⇒ 発散
x = rでは収束発散両方の場合がある

Pr. n十分大 → |anxn| < 1, |anxn| = |anxn||x/x0|n ≤ |x/x0|, |x| < x0

Σ|anxn| 収束 → Σanxn, |x| = x0 絶対収束
Σanx1n 発散 → |x1| < |x|, Σanxn 発散 //

Th. 収束半径判別法 Σanxn 整級数, limn→∞|an/an+1| = r

⇒ 収束半径 = r (r = ∞でも成立)

Pr. r ≠ 0, ∞, limn→∞|an+1xn+1/anxn| = limn→∞|an+1/an||x| = |x|/r

∴ |x|/r < 1 → 収束, |x|/r > 1 → 発散
r = 0, x = 0を除く全ての実数xに対して発散
r = ∞, 全ての実数xに対して収束 //

Th. f(x) = Σanxn = r > 0 ⇒ f(x), (–r, r) continuous
Pr. |x0 + h| < ρ < r, Σn=0anρn 絶対収束

Σn=N+1|anρn| < εなるNを決めることができる
|f(x0 + h) – f(x0)| = n=0an(x0 + h)nΣn=0anx0n|
n=0Nan{(x0 + h)nx0n}| + Σn=N+1|an(x0 + h)n| + Σn=N+1|anx0n|

n=0Nan{(x0 + h}nx0n| + 2ε

Σn=0Nan: xに関するN次多項式 → x = x0 continuous
→ 十分小さい|h| → |Σan{x0 + h}nx0n| < ε //

Th. Σanxn, r = 1/limn→∞n√|an| 確定(r = ∞でも成立) ⇒ 収束半径 = r
Th. Σn=0an 収束 ⇒ Σn=0anxn 一様収束 [0, 1]

⇒ Abelの連続性定理 (Abel's continuity theorem)

Th. 項別積分定理 term-by-term integration theorem
f(x) = Σn=0anxn, r > 0, |u| > r

0uf(x)dx = 0uΣ(anxn)dx = Σ∫0uanxndx = Σn=0(an/(n + 1))un+1

Pr. f(x) = Σn=0anxn = Σn=0Nanxn + Σn=N+1anxn = Σn=0Nanxn + RN(x)

0uf(x)dx = 0un=0Nanxn)dx + 0uRN(x)dx
0uf(x)dxΣn=0Nan/(n + 1)·un+1 = 0uRN(x)dx
|∫0uRN(x)dx|0|u|Σn=N+1|anxn|dx
0 ≤ x ≤ 0, |u| < ρ < r

Σn=N+1|anxn| ≤ Σn=N+1|anxnun| ≤ Σn=N+1|anxn|ρn

→ 収束 → Σn=N+1|anxn| < ε
|∫0uRN(x)dx|ε|u|, uは定まった数であり、

εNが大きくなればいかほどでも小さくできる

limN→∞{∫0uf(x)dxΣn=0Nan/(n + 1)·un+1} = limN→∞0uRN(x)dx = 0

0uf(x)dx = Σn=0(an/(n + 1))un+1

|x1| > r, if Σn=0an/(n + 1)·x1n+1 → 収束

→ |an/(n + 1)·x1n+1| < M (n = 1, 2, …)

|x| < |x1|
∴ |anxn| = |anx1n+1/(n + 1)|·|x/x1|n·(n + 1)/|x1|

M·(n + 1)/|x1|·|x/x1|n

|x/x1| < 1 ∴ Σ(n + 1)/|x/x1|n → 収束

r < |x| < |x1|, rΣanxnの収束半径であることと矛盾

Σan/(n + 1)xn+1r < |x|なるxに対して発散
|x| < r → |an/(n + 1)·xn+1| = |anxn|·|x|/(n + 1), n十分大 → |x|(n + 1) < 1
→ |an/(n + 1)·xn+1| ≤ |anxn|はある数字より大きい全てのnについて成立
|anxn| → 収束 ∴ Σan/(n + 1)xn+1 → 収束 //

Th. 項別微分定理 term-by-term differentiation theorem
f(x) = Σn=0anxn, r (> 0), |x| < rf'(x) = Σn=0nanxn-1
Pr. g(x) := Σn=0nanxn–1, 項別積分定理より0ug(x)dx = f(u) – a0

→ 両辺をuについて微分: g(u) = f'(u), u = x //

拡張 f(k)(x) = Σn=0n(n – 1) … (nk + 1)anxn-k, |x| < r
Th. |f(n)(x)| < M for nf(x) = Σn=0(f(n)(0)/n!)xn (Mxに依存)
Pr. |(f(n)(ξ)/n!)·xn| < M·|xn|/n!, 整級数 Σn=0(xn/n!)の収束半径 = ∞

limn→∞(xn/n!) = 0 ∴ limn→∞(f(n)(ξ)/n!)·xn = 0 //

Th._sinx = xx3/3! + … + (-1)n–1{x2n–1/(2n – 1)!} + …,

cosx = x2/2! + … + (-1){x2n/(2n!)} (-∞ < x < ∞)

Pr. f(x) = sinx, f(n)(x) = sin(x + /2), f(0) = sin(/2)

f(n)(0) = 1, 0, -1, 0, 1, 0, -1, … (n = 1, 2, 3, …)
|f(n)(x)| = |sin(x + /2)| ≤ 1 (n = 1, 2, …) (–∞ < x < ∞)
前定理より展開できる //

フーリエ級数 (Fourier series)


周期性をもつ関数の(無限の)和により表したもの →
フーリエ解析: フーリエ級数を用いた解析 Ex. フーリエ変換

画像処理、データ圧縮、CT、MRI等の基礎技術として発展

Ax. (三角関数の直交性) m, nN

-ππsinmxcosnxdx = 0 … (1)
-ππsinmxsinnxdx = -ππcosmxcosnxdx

= π (m = n)
= 0 (m ≠ 0) … (2)

Pr. sinmx = odd function, cosnx = even f. ∴ sinmxcosnx = odd f.

∴ (1) trivial
sinmxsinnx = 1/2(cos(m - n)x - cos(m + n)x) … even f.,
cosmxcosnx = 1/2(cos(m + n)x + cos(m - n)x) … even f. →
-ππsinmxsinnxdx = 0π{cos(m - n)x - cos(m + n)x}dx
-ππcosmxcosnxdx = 0π{cos(m + n)x + cos(m - n)x}dx

i) m = n-ππsin2mxdx = 0π(1 - cos2mx)dx = [x - sin2mx/2m]0π = π

_ -ππcos2mxdx = 0π(1 + cos2mx)dx = [x + sin2mx/2m]0π = π

ii) mn-ππsinmxsinnxdx

= [sin(m - n)x/(m - n) - sin(m + n)x/(m + n)]0π = 0

_ -ππcosmxcosnxdx

= [sin(m + n)x/(m + n) + sin(m - n)x/(m - n)]0π = 0 //

偏微分法 (partial differentiation)


partial derivative Def. 領域 domain: (a1, a2), (b1, b2) ∈ A, [0, 1],

φ(t), ψ(t) continuous ⇒
(φ(0), ψ(0)) = (a1, a2), (φ(1), ψ(1)) = (b1, b2),
(φ(t), ψ(t)) ∈ A (0 ≤ t ≤ 1) exist

Def. 開領域: 境界を全く含まない領域 Ex. x2 + y2 > 0 ⇔
Def. 閉領域: 境界を全て含む領域 Ex. x2 + y2 ≥ 0
Def. 極限値 Pn(xn, yn), 点列{Pn}, {Pn}
A(a, b) ⇔ ε, n0 < 0, lim(x, y)→(a, b)f(x, y) = c
Pr. √{(axn)2 + (byn)2} < ε → |axn| < ε and |byn| < ε

逆に|axn| < ε, |byn| < ε → √{(axn)2 + (byn)2} < ε < √2·ε < 2ε
PnA, limybf(y) = b [yについて偏diff.]
{Pn(x, y)}: 点集合 ∋ P, |x| < M = const, |y| < M = const → 有界
P(x, y), A(a, b) → f(x, y) = f(P)
0 < |xa| + |yb| < δ → |f(P) - c| < ε

[閉集合・開集合 closed set/open set]
Th1. 部分集合K (≠ ∅)が閉集合 (N.S.C.) ⇒ xnK, limn→∞xn = aaK
Th2. 部分集合K (≠ ∅)が閉集合 (N.S.C.) ⇒ Kc 開集合
Def. 類似極限 repeated limit: limyb{limxaf(x, y)}, limxa{limybf(x, y)}
Def. 偏微分Aδ近傍内全ての点Pについて進路に無関係

(= A点からP点に近づく方法に無関係)
x = aのときy = bになる任意の曲線y = f(x)について
lim(x, y)→(a, b)f(x, y) = a, f(P)は点Aでcontinuous,
つまりlim(x, y)→(a, b)f(x, y) = f'(a, b) = f(A)

Pr. ε > 0, δ > 0, |PA| < δ → |f(P) – f(A)| < ε

limh→0{f(a + h, b) – f(a, b)}/h = fx(a, b) = ∂f/∂x

f(x, y)の(a, b)におけるxに関する偏微(分)係数

limh→0{f(a, b + h) – f(a, b)}/h = fy(a, b) = ∂f/∂y

f(x, y)の(a, b)におけるyに関する偏微(分)係数

Ex. f(x, y) = x/(x + y)のfx, fy: fx (fy)を求める間はy(x)を定数と考える

fx = y/(x + y)2, fy = -1/(x + y)2

Def. fx, fy exist ⇒ 偏微分可能
Def. 方向微分 limr→0{f(a + rcosθ, b + rsinθ) – f(a, b)}/r = df/drθ (0 ≤ θ < π)

θなる方向についての偏微分係数 in (a, b) →
θ = 0のときfxと等しく、θ = 2/πのときfyと等しい

偏導関数 (partial derivative)


Def. C1-class: f(x, y) on D 偏微分可能, fx(x, y), fy(x, y) continuous

f(x, y)はDC1-class

微分間の関係

Relations
Relations (1) 連続で偏微分可能ではない 凡例 z = |x|
(2) 方向微分可能 ⇒ 偏微分可能

明らかにθ = 0, θ = π/2のときそれぞれの方向微分値が偏微分となる
× → (逆は必ずしも真ではない) = 不連続、偏微分可能
凡例 f(x, y) = {2xy/(x2 + y2) for (x, y) ≠ (0, 0), = 0 for (x, y) = (0, 0)}
原点(0, 0)で不連続、偏微分可能。方向微分は可能ではない

y = mxに沿って(0, 0)に近づく

limx→0f(x, mx) = limx→0{2mx2/(1 + m2)}

= 2m/(1 + m2) ≠ 0 (m ≠ 0のとき)不連続 in (0, 0)

Relations (3) 方向微分可能、不連続
凡例 f(x, y) = 2xy/(x2 + y2) for (x, y) ≠ (0, 0), or
_________= 0 for (x, y) = (0, 0) in (0, 0)
Pr._limh→0(f(h, 0) – f(0, 0))/h = limh→0(0 – 0)/h = 0

0 = fx(0, 0), 0 = fy(0, 0) ∴ fx(0, 0), fy(0, 0) → parted diff.
limr→0{f(rcosθ, rsinθ) – f(0, 0)}/r = limr→01/r·(2r2cosθsinθ)/r2

= limr→02sinθcosθ/r = limr→0sin2θ/r

θ = 0, π/2のとき方向微分可能ではない

lim(h, k)→(0, 0)f(x, y) = 0, f(0, 0) = 0

Def. 全微分可能 totally differential

f(x, y): D ∋ (a, b), (a + h, b + k),

f(a + h, b + k) - f(a, b) = A(a, b)h + B(a, b)k + ε(h, k)·√(h2 + k2),

lim(h, k)→(0, 0)ε(h, k) = 0 ⇒ f(x, y) is totally differential in (a, b)

Th. [1-3] f(x, y), (a, b) totally diff. ⇒
[1] (a, b) continuous, [2] fx, fy exist, [3] f(x, y) (a, b) 方向微分可能
Pr._[1] trivial

[2] k := 0, (f(a + h, b) – f(a, b))/h = A(a, b) ± ε(h, 0)

h → 0, fx(a, b) = A(a, b), k = 0, fy(a, b) = B(a, b)
同様にh := 0としfyの存在を示す

[3] h = rcosθ, k = rsinθ,

f(a + rcosθ, b + rsinθ)/r

= fx(a, b)cosθ + fy(a, b)sinθ ± ε(rcosθ, rsinθ)

h → 0 → ε → 0, df/drθ=fx(a, b)cosθ + fy(a, b)sinθ

Ex. 全微分可能でないもの f(x, y) = √|xy| in (0, 0)
Pr._limh→0(f(0 + h, 0) - f(0,0))/h = limh→0(0 - 0)/h = 0

∴ (0, 0)でxについてfx(0, 0) = 0となり偏微分可能。一方、
f(x, y) – f(0, 0) = fx(x, y)x + fy(0, 0) + ε(x, y)√(x2 + y2)
f(x, y) = ε(x, y)√(x2 + y2) ∵ f(0, 0) = 0
ε(x, y) = f(x, y)/√(x2 + y2) = √|xy|/√(x2 + y2) = √[|xy|/√(x2 + y2)]
y = mxによって(x, y) → (0, 0)にすると

ε(x, mx) = √(m/(1 + m2)) = √(1/2) (m = 1)

よって全微分可能ではない

Th. fx, fy exist, fx, fy continuous (= C1-class) ⇒ f(x, y)は全微分可能
Pr._f(a + h, b + k) – f(a, b)

= f(a + h, b + k) – f(a, b + k) + f(a, b + k) – f(a, b)
= hfx(a + θ1h, b + k) + kfy(a, b + θ2k) (0 < θ1, θ2 < 1) ∵ 平均値の定理
C0-class与式: fx(a + θ1h, b + k) = fx(a, b)continuous + ε1 (ε1 → 0)
fy(a + h, b + θ2k) = fy(a, b) + ε2 (ε2 → 0)

= fx(a, b)h + fy(a, b)k + {(ε1h + ε2k)√(h2 + k2)}/√(h2 + k2)

(ε1h + ε2k)/√(h2 + k2) = ε(h, k),

|ε1h/√(h2 + k2)| < |ε1|·|h|/√(h2 + k2) < |ε1|

|(ε1h + ε2k)/√(h2 + k2)| < |ε1h/√(h2 + k2)| + |ε2k/√(h2 + k2)| <

|ε1| + |ε2| → 0

(h, k) → (0, 0)

Ex. z = f(x, y) = x2 + y2, A(1, 2, f(1, 2))における接平面の方程式

C1-class - totally diff., fx(x, y) = 2x, fy(x, y) = 2y
fx(1, 2) = 2, fy(1, 2) = 4, f(1, 2) = 5
z = 5 + 2(x - 1) + 4(y - 2) = 2x + 4y - 5

高次導関数 (high-order derivatives)

limh→0(fx(a + h, b) – fx(a, b))/k = fxx(a, b),
limh→0(fx(a, b + k) – fx(a, b))/k = fxy(a, b)などfxx, fxy, fyx, fyyを考える
更に、高次偏導関数を考える (一般にfxyfyx)
Ex. f(x, y) = [xy(x2y2)]/(x2 + y2) in (x, y) ≠ (0, 0), = 0 in (x, y) = (0, 0)
Def. Δf(x, y) = fxx(x, y) + fyy(x, y) ⇒ f(x, y)のラプラシアン (Laplacian)
Def. 調和関数: 定義域でΔf(x, y) = 0を満たす関数
Th. fxy = fyxの十分条件 [1-4]
[1] f(x, y): C2-class__[2]の拡張 ∵ [1] fxx, fxy, fyx, fyy: exist, continuous
[2] Youngの定理 fxy, fyx exist, continuous
[3] fx, fy exist, totally diff.__偏微分可能 → fxx, fxy, fyx, fyy 存在

(連続の仮定なし)

[4] Schwarzの定理 fx, fy, fxy exist, fxy continuous
Pr._ [2] Δ = f(a + h), b + k) - f(a + h, b) - f(a, b + k) + f(a, b)

φ(x) ≡put f(x, b + k) - f(x, b)
Δ = φ(a + h) - φ(a) = '(a + θ1h) 平均値の定理

= h[fx(a + θ1h, b + k) - fx(a + θ1h, b)] = hkfxy(a + θ1h, b + θ2k)

lim(h, k)→(0, 0)Δ/hk = lim(h, k)→(0, 0)fxy(a + θ1h, b + θ2k) = fxy(a, b) fxy: 連続
φ(y) ≡put f(a + h, y) - f(a, y)
Δ = φ(b + k) - φ(b) = '(b + θ3k) = k[fy(a + h, b + θ3k) - fy(a, b + θ3k)]

= hkfyx(a + θ4h, b + θ3k) = fyx(a, b)

fyx continuous

Pr._[3] h := k → Δ = h[fx(a + θ1h, b + k) - fx(a + θ1h, b)] fx: to (a + h, b + k)

fx(a + θ1h, b + h) – fx(a, b)

= fxx(a, b)θ1h + fxy(a, b)h + ε(a1h, h)√{(θ1h)2 + h2} … (1)

fx(a + θ1h, b) – fx(a, b)

= fxx(a, b)θ1h + fxy(a, b)·0 + ε(a1h, 0)√{(θ1h)2 + 02} … (2)

(1) – (2): Δ = h2[fxy(a, b) ± ε(θ1h, h)√(θ12 + 1) ∓ ε(θ1h, 0)|θ1|]
limh→0Δ/h2 = fxy(a, b) … (3)
(3)と同様にlimk→0(Δ/k2) = fyx(a, b) … (4)

Pr._[4] Δ/hk = 1/h[{f(a + h, b + k) – f(a + h, b)}/k – {f(a, b + k) – f(a, b)}/k]

limk→0(Δ/hk) = 1/h·[fy(a + h, b) – fy(a, b)]
limh→0(limk→0Δ/hk) = fxy(a, b) =Def fxylim(h, k)→(0, 0)(Δ/hk) = fxy(a, b) //

Th. [1]のn次への拡張: f(x, y) Cn-class ⇒ n次偏導関数は

nf/∂xn, nf/∂xn-1∂y, nf/∂xn-2∂y2, …, nf/∂x∂yn-1, nf/∂yn

の何れかと一致

高次偏導関数 (high-order partial derivative)

(1) z = f(x, y), x = x(t), y = y(t) ⇒ z = f(x(t), y(t)) cp. Taylorの定理
Pr._dz/dt = ∂f/∂x·dx/dt + ∂f/∂y·dy/dt

d2z/dt2 = (fxx·dx/dt + fxy·dy/dt)dx/dt + fx·d2x/dt2

+ (fyx·dx/dt + fyy·dy/dt)dy/dt + fy·d2y/dt2

= fxx(dx/dt)2 + 2fxy(dx/dt·dy/dt) + fyy(dy/dt)2 + fx(d2x/dt2/) + fy(d2y/dt2)

(2) z = f(x, y), y = y(x) → z = f(x, y(x))
Pr._dz/dx = ∂f/∂x·dx/dx + ∂f/∂y·dy/dx = fx + fy·(dy/dy)

d2z/dx2 = fxx·dx/dx + fxy·dy/dx + (fyx·dx/dx

+ fyy·dy/dx)dy/dx + fy·d2y/dx2

= fxx + 2fxy·dy/dx + fyy(dy/dx)2 + fy·d2y/dx2

(3) z = f(x, y), x = x(t), y = y(t) → dz/dt = fx·dx/dt = fy·dy/dt

d2z/dt2 = fxx(dx/dt)2 + 2fxy·dx/dt·dy/dt + fyy(dy/dt)2

Partial

+ fx·d2x/dt2 + fy·d2y/dt2

{z = f(x, y), x = a + ht, y = b + kt}
dz/dt = ∂z/∂x·h + ∂f/∂y·k = (∂h/∂x + ∂k/∂y)
d2z/dt2 =(2f/∂x2)h2 + 2(2f/∂y∂x)hk + (2f/∂y2)k2 = (∂h/∂x + ∂k/∂y)
dnz/dtn = Σr=0nnCrhnrkr(nf/(∂xnr∂yr))

Th. Taylorの展開定理: 凸領域において

Cn-classでD ∋ (a, b), (a + h, b + k)に対して
f(a + h, b + k) = f(a, b) + (h·/∂x + k·/∂yf(a, b)

+ 1/2!·(h·/∂x + k·/∂y)2·f(a, b) + …
+ 1/(n - 1)!·(h·/∂x + k·/∂y)n-1·f(a, b)
+ 1/n!·(h·/∂x + k·/∂y)n·f(a + θk, b + θh)

(a, b) := (0, 0), (h, k) := (x, y) → Maclaurinの展開

Pr. x := a + ht, y := b + kt,

f(a + ht, b + kt) ≡put φ(t): Maclaurin条件を満足する
φ(t) = φ(0) + φ1(0)t + (φ2(0)/2!)t2 + …

+ (φn–1(0)/(n – 1)!)/tn-1 + (tn/n!)φ(θt)

φ(n)(t) =(h·/∂x + k·/∂y)n·f(a + ht, b + kt) ここでt := 1

Th. (漸近展開) f(x, y) Cn-class in nbh (a, b), (x, y) ⇒

f(x, y) = Σl=0nΣj=0l(lCj/l!)·(lf/∂xj∂yl-j)(a, b)(x - a)j(y - b)l-j + ο(rn)

(r = √((x - a)2 + (y - b)2) → 0)

Ex. n = 2, (0, 0) →

f(x, y) = f(0, 0) + fx(0, 0) + fy(0, 0)

+ 1/2{fxx(0, 0)x2 + 2fxy(0, 0)xy + fyy(0, 0)}y2} + ο(r2)

(r = √((x - a)2 + (y - b)2) → 0)

Th. α: Real, t ≠ 0, f(tx, ty, tz) = tαf(x, y, z) 三次

f(x, y, z) α次の斉次(同次)関数 homogenous function

Ex. fx(x, y) ≡ fy(x, y) ≡ 0 → f(x, y) constant
Pr. f(x, y) = f(0, 0) + xfx(θx, θy) + yfy(θx, θy) = f(0, 0) = constant
Th. Eulerの定理: (0, 0, 0)を含まない領域 f(x, y, z) α次のhomog. fn

x·∂f/∂x + y·∂f/∂y + z·∂f/∂z = αf(x, y, z)

xtx, yty, ztzxfx + yfy + zfz = αf(x, y, z)

Pr. N.C. f(tx, ty, tz) = tαf(x, y, z): tでdiff.,

fx(tx, ty, tz)x + fy(tx, ty, tz)y + fz(tx, ty, tz)z = αtα-1f(x, y, z)
t := 1 → S.C.t ≠ 0
(f(tx, ty, tz)/tαφ(t) = (f(x, y, z)/tφ(t) → φ(t) = constant (tに無関係)
φ(t) = φ(1) = f(x, y, z) degreeのhomog. fn
φ'(t) = 0 //

合成関数の偏微分 (partial differentiation of composite function)

偏導関数の応用 (application of partial derivative)

Def. δ > 0 十分小さいδ, √(h2 + k2) < δ, h, k

f(a + h, b + k) < f(a, b) at U(a, b) → 極大 ⇒ 極大値 ≡ f(a, b)
f(a + h, b + k) > f(a, b) → 極小 ⇒ 極小値 ≡ f(a, b)

Def. 極値 = 極大値 + 極値
Def. f(x, y) ≤(≥) f(a, b) ⇒ f(x, y) ≡ 広義の極大(小)
Th. 極値のN.C. fx, fy continuous, f(x, y) 極大or 極小 in (a, b)

fx(a, b) = 0, fy(a, b) = 0

Pr. x = a 極値 ∴ fx(x, b) = 0 at x = afx(a, b) = 0, fy(a, b)も同様
Th. fx(a, b) = fy(a, b) = 0 ⇒ 滞留点(臨界点) ≡ (a, b) ⇔ 鞍点 ≠ 極小/極大
Th. 極値のS.C. fx(a, b) = fy(a, b) = 0, fxx, fxy, fyy continuous,

Dh2fxx(a, b)A + 2hkfxy(a, b)B + k2fyy(a, b)C

i) ACB2 < 0,

AD = A2h2 + 2ABhk + ACk2

= (Ah + Bk)2[> 0] + (ACB2)[> 0]k2[> 0]AD > 0

A > 0, D > 0, f(a, b) 極小
A < 0, D < 0, f(a, b) 極大

ii) ACB2 < 0, AD = (Ah + Bk)2[> 0] + (ACB2)[< 0]k2[> 0]

Ah + Bk ≠ 0, k = 0
AD > 0, or Ah + Bk = 0, k ≠ 0
AD < 0 (i.e., A ≠ 0 or C ≠ 0) 極値とならない

Ex. x + y + z = a (a > 0)の極値
A. z = axy, xyz = xy(axy), f(x, y) := xy(axy),

fx = y(a – 2xy) = 0, fy = x(ax – 2y) = 0
fxx = –2y, fxy = a – 2xy, fyy = –2x
1) x = 0, y = 0, ACB2 = –a2 < 0,
2) x = 0, y = a, fxx ≠ 0,
3) x = a, y = 0, fyy ≠ 0 いずれも極値とならない
4) x = a/3, y = a/3, ACB2 = a2/3 > 0, fxx = -2a/3 < 0
x = y = z = a/3でa3/27が極大値 //

Th. 陰関数の微分 F(x, y) = 0, F, Fx, Fy cont., F(a, b) = 0, Fy(a, b) ≠ 0

y = f(x) 一価関数 exist in nbh at x = a, b = f(a), F(x, f(x)) = 0,

dy/dx = -Fx/Fy

Ex. x2 + 2xy + 2y2 – 1 = 0, dy/dx

f(x, y) = x2 + 2xy + 2y2 – 1, fx = 2x + 2y, fy = 2x + 4y
dy/dx = -(x + y)/(x + 2y)

Q. 2x3 + 6x2y - 4xy + 3y2 - 10 = 0の(1, -2)における接線の方程式
A. xで微分 → 6x2 + 12xy + 6x2y' - 4y - 4xy' + 6yy' = 0

∴ 2(3x2 - 2x + 3y)y'= -6x2 - 12xy + 4y
If 3x2 - 2x + 3y ≠ 0, then y' = (-3x2 - 6xy + 2y)/(3x2 - 2x + 3y)
(x, y) = (1, -2) → y' = (-3 + 12 - 4)/(3 - 2 - 6) = 5/(-5) = -1
y - (-2) = -1(x - 1) ∴ y = -x - 1

特異点と曲面

f(x, y) = 0, fx, fy continuous in (a, b) ⇒
Def. 通常点 (ordinary point) fx(a, b) ≠ 0

→ (dy/dx)x=a = -(fx(a, b)/fy(a, b)), fy(a, b) ≠ 0
→ (dx/dy)y=a = -(fy(a, b)/fx(a, b))

Def. 特異点 (singular point) fx(a, b) = 0, fy(a, b) = 0
Case. (0, 0)が特異点 f(0, 0) = 0, fx(0, 0) = 0, fy(0, 0) = 0

Maclaurin展開
f(x, y) = 1/2!(x·/∂y + y·/∂y)2f(0, 0)

+ 1/3!(x·/∂y + y·/∂y)3f(0, 0) + … [類推]

f(a, b) = ax2 + 2bxy + cy2 + px3 + 3qx2y + 3rxy2 + sy3 + …

(a = 1/2gxx(0, 0), b = 1/2gxy(0, 0), c = 1/2gyy(0, 0) …)

媒介変数表示 x = lt, y = mt
f(a, b) = (al2 + 2blm + cm2)t2 + (pl3 + 3ql2m + 3rlm2 + sm3)t2 + …
t = 0の根数 → n重点(重複点) Ex. 3重点 a = b = c = 0

Th. ラグランジュの未定乗数法 (method of Lagrange multipliers)
φ(x, y) diff. in nbh (a, b), C1-class, φ(0, 0) = 0, (a, b) ≠ φ(x, y)の特異点

φ(x, y) = 0, f(x, y) (a, b)で極値 totally diff. ⇒ λ exist,

fx(a, b) + λφx(a, b) = 0, fy(a, b) + λφy(a, b) = 0

Pr. φx(a, b) ≠ 0 or φy(a, b) ≠ 0 (∵ (a, b) ≠ φ(x, y) = 0の特異点)
Case. φx(a, b) ≠ 0: φ(x, y) = 0 , y = η(x) C1-class in nhb a, b = η(a) exist

g(x) = f(x, η(x)) → x = aで極値 → g'(a) = 0
g'(x) = fx(x, η(x)) + fy(x, η(x))η'(x)

= fx(x, η(x)) - fy(x, η(x))·{φx(x, η(x))/φy(x, η(x))}

b = η(a) → g'(a) = fx(a, b) - fy(a, b)/φy(a, bφx(a, b) = 0
λ := fy(a, b)/φy(a, b) →

fx(a, b) + λφx(a, b) = 0, fy(a, b) + λφy(a, b) = 0 //

Def. ラグランジュの未定乗数 ≡ λ
Def'. 特異点 (singular point) φ(x) C1-class, φ(x) = 0,

φ(a) = φx1(a) = φx2(a) = … = φxn(a) = 0 ⇒
φ(a) = 0の特異点 ≡ a = (a1, a2, … an)

Th'. ラグランジュの未定乗数法: φ(x) C1-class in nbh a,

aφ(x)の特異点, f(x) aで極値, totally diff. (φ(x) = 0) ⇒
λ exist (constant), fxj(a) + λφxj(a) = 0 (j = 1, 2, … n)

Def. f(x1, x2, … xn) C1-class ⇒

f(x1, x2, … xn) = (fx1(x1, x2, … xn), fx2(x1, x2,

xn), … fxn(x1, x2, … xn)) ⇒

f(x1, x2, … xn)の勾配 [∇f(x)またはgradf(x)と表記]

Def''. 特異点(複数): φ1(x), φ2(x), … φm(x) C1-class, a = (a1, a2, … an)

φ1(a) = φ2(a) … φm(a) = 0, ∇φ1(a), ∇φ2(a), …, ∇φm(a) (1次従属)
aφ1(a) = φ2(a) … φm(a) = 0の特異点

Th''. ラグランジュの未定乗数法: φ1(x), φ2(x), …, φm(x)

C1-class in nbh a = (a1, a1, …, an), φ1(a) = φ2(a) = … = φn(a) = 0
aφ1(x) = φ2(x) = … = φn(x) = 0の特異点
f(x) → aで極値, totally diff. ⇒ λ1, λ2, …, λm exist

∂f/∂xj·(a) = Σi=1m{λ∂φi/∂xj·(a)} = 0 (j = 1, 2, …, n)

Ex. x2 + y2 = 2, f(x, y) = xyの最大値と最小値
Ex. 半径rの球に内接する直方体の体積の最大値
2重点 al2 + 2blm + cm2 = 0, D = b2ac

points
[I] 結節点 D > 0 [II] 孤立点 D < 0 [III] 尖点 D = 0_______自接点

無限分枝: 動点Pが原点から無限に遠ざかり得る曲線は無限分枝を持つ
漸近線: 無限分枝に沿い原点から無限に遠ざかる、この点から一定直線gまでの距離PH → 限りなく0となる曲線
points g:y = mx + cy軸に平行でない漸近線
PH = |ymxc|/√(1 + m2)

Pが曲線上を原点から無限に遠ざかる →
____lim|x|→∞|ymxc|/√(1 + m2) = 0,
____lim|x|→∞(ymxc) = 0

lim|x|→∞y/x = mmが求まった ∴ c = lim|x|→∞(ymx) → cが求まった
y軸に平行な漸近線: x = a, y → ±∞ → 漸近線存在

Ex. x3 – 3axy + y3 = 0 (a > 0)の漸近線を求め図を書く
points A._1 – 3ay/x2 + (y/x)3 = 0, m = lim|x|→∞y/x

lim|x|→∞(1 – 3a/x·m + m3) = 0 ∴ m = –1
c = lim|x|→∞(ymx) = lim|x|→∞|x + y|,
x + y = 3axy/(x2xy + y2) = (3a·y/x)/{1 – y/x + (y/x)2}
c = 3am/(1 – m + m2) = -3a/3 = –a
y = –xaが求める漸近線
原点は2重点であり、この点における接線はxy = 0 → 原点は結節点
y := mxx = 3am/(1 + m3), y = 3am2/(1 + m3)
dy/dm = 3am(2 – m3)(1 + m3)2

m = 32y = 3√4·a (極大), m = 0 → y = 0 (極小)

Ex. 柱面の一般方程式: F(ymx, znx) → 接平面は全て一定直線に平行 Pr._α = ymx, β = znx, (x, y, z)における接平面

→ (-mFαnFβ)(Xx) + Fα(Yy) + Fβ(Zz) = 0
方向比 1:m:nなる直線に平行

二重積分と空間 (double integral and space)


二重積分(2変数関数の積分)

Def. f(x) [a, b], Δ: a = x0 < x < … < xn-1 < xn = b, [xi–1, xi] ∋ ξi,

xixi-1 = ξi, limδ→0Σi=0nf(ξi)δi = I exist
f(x) integrable, abf(x)dx = SΔ = sΔ, SΔ = ΣMiδi

Mi max, sΔ = Σmiδimi min, S(infSΔ) = s(supsΔ)

Th. f(x): [a, b] continuous ⇒ 一様連続 ⇒ integrable
Pr. Δ分割, 細区間δiにおける振動量(Mimi = vi)

points
I = kf(x, y)dxdy
_= kf(p)

SΔ – sΔ = Σiviδi, limδ→0Σiviδi = limδ→0(SΔ – sΔ) = Ss
s = S ⇔ limΣiviδi = 0
z = f(x, y): D = (axb, cy ≤ d)
∫∫(D)f(x, y)dxdy: k
limδ→0Σi,jf(ξi, ei)ωi = I exist → f(x, y) integrable at k

Th. 1 累次積分 f(x, y) kでcontinuous, kf(x, y)dxdy exist

kf(x, y)dxdy = abcdf(x, y)dydx = cdabf(x, y)dxdy
累次積分 → 1変数の積分を2回繰り返していることに帰着

Th. 2 Lemma f(x, y) continuous ⇒ integrable
Pr. f(x, y) continuous → 一様連続

ωijを十分小さくとると(ωij < εにとれる)

Mijmij = vij (ωijにおける振動量)

Σi,jvijωij < Σi,jεωij = ε(ba)(dc) → εであるからlimΣvijωij = 0
∴ integrable //

Th. 1 f(x, y)でxをfixed → yの関数でcontinuous, i.e., yiyjf(ξ, y)dy exist
Pr. 平均値の定理よりmij(yjyj–1)

yi-1yjf(ξi, y)dyMij(yjyj–1), Σjmij(yjyj–1)
cdf(x, y)dyΣjMij(yjyj–1)
cdf(x, y)dy, x continuous → integrable
Σijmij(yjyj–1)(xixi–1sΔ ≤ abcdf(x, y)dydx

ΣijMij(yjyj–1)(xixi–1SΔ

supsΔ·sabcdf(x, y)dydxS·infSΔ
f(x, y) continuous at k

S = s = kf(x, y)dxdy yをfixedすると同様に得られる //

Th. f(x, y) continuous

F(x) = cdf(x, y)dy continuous, f(x, y) continuous → 一様連続
i.e., ε > 0, |Δx| < δ(ε) → |f(x + Δx, y) – f(x, y)| < ε,

F(x) = limΔx→0F(x + Δx, y)

空間問題 (space)

一般有界閉領域 D = (axb, g1(x) ≤ yg2(x))
space

ωijが完全にDに含まれているとき Mij = mij = 1
ωijが完全にDの外部にあるとき Mij = mij = 0
ωijDの臨界領域にあるとき Mij = 1, mij = 0
sΔ·Σmijωij, SΔ·ΣMijωijsΔ ≤ SΔ' → s < S
∫∫(D)f(x, y)dxdy = ∫∫(D1+D+D2)f(x, y)dxdy: integrable
∫∫(D)f(x, y)dxdy = abf(x)[cdf(x, y)dy]dx

= cdf(x)[abf(x, y)dx]dy 累次積分

置換積分 x = u(x, y), y = v(x, y)
(x, y)の(u, v)に関するヤコビアン Jacobian (関数行列式)
J = xξyτxτyξ = 0 … (1)
x = u(ξ, τ), y = v(ξ, τ) … (2)
(2)を(1)に施す(代入する)とx = x(u(ξ, τ), v(ξ, τ)), y = y(u(ξ, τ), v(ξ, τ))
space
(x, y)/(ξ, τ) = ∂(x, y)/(u, v(u, v)/(ξ, τ)
if J ≠ 0 → 逆変換成立 x(u(x, y), v(x, y)) = x, y(u(x, y), v(x, y)) = y

→ 全て点は不変 invariant

(x, y)/(u, v(u, v)/(ξ, τ) = (x, y)/(ξ, τ)
特にS = sのときDは面積確定, S = sDの範囲 area

S - s = SΔ – sΔ = 臨界領域の面積 K

Def. 面積確定: δ → 0 → KΔ

≡ 0 臨界領域が幾らでも小さい面積中に含められる

Ex. Dの境界が滑らかな曲線 smooth curve あるいはそれらの有限の接合

x = φ(t), y = Ψ(t), φ'(t), Ψ'(t): continuous (C1-class), φ'(t)2 + φ'(t)2 ≠ 0

→ 面積確定

K: axb, cyd f(x, y) → S = ∫∫Kf(x, y)dxdy
space

面積系 area element_______________体積系 volume element
space

S = ab(φ2(x) – φ1(x))dx________V = cd(ψ2(x) – ψ1(x))dx

Ex 1.________________________Ex 2.

space____ space

Ex. 0a{(1 – x/ab}dx = 0b{(1 – y/b)·a}dy = 1/2·ab [積分順序の変更]
Ex. 0c{(1 – x/ac – (1 – b/xc}dy
Ex. 2次元 two-dimension における一次変換 linear transform

x = au + bv, y = cu + dv (a, b, c, d: constant)
xy: Ax + By + C = 0, uv: A'u + B'v + C' = 0
J = AB' – A'B

if J ≠ 0 → xy平面全体とuv平面全体とは1:1

Cf. 直線は直線、平行線は平行線に移る
A' = aA + cB, B' = bA + dB

前式に代入
(aA + cB)u + (bA + dB)v + C = 0
(aA' + cB')u + (bA' + dB')v + C = 0 - 直線は直線に移る
space = 0 → space - 平行線は平行線に移る
space

S'' = ad - bc = JJに変換される

Ex. 円柱座標 z = f(r, q) (r, θ, z) space

τ = acosθ, 0 ≤ θπ/2
∫∫(τ)√(a2 + x2)rdrdθ
変数変換で用いる形

J = (x, y, z)/(r, θ, z) = r

Ex. 極座標: 自由変数と従属変数との両方を変換する方法 space

f(x, y) = f(rcosθ, rsinθ), f(rcosθ, rsinθσij (r, θ) ∈ (Δσij)
∫∫(σ)f(x, y)dxdy
= α1α2ρ1ρ2f(rcosθ, rsinθ)rdrdθ
= ρ1ρ2α1α2f(rcosθ, rsinθ)rdθdr

Def. D = (axb, cyd), I = limΣΔsij exist → z = f(x, y)の曲面積
Pr. f(x, y), ∂f/∂x, ∂f/∂y continuous
zzij = (xξi)fx(ξi, ηj) + (yηj)fy(ξi, ηj) 接平面をとったもの

この接平面とxy平面のなす角をrij (0 ≤ rijπ/2)
secrij = √(1 + fx(ξi, ηj)2 + fy(ξi, ηj)2)

→ Δσij = √(1 + fx(ξi, ηj)2 + fy(ξi, ηj)2)·ΔxiΔyj

ΣΔσij = Σj=1nΣi=1m√{1 + fx(ξi, ηj)2 + fy(ξi, ηj)2}·ΔxiΔyj

(1) surface z = f(x, y): S = ∫∫(D)√(1 + (∂z/∂r)2 + (∂z/∂θ)2)rdrdθ
(2) 曲面 円柱座標: z = f(x, y), x = rcosθ, y = rsinθ 変数変換

zr = fxfr + fyfr = fxcosθ + fycosθ, 1/r· = fx(-sinθ) + fy(cosθ)
Clanbelの公式で解く

S = ∫∫D'√(1 + (∂z/∂r)2 + (1/r2)·(∂z/∂θ)2)rdrdθ, D': 極座標表示

(3) 曲面 x = f(x, y), y = x(u, v), z = y(u, v)

[zu = zxxu + zyyu, zv = zxxv + zyyv, zx = fx]
E = xu2 + yu2 + zu2, F = xv2 + yv2 + zv2, G = xuxv + yuyv + zuzv

第2基本量

Ex. x2 + y2 + z2 = a2の表(曲)面積 =

z = √(a2x2y2),
∂z/∂x = -x/√(a2x2y2) = -x/z, ∂z/∂y = -y/√(a2x2y2) = -y/z
円柱座標に変数変換 → x = rcosθ, y = rsinθ
S = 80π/20a(a/z)rdrdθ = 80π/20ar/√(a2r2)drdθ

= 80π/2[–√(a2r2)]0a = 8a0π/2adθ = 4πa2

Ex. 楕円 x2/a2 + y2/b2 = 1 (0 < b < a)をx軸の回りに回転したときの表面積

y2 = b2/a2·(a2x2), 2y·dx/dy = –2(b2/a2)x
y√(1 + y'2) = √(y2 + (yy')2), y2 + (yy')2 = b2 – (b2/a2)(1 – b2/a2)x2
[√(a2b2)]/a := e, √(y2 + (yy')2) = be/a·√(a2/e2x2) S = 2π(be/a)√{(a/e)2x2}dx

= 2πbe/a[1/2·x√{(a/e)2x2} + (a2/ae2)sin–1(ex/a)]aa
= 2πb2 + 2πab/e·sin-1e

If a = bS = 4πa2

多重積分/n重積分 (multiple integration)

W = f(x, y, z), D (axb, cyd, ezf)

Σk=1nΣj=1mΣi=1l(ξi, ηj, ςkijk((ξ, η, ς) ∈ ΔDijk) → 0
= ∫∫∫(D)f(x, y, z)dxdydz 三重積分 (dxdydz 体積要素)
= ∫∫∫(D)f(rcosθ, rsinθ, z)drdqdz, (r, θ, φ) – (x, y, z)

⇑ 円柱座標 x = rsinθcosθ, y = rsinθsinθ, z = rcosθ

Ex.x2 + y2 + z2 = a2 の表(曲)面積と体積
A. S = 4a2π, V = 4/3·πa3

微分方程式 (differential equation)


独立変数、従属変数、従属変数の微分係数を含む式
さまざまな自然現象や社会現象を記述
Ex. 落下運動の方程式

d2x/dt2 = -g, x: 物体の高さ, g: 重力加速度

Ex. 振子の運動方程式

l·d2θ/dt2 = -gsinθ, l: 紐の長さ, θ: 下向き垂直方向からの角度

Ex. 惑星の運動方程式
Ex. ロジスティック式(個体群成長)
Ex. Lotka-Volterra式 (捕食者-被捕食者モデル)
Def. ラプラス変換 Laplace transform, F(s): f(t) [0, ∞] integrable,

F(s) = 0f(t)e-stdt converge on sL[f(t)](s) = F(s)

Def. 逆ラプラス変換, L-1: L[f(t)](s) = F(s) ⇒ L-1[F(s)](t) = f(t)
Th. ラプラス変換可能十分条件

[0, ∞], f(t) continuous, α, M > 0 → |f(t)| ≤ Meαt (t ≥ 0) ⇒
s > αであるsに対しL[f(t)](s)は定義される

Pr. |f(t)e-st| = |f(t)|e-stMeαte-st = Me-(s-α)t → (s > α) 右辺の広義積分:

0Me-(s-α)tdt = limR→∞[-M/(s - αe-(s-α)t]t=0t=R
= limR→∞M/(s - α)·(1 - e-(s-α)R) = M/(s - α) → convergence
0f(t)e-stdt → convergence (比較判定法) //

Th. 線形性: [0, ∞), f(t) g(t) continuous, s > αでラプラス変換可能, a, b

L[af(t) + bg(t)](s) = aL[f(t)](s) + bL[g(t)](s)

Th. 導関数: f(t) [0, ∞) C1-class, α, M > 0 → |f(t)| ≤ Meαt (t ≥ 0)

L[f'(t)](s) = sL[f(t)](s) - f(0)

スツルムリウビル方程式(Sturm-Liouville equation)の特別例
Ax. (ルジャンドル多項式 Legendre polynomial), N

Pn(x) := 1/2nndn/dxn(x2 - 1)n
1. Pn(x) → n次多項式
2. (1 - x2)Pn''(x) + 2xPn'(x) + n(n + 1)Pn(x) = 0
3. -11Pm(x)Pn(x)dx = 2/(2n + 1) (m = n), or 0 (mn)

Ax. (ラゲール多項式 Laguerre polynomials), N

Ln(x) := ex·dn/dxn(xne-x) ⇒
1. Ln(x) → n次多項式
2. xLn''(x) + (1 - x)Ln'(x) + nLn(x) = 0
3. 0Lm(x)Ln(x)e-xdx = (n!)2 (m = n), or 0 (mn)

Ax. (エルミート多項式 Hermite polynomials), N

Hn(x) := (-1)nex2·dn/dxne-x2
1. Hn(x) → n次多項式
2. Hn''(x) - 2xHn'(x) + 2nHn(x) = 0
3. -∞Hm(x)Hn(x)e-x2dx = 2nn!√π (m = n), or 0 (mn)

Pr. 3多項式共にライプニッツの法則を使う

常微分方程式 (ordinary differential equation)


Def.(常)微分方程式 F(x, y, y', y'' … y(n)) = 0

独立変数x、未知関数y及びその導関数y', y'' …, y(n)を含む方程式
y最高次の導関数がn次導関数 ≡ n階(常)微分方程式

Cf. 偏微分方程式: x·∂z/∂x + y·∂z/∂y = z, x, y 独立変数, z(x, y) 未知関数
Def. 解 = 常微分方程式を満足する関数y(x)

一般解 Ex. dy/dx = y, d2y/dx2 + y = 0 →

dy/dx = y = Cex, y = Asinx + Bcosx (A, B, C: 任意定数, 積分定数)

初期条件 x = x0, y = y0

特別解(特殊解): 一般解の積分定数に特別な値を得て得られる解

基本型
  1. dy/dx = f(x) → [yf(x)の原始関数] 一般解 y = f(x)dx + C
  2. g(ydy/dx = 1 → [g(ydy/dx·dx = g(ydy = x + C] g(y)dy = x + C
  3. g(ydy/dx = f(x) → [(左辺) = (2)] g(y)dy = f(x)dx

    G(y) = F(x) + C

Def. 変数分離形: F(x, y, dy/dx) = 0, dy/dx = f(x, y) = P(x)Q(y)

If Q(y) ≠ 0, 1/Q(ydy/dx = P(x) →
(1/Q(y))·(dy/dx)dx = (1/Q(y))dy = P(x)dx + C

Ex. dy/dx = x/yydy = xdx + C, y2/2 = x2/2 + C

x2y2 = A (A: 任意定数) → 一般解

Def. 同次(斉次)微分方程式 dy/dx = f(y/x)

dy/dx = f(x, y)の右辺がx, yの0次同次関数
y/x := uy = ux, dy/dx = x·du/dx + u
du/dx = (f(u) - u)/x (変数分離型) ⇒ (dy/(f(u) - u)) = dx/x

Q. dy/dx = (x + y)/(x - y)
A. dy/dx = (1 + y/x)/(1 - y/x), y := xuu + x·du/dx = (1 + u)/(1 - u)

x·du/dx = (1 + u)/(1 - u) - u = (1 + u2)/(1 - u)
(1 - u)/(1 + u2)du = dx/x(1 - u)/(1 + u2)du = dx/x
tan-1u - 1/2·log(1 + u2) = log|x| + c
tan-1(y/x) = 1/2·log(1 + y2/x2) + log|x| + c
2tan-1(y/x) = log(x2 + y2) + c

線型微分方程式 (linear differential equation)


Def. 線型微分方程式: y'とyについての一次式

p(x), q(x), dy/dx = p(x)y = q(x)

y := u(x)v(x), dy/dx = du/dx·v + u·dv/dx, du/dx·v + u(dv/dx + pv) = q
v(x): dv/dx + pv = 0, du/dx·v = qv = Ae-pdx
du/dx = q/v = q/A·epdxu = 1/A·qepdxdx + B

y = uv = Aepdx(1/Aqepdxdx + B) = epdx(qepdxdx + C)
定数変化法
dy/dx = dA/dx·e-pdxApe-pdx
dA/dx·e-pdxApe-pdx + Ape-pdx = p(x)
dA/dx = q(x)epdx

A = q(x)epdxedx + Cy = Aepdx(q(x)epdxdx + C)
Ex. dy/dx + p(x)y = q(x)yn, n ≠ 0, 1: Bernoulliの方程式

y1 – n := v
1/yn·dy/dx = 1/(1 – ndv/dx
1/(1 – ndv/dx + p(x)v = q(x)

X(x, y) + Y(x, ydy/dx = 0
X(x, y), Y(x, y) → ∂X/∂y = ∂Y/∂x → 完全微分形
Ex. 3x2 + 6xy2 + (6x2y + 4y2dy/dx = 0

x0x(3x2 + 6xy2)dx + y0y(6x02y + 4y2)dy = C
x0 = y0 = 0
x3 + 2x2y2 + 4/3·y3 = C

クレローの常微分方程式 Clairaut's equation
Def. y = x·dy/dx + f(dy/dx
dy/dx := py = px + f(p)
両辺をxで微分

p = x·dp/dx + p + f'(pdp/dx
0 = dp/dx·(x + f'(p))
dp/dx = 0 or x = pf'(p)

i) dp/dx =0 → p = dy/dx = C ⇒ 一般解 (直線の族)
ii) x = pf'(p) → y = f(p) - pf'(p)

x = -f'(p), and y = f(p) - pf'(p) → p := C
x = -f'(C), and y = f(C) - Cf'(C) ⇒ 特異解 (包絡線)

Ex. y = x·dy/dx + 1/(dy/dx): f(p) := 1/p, y = Cx + 1/C, y2 = 4x<
ダランベールの常微分方程式 d'Alembert's equation
Def. y = xf(y') + g(y')

n階線形微分方程式

Def. pn(xdny/dxn + pn–1(xdn–1y/dxn–1 + … + p1(xdy/dx + p0(x)y = f(x)

p0(x) … pn(x) 関数

Def. 線形同次微分方程式: f(x) = 0 ⇒ Def. 同次形(斉次形): f(x) ≡ 0
Def. 線形非同次微分方程式: f(x) ≠ 0
Def. n階線形微分方程式: 線形微分方程式に含まれる導関数の最高次数がn

Ex. x3y''' + x2y'' + xy' + y = 0 → 3階線形微分方程式

Th. 一次独立な2つの解は基本解を形成する
Pr. y1(x), y2(x) 一次独立な2つの解

W(x): ロンスキー(Wronski or Wronskian)行列式を用いる

Q. y'' + y = secxを解く
A. y := eλx, λ2 + 1 = 0, λ = ±ie±ix, cosx, sinx 基本解

y = c1(x)cosx + c2(x)sinx, y' = -c1(x)sinx + c2(x)cosx,

c1'(x)cosx + c2'(x)sinx = 0

y'' = -c1'(x)sinx + c2'(x)cosxc1(x)cosxc2(x)sinx 原式に戻す
-c1'(x)sinx + c2'(x)cosx = secx 積分,
c1(x) = A1 + log|cosx|, c2(x) = A2 + x
∴ 一般解 y = A1cosx + A2sinx + cosxlog|cosx| + xsinx

連立線形微分方程式 system of linear differential equation

n個の未知関数をもつ微分方程式の系
X' = AX + F

X = slde, A = slde, F = slde

Def. この微分方程式を満たすn個のdiff.な関数 x1, x2, … xnをこの方程式の解 solution という

Def. 微分方程式 F = 0 → 同次方程式 homogeneous equation

2つのn次正方行列 A, B, B = P-1AP

Aは正則行列Pによって変換された
ABは互いに相似
P-1AP = Λ = sldeAP =
P = (p1, p2, …, pn)

Def. |A - λE|はλについてのn次多項式

Aの固有多項式 → Def. 固有値(特性値): Aの根
固有ベクトル

Def. 固有値 Case 一次式

Y = AX + B, 2Y = 2AX+ 2Bは等式 → Y = 2Y → 固有値 = 2
固有値は、その方向を変えないベクトルがその1次変換で大きさが何倍になるかを表す

Ex. X = x + 2y, Y = 3x + 4yslde = sldeslde

X := Y (つまり平行) x + y = 0 → slde = slde × Aslde,

A: 固有値, (1,-1): 固有ベクトル

Def. 相似な行列: BP-1AP (A: n次の正方行列, P: n次の正則行列)
Th. 固有多項式: 相似な2つの行列は同じ固有多項式を持つ 逆は正しくない

φA(λ) = φP-1AP(λ)

Pr. φP-1AP(λ) = det(P-1AP - λI) = det(P-1AP - λP-1IP)

= det(P-1AP - P-1λIP) = det(P-1(A - λI)P) = |P-1||AP - λI||P|
= det(A - λI) = φA(λ) //

Th. 相似な行列は固有値が等しい
Pr. trivial (固有多項式が等しい)
Def. 正値定符号と負値定符号: q

> 0 (正) - 正値定符号 positive definite
≥ 0 (正負) - 半正値定符号 positive semidefinite
≤ 0 (非正) - 半負値定符号 negative semidefinite
< 0 (負) - 負値定符号 negative definite

線形偏微分方程式


変数変換法

複素数 complex number (関数論)


Def. 虚数: x2 + 1 = 0 → x2 = –1 → x = √-1 ≡ i (i, 虚数単位)

i2 = -1, i3 = i × i2 = -i, i4 = 1, …

Def. 複素数(z): z = x + iy (i2 = -1) → x: 実数部 real part, . y: 虚数部 imaginary part,

整数部: Re(z) or z = x, 虚数部: Im(z) or z = y
complex → ガウス平面: 複素数を表わすため用いる座標平面

Def. 複素共役 complex conjugate, z*: 複素数の虚数部分の符号を変えたもの (y → –y) → z* = xiy
Def. トレース trace: zz* = x2 + y2 (実数)
Def. zz* = 2iy (純虚数)
Def. 絶対値absolute value: |z| ≡ √zz* = √(x2 + y2)
Def. 複素数の四則演算: z1 = x1 + y1i, z2 = x2 + y2i

加減則 (シュプール spur): z1 ± z2 = (x1 ± x2) + (y1 ± y2)i
乗法即: z1z2 = (x1x2y1y2) + (x1y2 + x2y1)i

Re(z1z2) ≠ Re(z1)Re(z2)

除法即 (ノルム norm): z1/z2 =

(x1x2 + y1y2)/(x22 + y22) + (–x1y2 + y1x2)/(x22 + y22)i

Th. z1 = z2x1 = x2, y1 = y2
→ 複素数全体は加減乗除演算に閉じた集合
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