テイラーの定理

ブルック・テイラー（Brook Taylor, 1685年8月18日 - 1731年12月29日） †

イギリスの数学者.生まれは現在のエドモントン。ケンブリッジ大学のセント・ジョンズ・カレッジで学ぶ。 マクローリン Colin Maclaurin (1698年2月～1746年6月14日)は天才数学者で英国スコットランドの大学教授。コリン・マクローリン(1698-1746)スコットランドの数学者。テイラーはカナダ生まれでケンブリッジ大学のカレッジ卒の数学者。年下のマクローリンは天才で19歳で大学教授に就任、テイラーはカナダから英国のケンブリッジ大のカレッジに留学した普通の大学卒の数学者のようで、英国舞台で同じ時代に活躍した

Brook Taylor †

Brook Taylor is an English mathematician, who is noted for his contribution to the development of calculus. He was born on 18 August 1685 in Edmonton to John Taylor and Olivia Tempest. Brook Taylor married Miss Brydges of Wallington, Surrey, in 1721 against his father’s wishes, leading to an estrangement from his father. His wife died in 1723 while giving birth to a son, who also died. His second marriage to Sabetta Sawbridge of Olantigh was also short lived as she too died in childbirth in 1730, leaving behind a daughter, Elizabeth. By this time, his health began to deteriorate and he died on 30 November 1731, at Somerset House. Brook Taylor was buried in London on 2 December 1731, near his first wife, in the churchyard at St Anne's, Soho.

Brook Taylor was born at Edmonton (at that time in Middlesex) to John Taylor of Bifrons House, Kent, and Olivia Tempest, daughter of Sir Nicholas Tempest, Bart., of Durham. Brook entered St John's College, Cambridge as a fellow-commoner in 1701, and took degrees of LL.B. and LL.D. in 1709 and 1714, respectively. Having studied mathematics under John Machin and John Keill, in 1708 he obtained a remarkable solution of the problem of the "centre of oscillation," which, however, remaining unpublished until May 1714 (Phil. Trans., vol. xxviii. p. x1), his claim to priority was unjustly disputed by Johann Bernoulli. Taylor's Methodus Incrementorum Directa et Inversa (1715) added a new branch to the higher mathematics, now designated the "calculus of finite differences." Among other ingenious applications, he used it to determine the form of movement of a vibrating string, by him first successfully reduced to mechanical principles. The same work contained the celebrated formula known as Taylor's theorem, the importance of which remained unrecognized until 1772, when J. L. Lagrange realized its powers and termed it "le principal fondement du calcul différentiel" ("the main foundation of differential calculus").

In his 1715 essay Linear Perspective, Taylor set forth the true principles of the art in an original and more general form than any of his predecessors; but the work suffered from the brevity and obscurity which affected most of his writings, and needed the elucidation bestowed on it in the treatises of Joshua Kirby (1754) and Daniel Fournier (1761).

Taylor was elected a fellow of the Royal Society early in 1712, and in the same year sat on the committee for adjudicating the claims of Sir Isaac Newton and Gottfried Leibniz, and acted as secretary to the society from 13 January 1714 to 21 October 1718. From 1715 his studies took a philosophical and religious bent. He corresponded, in that year, with the Comte de Montmort on the subject of Nicolas Malebranche's tenets; and unfinished treatises, On the Jewish Sacrifices and On the Lawfulness of Eating Blood, written on his return from Aix-la-Chapelle in 1719, were afterwards found among his papers. His marriage in 1721 with Miss Brydges of Wallington, Surrey, led to an estrangement from his father, which ended in 1723 after her death in giving birth to a son, who also died. The next two years were spent by him with his family at Bifrons, and in 1725 he married this time with his father's approval, Sabetta Sawbridge of Olantigh, Kent, who also died in childbirth in 1730 ; in this case, however, the child, a daughter, survived. Taylor's fragile health gave way; he fell into a decline, died at Somerset House, and was buried at St Ann's, Soho. By the date of his father's death in 1729 he had inherited the Bifrons estate. As a mathematician, he was the only Englishman after Sir Isaac Newton and Roger Cotes capable of holding his own with the Bernoullis, but a great part of the effect of his demonstrations was lost through his failure to express his ideas fully and clearly.

According to the Oxford Dictionary of National Biography, Taylor died "on 30 November 1731 in Somerset House, London. He was buried in London on 2 December 1731, near his first wife, in the churchyard of St Anne's, Soho."

• 何故　テーラー展開と呼ばれたか？ 今日，名前の残っている テイラー展開 は彼の『増分法』という本の中に

  f(x+h)　=　f(x)　 +　 f'(x)h　 +　 f''(x)h^2 /2!　+　f'''(x)h^3 /3!　... 

  が出ています． これは既に グレゴリ に知られていました．もちろんテイラーはそんなことは知らなかった． マクローリン展開 の方も マクローリン が本を出版する6年以上も前に スターリング の『微分法』という本の中に載っていました． 定理の命名に関しては，歴史はしばしば気まぐれだったようです．

テイラーの定理（Taylor's theorem） †

関数をある一点における高階の微分係数を用いて近似するものである。イギリスの数学者ブルック・テイラーによって1712年に述べられた.

関数 f が閉区間 [a, x] で n 回微分可能であるとき、

が成り立つ。ここで、Rn は剰余項（じょうよこう、residue） たとえば開区間 (a, x) に存在する c を用いて

のように書ける テイラーの定理は平均値の定理を一般化したものになっている。実際、上の式において n = 1 としたもの、つまり

f(x) = f(a) + f'(c)(x − a) 

は平均値の定理に他ならない

• 数学的帰納法で証明できる。

定理の証明 †

部分積分の公式を使う。

• もうひとつの証明：平均値の定理を使う
  

関数f(x)を無限級数で表示 †

n→∞とした場合において、Rn→0となってくれることが証明できた場合、

　　　　　　　 という級数展開が正当化され、この右辺を「f(x)のx=aにおけるテイラー展開」"Taylor Series at x=a"　"Taylor Expansion around x=a"　などと呼ぶ。

EXP(x)log(x)の展開 †

exp( x )＝１＋ x ＋ x2/２! ＋…＋ xn/n! ＋…　（| x |＜∞）
log ( 1+ x )= x － x2/２ ＋ x3/３ －…＋(－１)n－1 xn/n＋…　（| x |＜1）

テーラー展開の例 †

オイラーの公式 †

j^2=-1を使って、オイラーの公式が理解できる

簡単な説明 †

• sinx＝a0+a1x+a2x2+a3x3+a4x4+…とする（とできるとする）。 　　　x＝0を代入すると、a0＝0　…(1) 　　　両辺を微分すると、cosx＝a1+2a2x+3a3x2+… 　　　x＝0を代入すると、cos0＝1＝a1　…(2) 　　　この(1)(2)を繰り返すと、sinx＝x-(1/2･3)x3+(1/2・3・4・5)x5-…と展開できます。 　　　
• これを一般化したのがマクローリンの展開式　ｆ（ｘ）＝ｆ(0)＋ｘf'(0)+(x2/2･1)f(x)+(x3/3･2･1)f'(x)+…

log(1+x)のテーラー展開と収束性 †

f(x)=log(x+1)についてx～0の周りで展開する。 n次導関数を正負符号に気を付けてテーラーの式に入れていくと

log(x+1)～x - x2/2 + x3/3 - x4/4 + …（但しx～0）  eq.1

• ここで　xの整式

  Σan･xn = a0 + a1x1 + a2x2 + a3x3 + …

  について、nが十分に大きくなったとき、それが収束するか判定する問題を考える。 適当な自然数nについて

  R=lim[n→∞]an/an+1

  とした時、

   |x| < R　⇒　収束する
   |x| = R　⇒　一般には定まらない(x=Rとx=-Rで違ったりもする)
   |x| > R　⇒　発散する
  このRを収束半径と呼ぶ。

eq.1の右辺について収束半径Rを求めると an/an+1 = -(n+1)/n なので R→1(n→∞のとき),すなわち

|x|<1の時しかlog(x+1)のテイラー展開は収束しないことになる。

これを微分するとちゃんと 1/(1 + x) の Taylor 級数になっている。

f(x) = (1 + x)^α. α は任意の実数 †

n回微分は、

f(n)(x) = α(α - 1)(α - 2)…(α - n + 1)(1 + x)^(α - n)=αCn・n!・(1 + x)^(α - n)

であるので、テーラー展開は

f(x)=(1 + x)^α = Σ αCn xn

これを使うと任意の実数乗の数　も求められる。

演習例題 †