Exercise 7, § 3

Functions of a Real Variable, I, 41

Let $f$ be a finite real function $n$ times differentiable at the point $x_0$ , and $g$ a vector function which is $n$ times differentiable at the point $y_0 = f(x_0)$ . Let

f(x_0 + h) = a_0 + a_1 h + \cdots + a_n h^n + r_n(h)

g(y_0 + k) = b_0 + b_1 k + \cdots + b_n k^n + s_n(k)

be the Taylor expansions of order

n

f

and

g

at the points

x_0

and

y_0

respectively. Show that the sum of the

n+1

terms of the Taylor expansion of order

n

of the composite function

g \circ f

at the point

x_0

is equal to the sum of the terms of degree

\leq n

in the polynomial

b_0 + b_1 (a_1 h + \cdots + a_n h^n) + b_2 (a_1 h + \cdots + a_n h^n)^2 + \cdots + b_n (a_1 h + \cdots + a_n h^n)^n.

Deduce the two following formulae:

D^n(g(f(x))) = \sum \frac{n!}{m_1! m_2! \ldots m_q!} g^{(p)}(f(x)) \left( \frac{f'(x)}{1!} \right)^{m_1} \cdots \left( \frac{f^{(q)}(x)}{q!} \right)^{m_q}

the sum being taken over all systems of positive integers

(m_i)_{1 \leq i \leq q}

such that

m_1 + 2m_2 + \cdots + qm_q = n

where

p

denotes the sum

m_1 + m_2 + \cdots + m_q

D^n(g(f(x))) = \sum_{p=1}^n \frac{1}{p!} g^{(p)}(f(x)) \left( \sum_{q=1}^p \binom{p}{q} (-f(x))^{p-q} D^n((f(x))^q) \right).

Solution

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