Exercise 12, § 2

Functions of a Real Variable, VI, p. 25

For every polynomial $Q$ with real coefficients, such that $Q(0) = Q(1)$ , denote by $\tilde{Q}$ the continuous function in $\mathbf{R}$ such that $\tilde{Q}(x) = Q(x)$ for $0 \leq x \leq 1$ and $\tilde{Q}(x+1) = \tilde{Q}(x)$ for all $x \in \mathbf{R}$ . a) For every integer $m \geq 2$ , the Bernoulli polynomial $B_m$ is the unique polynomial with real coefficients, $Q$ of degree $m$ , monic and such that one has $\int_0^1 Q(x) dx = 0$ , $Q(1) = Q(0)$ and that the function $\tilde{Q}$ is $m - 2$ times differentiable in $\mathbf{R}$ and has a continuous $(m - 2)$ -th derivative. (Reason by induction on $m$ .) b) Prove that one has, for $m \geq 1$

\int_0^1 B_m(x) e^{-2\pi inx} dx = \begin{cases} 0 & \text{if } n = 0 \\ -\frac{m!}{(2\pi i)^m} & \text{if } n \neq 0 \text{ in } \mathbf{Z}. \end{cases}

\* c) Deduce from b) that, for $0 \leq x \leq 1$ and $m \geq 2$ , one has

B_m(x) = -\frac{m!}{(2\pi i)^m} \sum_{n \in \mathbf{Z} - \{0\}} \frac{e^{2\pi inx}}{n^m}

the series being normally convergent.*

Solution

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