Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
2026-04-03 12:40:13.1775220013
Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$
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A few possible methods:
Apply Cauchy's integral formula
Put $\displaystyle e^{\lambda t} = \sum\limits_{n=0}^\infty \frac{t^n}{n!} \lambda^n$ and integrate term-wise.
Integrate by parts. (Leads to induction on $k$)
Differentiate with respect to $t$.