Integral Representation of Bernoulli Number

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Theorem

Bernoulli numbers can be expressed in integral form as follows:

$\ds \size {B_{2 n} } = 4 n \int_0^\infty \frac {t^{2 n - 1} } {e^{2 \pi t} - 1} \rd t$

where:

$B_n$ are the Bernoulli numbers
$n$ is a positive integer.


Proof

\(\ds \map \zeta s \map \Gamma s\) \(=\) \(\ds \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t\) Integral Representation of Riemann Zeta Function in terms of Gamma Function
\(\ds \leadsto \ \ \) \(\ds \map \zeta {2 n} \map \Gamma {2 n}\) \(=\) \(\ds \int_0^\infty \frac {t^{2 n - 1} } {e^t - 1} \rd t\) setting $s:= 2n$
\(\ds \leadsto \ \ \) \(\ds \paren {-1}^{n + 1} \frac {B_{2 n} 2^{2 n - 1} \pi^{2 n} } {\paren {2 n}!} \paren {2 n - 1}!\) \(=\) \(\ds \int_0^\infty \frac {t^{2 n - 1} } {e^t - 1} \rd t\) Riemann Zeta Function at Even Integers, Gamma Function Extends Factorial
\(\ds \leadsto \ \ \) \(\ds \frac {\size {B_{2 n} } 2^{2 n} \pi^{2 n} } {4 n}\) \(=\) \(\ds \int_0^\infty \frac {t^{2 n - 1} } {e^t - 1} \rd t\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {\size {B_{2 n} } \paren {4 \pi^2}^n} {4 n}\) \(=\) \(\ds \int_0^\infty \frac {\paren {2 \pi t}^{2 n - 1} } {e^{2 \pi t} - 1} \paren {2 \pi \rd t}\) $t \to 2 \pi t$ and $\rd t \to 2 \pi \rd t$ and Power of Product
\(\ds \leadsto \ \ \) \(\ds \size {B_{2 n} } \paren {4 \pi^2}^n\) \(=\) \(\ds \paren {4 n} \paren {4 \pi^2}^n \int_0^\infty \frac {t^{2 n - 1} } {e^{2 \pi t} - 1} \rd t\)
\(\ds \leadsto \ \ \) \(\ds \size {B_{2 n} }\) \(=\) \(\ds 4 n \int_0^\infty \frac {t^{2 n - 1} } {e^{2 \pi t} - 1} \rd t\)

$\blacksquare$


Sources

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