Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x
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Theorem
- $\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$
where:
- $a$ and $n$ are positive real numbers
- $\Gamma$ denotes the gamma function
- $\zeta$ denotes the Riemann zeta function.
Proof
\(\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x\) | \(=\) | \(\ds \frac 1 a \int_0^\infty \frac {\paren {\frac u a}^n} {\sinh u} \rd u\) | substituting $u = a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {a^{n + 1} } \int_0^\infty \frac {u^n e^{-u} } {1 - e^{-2 u} } \rd u\) | Definition of Hyperbolic Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {a^{n + 1} } \int_0^\infty u^n e^{-u} \paren {\sum_{N \mathop = 0}^\infty \paren {e^{-2 u} }^N} \rd u\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {a^{n + 1} } \sum_{N \mathop = 0}^\infty \int_0^\infty u^n e^{-\paren {2 N + 1} u} \rd u\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {a^{n + 1} } \map \Gamma {n + 1} \sum_{N \mathop = 0}^\infty \frac 1 {\paren {2 N + 1}^{n + 1} }\) | Laplace Transform of Real Power |
We have:
\(\ds \sum_{N \mathop = 1}^\infty \frac 1 {N^{n + 1} }\) | \(=\) | \(\ds \map \zeta {n + 1}\) | Definition of Riemann Zeta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{N \mathop = 1}^\infty \frac 1 {\paren {2 N}^{n + 1} } + \sum_{N \mathop = 0}^\infty \frac 1 {\paren {2 N + 1}^{n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2^{n + 1} } \sum_{N \mathop = 1}^\infty \frac 1 {N^{n + 1} } + \sum_{N \mathop = 0}^\infty \frac 1 {\paren {2 N + 1}^{n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2^{n + 1} } \map \zeta {n + 1} + \sum_{N \mathop = 0}^\infty \frac 1 {\paren {2 N + 1}^{n + 1} }\) | Definition of Riemann Zeta Function |
So:
\(\ds \sum_{N \mathop = 0}^\infty \frac 1 {\paren {2 N + 1}^{n + 1} }\) | \(=\) | \(\ds \paren {1 - \frac 1 {2^{n + 1} } } \map \zeta {n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2^{n + 1} - 1} {2^{n + 1} } \map \zeta {n + 1}\) |
giving:
- $\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Hyperbolic Functions: $15.115$