User:Caliburn/Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x/test

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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
\(\ds \) \(=\) \(\ds \frac 2 {a^{n + 1} } \int_0^\infty \sum_{N \mathop = 0}^\infty u^n e^{-\paren {2 N + 1} u} \rd u\)

Let $\mu$ be the Lebesgue measure on $\R$.

For each integer $N \ge 0$, define $f_N : \R \to \R$ by:

$\ds \map f u = u^n e^{-\paren {2 N + 1} u}$

We have that:

$f_N$ is continuous for each $N \ge 0$.

So, from Continuous Mapping is Measurable:

$f_N$ is Borel measurable for each $N \ge 0$.

From Integral of Series of Positive Measurable Functions, we have:

$\ds \frac 2 {a^{n + 1} } \int_0^\infty \sum_{N \mathop = 0}^\infty f_N \rd \mu = \frac 2 {a^{n + 1} } \sum_{N \mathop = 0}^\infty \int_0^\infty f_N \rd \mu$

That is, from Lebesgue Integral Coincides with Riemann Integral: (need positive function Lebesgue integrable iff Riemann integrable too)

$\ds \frac 2 {a^{n + 1} } \int_0^\infty \sum_{N \mathop = 0}^\infty u^n e^{-\paren {2 N + 1} u} \rd x = \frac 2 {a^{n + 1} } \sum_{N \mathop = 0}^\infty \int_0^\infty u^n e^{-\paren {2 N + 1} u} \rd x$

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:

$\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$


Sources

Category:Definite Integrals involving Hyperbolic Sine Function Category:Gamma Function Category:Riemann Zeta Function

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