Primitive of Hyperbolic Cotangent of a x over x
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Theorem
\(\ds \int \frac {\coth a x \rd x} x\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\frac 1 {a x} + \frac {a x} 3 - \frac {\paren {a x}^3} {135} + \cdots + C\) |
where $B_k$ denotes the $k$th Bernoulli number.
Proof
\(\ds \coth x\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}\) | Power Series Expansion for Hyperbolic Cotangent Function | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 x + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, x^{2 k - 1} } {\paren {2 k}!}\) | extracting the first term, just in case | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\coth a x} x\) | \(=\) | \(\ds \dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} \, \paren {a x}^{2 k - 1} } {x \paren {2 k}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!}\) | |||||||||||||
It is noted that the exponent of $x$ is always even, so there is no need to consider the special case where $x^{-1}$. | |||||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\coth a x \rd x} x\) | \(=\) | \(\ds \int \paren {\dfrac 1 {a x^2} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \dfrac 1 {a x^2} \rd x + \sum_{k \mathop = 1}^\infty \int \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 2} } {\paren {2 k}!} \rd x\) | |||||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 {a x} + \sum_{k \mathop = 1}^\infty \frac {2^{2 k} B_{2 k} a^{2 k - 1} x^{2 k - 1} } {\paren {2 k - 1} \paren {2 k}!} + C\) | Primitive of Power | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {2^{2 k} B_{2 k} \paren {a x}^{2 k - 1} } {\paren {2 k}!} + C\) | bringing the first term back inside the summation |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.623$