Convergent Integrals

SUPERSECTIONS

SECTIONS

exp(2πt) – 1 as the denominator

$\displaystyle \int_0^{\infty_+} \dfrac{\sin(t\ln a)}{e^{2\pi t}-1} \, dt = \dfrac{1}{2} \left(\dfrac{a}{a-1} - \dfrac{1}{2} - \dfrac{1}{\ln a}\right) = \dfrac{1}{4} \coth\!\left(\dfrac{\ln a}{2}\right) - \nolinebreak \dfrac{1}{2\ln a}$

$\displaystyle \int_0^{\infty_+} \dfrac{\sin(at)}{e^{2\pi t}-1} \, dt = \dfrac{1}{2} \left(\dfrac{e^a}{e^a-1} - \dfrac{1}{2} - \dfrac{1}{a}\right) = \dfrac{1}{4} \coth\!\left(\dfrac{a}{2}\right) - \nolinebreak \dfrac{1}{2a}$

$i\,\displaystyle \int_0^{\infty_+} \dfrac{\dfrac{\ln(a\!+\!it)}{a\!+\!it} - \dfrac{\ln(a\!-\!it)}{a\!-\!it}}{e^{2\pi t}-1} \, dt = \gamma_1(a) - \dfrac{\ln a}{2}\!\left(\dfrac{1}{a} + \ln a\right)$

SUBSECTIONS

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