Convergent Integrals

GENERA

$\displaystyle \int_0^1 \dfrac{1-t^r}{1-t} \, dt = H_r$

$\displaystyle \int_0^1 t^r \, \ln(1\!-\!t) \, dt = - \dfrac{H_{r+1}}{r+1}$

$\displaystyle \int_0^{\infty_+} t^{r-1} \ln(1\!+\!t) \, dt = \dfrac{\Gamma(r)\,\Gamma(1\!-\!r)}{r} = \dfrac{\pi\csc(\pi r)}{r}$

$\displaystyle \int_0^1 \dfrac{t^n \ln t}{1\!-\!t} \, dt = \displaystyle \int_0^1 \dfrac{(1\!-\!t)^n \ln(1\!-\!t)}{t} \, dt = H_{n,2} - \dfrac{\pi^2}{6}$

Exponential Integrals (r > 0)

$\displaystyle \int_0^{\infty_+} t^r \, e^{-t} \, dt = \Gamma(r\!+\!1)$

$\displaystyle \int_0^{\infty_+} t^r \ln t \, e^{-t} \, dt = \Gamma(r\!+\!1) \, \Psi^{(0)}(r\!+\!1) = \Gamma(r\!+\!1) \, (H_r - \gamma)$

$\displaystyle \int_0^{\infty_+} t^r (\ln t)^2 \, e^{-t} \, dt = \Gamma(r\!+\!1) \, \left((\Psi^{(0)}(r\!+\!1))^2 + \Psi^{(1)}(r\!+\!1)\right) \\ = \Gamma(r\!+\!1) \, \left((H_r - \gamma)^2 + \Psi^{(1)}(r\!+\!1)\right)$

Laplace Transforms (r > 0)

$\displaystyle \int_0^{\infty_+} e^{-rt} \, dt = \dfrac{\Gamma(1)}{r} = \dfrac{1}{r}$

$\displaystyle \int_0^{\infty_+} t \, e^{-rt} \, dt = \dfrac{\Gamma(2)}{r^2} = \dfrac{1}{r^2}$

$\displaystyle \int_0^{\infty_+} t^2 \, e^{-rt} \, dt = \dfrac{\Gamma(3)}{r^3} = \dfrac{2}{r^3}$

$\displaystyle \int_0^{\infty_+} t^3 \, e^{-rt} \, dt = \dfrac{\Gamma(4)}{r^4} = \dfrac{6}{r^4}$

$\displaystyle \int_0^{\infty_+} t^4 \, e^{-rt} \, dt = \dfrac{\Gamma(5)}{r^5} = \dfrac{24}{r^5}$

$\displaystyle \int_0^{\infty_+} t^5 \, e^{-rt} \, dt = \dfrac{\Gamma(6)}{r^6} = \dfrac{120}{r^6}$

$\displaystyle \int_0^{\infty_+} \dfrac{e^{-rt}}{t} \, dt = \dfrac{r^0}{0!} \, (H_0 - \ln r) = -\ln r$

$\displaystyle \int_0^{\infty_+} \dfrac{e^{-rt}}{t^2} \, dt = \dfrac{r}{1!} \, (\ln r - H_1) = r\,(\ln r - 1)$

$\displaystyle \int_0^{\infty_+} \dfrac{e^{-rt}}{t^3} \, dt = \dfrac{r^2}{2!} \, (H_2 - \ln r) = \dfrac{r^2}{2} \left(\dfrac{3}{2} - \ln r\right)$

$\displaystyle \int_0^{\infty_+} \dfrac{e^{-rt}}{t^4} \, dt = \dfrac{r^3}{3!} \, (\ln r - H_3) = \dfrac{r^3}{6} \left(\ln r - \dfrac{11}{6}\right)$

$\displaystyle \int_0^{\infty_+} \dfrac{e^{-rt}}{t^5} \, dt = \dfrac{r^4}{4!} \, (H_4 - \ln r) = \dfrac{r^4}{24} \left(\dfrac{25}{12} - \ln r\right)$

$\displaystyle \int_0^{\infty_+} \dfrac{e^{-rt}}{t^6} \, dt = \dfrac{r^5}{5!} \, (\ln r - H_5) = \dfrac{r^5}{120} \left(\ln r - \dfrac{137}{60}\right)$

Mellin Transforms (s > 0)

$\displaystyle \int_0^{\infty_+} t^{s-1} \, e^{-t} \, dt = \Gamma(s)$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, e^{-2t} \, dt = \dfrac{\Gamma(s)}{2^s}$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, e^{-3t} \, dt = \dfrac{\Gamma(s)}{3^s}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{e^t-1} \, dt = \Gamma(s) \, \zeta(s)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{e^{2t}-1} \, dt = \dfrac{\Gamma(s) \, \zeta(s)}{2^s}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{e^{3t}-1} \, dt = \dfrac{\Gamma(s) \, \zeta(s)}{3^s}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{e^t+1} \, dt = \Gamma(s) \, \eta(s)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{e^{2t}+1} \, dt = \dfrac{\Gamma(s) \, \eta(s)}{2^s}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{e^{3t}+1} \, dt = \dfrac{\Gamma(s) \, \eta(s)}{3^s}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}}{1-e^{-t}} \, dt = \int_0^{\infty_+} \dfrac{t^{s-1} \, e^t}{e^t-1} \, dt = \Gamma(s) \zeta(s,0) = \dfrac{(-1)^s}{s} + \Gamma(s) \zeta(s)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1} \, e^{-t}}{1-e^{-t}} \, dt = \int_0^{\infty_+} \dfrac{t^{s-1}}{e^t-1} \, dt = \Gamma(s) \, \zeta(s,1) = \Gamma(s) \, \zeta(s)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{s-1}\,e^{-t}}{1-e^{-2t}} \, dt = \int_0^{\infty_+} \dfrac{t^{s-1}\,e^t}{e^{2t}-1} \, dt = \dfrac{\Gamma(s) \, \zeta(s,\frac{1}{2})}{2^s} = (1-2^{-s}) \, \Gamma(s) \, \zeta(s)$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, \dfrac{e^{-3t}}{1-e^{-3t}} \, dt = \int_0^{\infty_+} t^{s-1} \, \dfrac{1}{e^{3t}-1} \, dt = \dfrac{\Gamma(s) \, \zeta(s,\frac{3}{3})}{3^s} = \dfrac{\Gamma(s) \, \zeta(s)}{3^s}$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, \dfrac{e^{-2t}}{1-e^{-3t}} \, dt = \int_0^{\infty_+} t^{s-1} \, \dfrac{e^t}{e^{3t}-1} \, dt = \dfrac{\Gamma(s) \, \zeta(s,\frac{2}{3})}{3^s}$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, \dfrac{e^{-t}}{1-e^{-3t}} \, dt = \int_0^{\infty_+} t^{s-1} \, \dfrac{e^{2t}}{e^{3t}-1} \, dt = \dfrac{\Gamma(s) \, \zeta(s,\frac{1}{3})}{3^s}$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, e^{-t} \, \ln t \, dt = \Gamma(s) \, \Psi^{(0)}(s) = \Gamma(s) \, (H_{s-1}-\gamma)$

$\displaystyle \int_0^{\infty_+} t^{s-1} \, e^{-t} \, (\ln t)^2 \, dt = \Gamma(s) \left((\Psi^{(0)}(s))^2 + \Psi^{(1)}(s)\right) = \Gamma(s) \left((H_{s-1}-\gamma)^2 + \Psi^{(1)}(s)\right)$

exp(2πt) – 1 as the denominator

$\displaystyle \int_0^{\infty_+} \dfrac{t^{n-1}}{e^{2\pi t}-1} \, dt = \dfrac{\Gamma(n)}{(2\pi)^n} \, \text{Li}_n(1) = \dfrac{(n\!-\!1)!}{(2\pi)^n} \, \zeta(n)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^n}{e^{2\pi t}-1} \, dt = \dfrac{\Gamma(n\!+\!1)}{(2\pi)^{n+1}} \, \text{Li}_{n+1}(1) = \dfrac{n!}{(2\pi)^{n+1}} \, \zeta(n\!+\!1)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{2n-1}}{e^{2\pi t}-1} \, dt = \dfrac{(2n\!-\!1)!}{(2\pi)^{2n}} \, \zeta(2n) = \dfrac{|B_{2n}|}{4n} = (-1)^{n-1} \dfrac{B_{2n}}{4n}$

$\displaystyle \int_0^{\infty_+} \dfrac{t^{2n}}{e^{2\pi t}-1} \, dt = \dfrac{(2n)!}{(2\pi)^{2n+1}} \, \zeta(2n\!+\!1)$

$\displaystyle i \int_0^{\infty_+} \dfrac{(it)^r - (-it)^r}{e^{2\pi t}-1} \, dt = \zeta(-r) + \dfrac{(-1)^r}{r\!+\!1}$

$\displaystyle i \int_0^{\infty_+} \dfrac{(1\!+\!it)^r - (1\!-\!it)^r}{e^{2\pi t}-1} \, dt = \zeta(-r) -\dfrac{1}{2} + \dfrac{1}{r\!+\!1}$

exp(2πt) + 1 as the denominator

$\displaystyle \int_0^{\infty_+} \dfrac{t^{n-1}}{e^{2\pi t}+1} \, dt = \dfrac{(1-2^{1-n})\,(n\!-\!1)!}{(2\pi)^n} \, \zeta(n) = \dfrac{(n\!-\!1)!}{(2\pi)^n} \, \eta(n)$

$\displaystyle \int_0^{\infty_+} \dfrac{t^n}{e^{2\pi t}+1} \, dt = \dfrac{(1-2^{-n})\,n!}{(2\pi)^{n+1}} \, \zeta(n\!+\!1) = \dfrac{n!}{(2\pi)^{n+1}} \, \eta(n\!+\!1)$

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