Divergent Integrals

SECTIONS

Exponential Integrals

$\displaystyle \int_0^{\infty_+} e^{at} \, dt = -\dfrac{1}{a}$

$\displaystyle \int_0^{\infty_+} e^{at+b} \, dt = -\dfrac{e^b}{a}$

$\displaystyle \int_0^{\infty_+} a^t \, dt = -\dfrac{1}{\ln a}$

$\displaystyle \int_0^{\infty_+} a^{bt} \, dt = -\dfrac{1}{b\ln a}$

$\displaystyle \int_0^{\infty_+} t\,a^t \, dt = \dfrac{1}{a^2}$

$\displaystyle \int_0^{\infty_+} t^2\,a^t \, dt = -\dfrac{2!}{a^3}$

$\displaystyle \int_0^{\infty_+} t^3\,a^t \, dt = \dfrac{3!}{a^4}$

$\displaystyle \int_0^{\infty_+} t^4\,a^t \, dt = -\dfrac{4!}{a^5}$

Trigonometric Integrals

$\displaystyle \int_0^{\infty_+} e^{iat} \, dt = \dfrac{i}{a}$

$\displaystyle \int_0^{\infty_+} \cos(at) \, dt = 0$

$\displaystyle \int_0^{\infty_+} \sin(at) \, dt = \dfrac{1}{a}$

$\displaystyle \int_0^{\infty_+} e^{i(at+b)} \, dt = \dfrac{ie^{ib}}{a}$

$\displaystyle \int_0^{\infty_+} \cos(at+b) \, dt = -\dfrac{\sin{b}}{a}$

$\displaystyle \int_0^{\infty_+} \sin(at+b) \, dt = \dfrac{\cos{b}}{a}$

exp(2πt) – 1 as the denominator

$\displaystyle \int_0^{\infty_+} \dfrac{e^{at}}{e^{2\pi t}-1} \, dt = -\dfrac{1}{2\pi}\,\Psi^{(0)}\!\left(1+\dfrac{a}{2\pi}\right) = \dfrac{1}{2\pi} \left(\gamma-H_\frac{a}{2\pi}\right)$

exp(at) – 1 as the denominator

$\displaystyle \int_0^{\infty_+} \dfrac{1}{e^{at}-1} \, dt = \int_0^{\infty_+} \dfrac{e^{-at}}{1-e^{-at}} \, dt = \dfrac{\zeta(1)}{a} = \dfrac{\gamma}{a}$

$\displaystyle \int_0^{\infty_+} \dfrac{1}{e^{-at}-1} \, dt = \int_0^{\infty_+} \dfrac{e^{at}}{1-e^{at}} \, dt = -\dfrac{\zeta(1)}{a} = -\dfrac{\gamma}{a}$

exp(at) + 1 as the denominator

$\displaystyle \int_0^{\infty_+} \dfrac{1}{e^{at}+1} \, dt = \int_0^{\infty_+} \dfrac{e^{-at}}{1+e^{-at}} \, dt = \dfrac{\ln 2}{a}$

$\displaystyle \int_0^{\infty_+} \dfrac{1}{e^{-at}+1} \, dt = \int_0^{\infty_+} \dfrac{e^{at}}{1+e^{at}} \, dt = -\dfrac{\ln 2}{a}$

Pages: 1 2 3 4 5 6 7 8 9 10 11