Gold 27

Incomplete gamma function

Gold ID

27

Link

https://sigir21.wmflabs.org/wiki/Incomplete_gamma_function#math.77.118

Formula

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\gamma (\frac{s}{2},z^2\pi )}{(z^2\pi {)}^{\frac{s}{2}}}{e}^{-2\pi ikz}\mathrm{d}z=\frac{\mathrm{\Gamma }(\frac{1-s}{2},k^2\pi )}{(k^2\pi {)}^{\frac{1-s}{2}}}$

TeX Source

\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Semantic LaTeX

Translation

\int_{-\infty}^\infty \frac{\incgamma@{\frac s 2}{z^2 \cpi}}{(z^2 \cpi)^\frac s 2} \expe^{- 2 \cpi \iunit k z} \diff{z} = \frac{\incGamma@{\frac {1-s} 2}{k^2 \cpi}}{(k^2 \cpi)^\frac {1-s} 2}

Expected (Gold Entry)

\int_{-\infty}^\infty \frac{\incgamma@{\frac s 2}{z^2 \cpi}}{(z^2 \cpi)^\frac s 2} \expe^{- 2 \cpi \iunit k z} \diff{z} = \frac{\incGamma@{\frac {1-s} 2}{k^2 \cpi}}{(k^2 \cpi)^\frac {1-s} 2}}

Mathematica

Translation

Integrate[Divide[Gamma[Divide[s,2], 0, (z)^(2)* Pi],((z)^(2)* Pi)^(Divide[s,2])]*Exp[- 2*Pi*I*k*z], {z, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1 - s,2], (k)^(2)* Pi],((k)^(2)* Pi)^(Divide[1 - s,2])]

Expected (Gold Entry)

Integrate[Divide[Gamma[Divide[s,2], 0, (z)^(2)* Pi],((z)^(2)* Pi)^(Divide[s,2])]*Exp[- 2*Pi*I*k*z], {z, - Infinity, Infinity}] == Divide[Gamma[Divide[1 - s,2], (k)^(2)* Pi],((k)^(2)* Pi)^(Divide[1 - s,2])]

Maple

Translation

int((GAMMA((s)/(2))-GAMMA((s)/(2), (z)^(2)* Pi))/(((z)^(2)* Pi)^((s)/(2)))*exp(- 2*Pi*I*k*z), z = - infinity..infinity) = (GAMMA((1 - s)/(2), (k)^(2)* Pi))/(((k)^(2)* Pi)^((1 - s)/(2)))

Expected (Gold Entry)

int((GAMMA((s)/(2))-GAMMA((s)/(2), (z)^(2)* Pi))/(((z)^(2)* Pi)^((s)/(2)))*exp(- 2*Pi*I*k*z), z = - infinity..infinity) = (GAMMA((1 - s)/(2), (k)^(2)* Pi))/(((k)^(2)* Pi)^((1 - s)/(2)))