Incomplete Gamma and Related Functions - 8.21 Generalized Sine and Cosine Integrals

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
8.21.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{+\mathrm{i}t}\mathrm{d}t=e^{+\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right)}}$ \int_{0}^{\infty}t^{a-1}e^{+\iunit t}\diff{t} = e^{+\frac{1}{2}\pi\iunit a}\EulerGamma@{a}	${\displaystyle{\displaystyle 0<\Re a,\Re a<1,\Re a>0}}$	int((t)^(a - 1)* exp(+ I*t), t = 0..infinity) = exp(+(1)/(2)*Pi*I*a)*GAMMA(a)	Integrate[(t)^(a - 1)* Exp[+ I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[+Divide[1,2]*Pi*I*a]*Gamma[a]	Successful	Aborted	-	Successful [Tested: 1]
8.21.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{-\mathrm{i}t}\mathrm{d}t=e^{-\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right)}}$ \int_{0}^{\infty}t^{a-1}e^{-\iunit t}\diff{t} = e^{-\frac{1}{2}\pi\iunit a}\EulerGamma@{a}	${\displaystyle{\displaystyle 0<\Re a,\Re a<1,\Re a>0}}$	int((t)^(a - 1)* exp(- I*t), t = 0..infinity) = exp(-(1)/(2)*Pi*I*a)*GAMMA(a)	Integrate[(t)^(a - 1)* Exp[- I*t], {t, 0, Infinity}, GenerateConditions->None] == Exp[-Divide[1,2]*Pi*I*a]*Gamma[a]	Successful	Aborted	-	Successful [Tested: 1]