q -Hypergeometric and Related Functions - 17.13 Integrals

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
17.13.E3	${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\alpha-1}\frac{\left(-tq^{\alpha+\beta};q\right)_{\infty}}{\left(-t;q\right)_{\infty}}\mathrm{d}t=\frac{\Gamma\left(\alpha\right)\Gamma\left(1-\alpha\right)\Gamma_{q}\left(\beta\right)}{\Gamma_{q}\left(1-\alpha\right)\Gamma_{q}\left(\alpha+\beta\right)}}}$ \int_{0}^{\infty}t^{\alpha-1}\frac{\qPochhammer{-tq^{\alpha+\beta}}{q}{\infty}}{\qPochhammer{-t}{q}{\infty}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{1-\alpha}\qGamma{q}@{\beta}}{\qGamma{q}@{1-\alpha}\qGamma{q}@{\alpha+\beta}}	${\displaystyle{\displaystyle\Re(\alpha)>0,\Re(1-\alpha)>0}}$	int((t)^(alpha - 1)*(QPochhammer(- t*(q)^(alpha + beta), q, infinity))/(QPochhammer(- t, q, infinity)), t = 0..infinity) = (GAMMA(alpha)*GAMMA(1 - alpha)*QGAMMA(q, beta))/(QGAMMA(q, 1 - alpha)*QGAMMA(q, alpha + beta))	Integrate[(t)^(\[Alpha]- 1)*Divide[QPochhammer[- t*(q)^(\[Alpha]+ \[Beta]), q, Infinity],QPochhammer[- t, q, Infinity]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]]*Gamma[1 - \[Alpha]]*QGamma[\[Beta],q],QGamma[1 - \[Alpha],q]*QGamma[\[Alpha]+ \[Beta],q]]	Error	Failure	-	Failed [26 / 30] Result: Plus[NIntegrate[Times[Power[t, -0.5], Power[QPochhammer[Times[-1, t], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], DirectedInfinity[1]], -1], QPochhammer[Times[Complex[-0.5000000000000001, -0.8660254037844386], t], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], DirectedInfinity[1]]] Test Values: {t, 0, DirectedInfinity[1]}], Times[-3.1415926535897936, Power[QGamma[0.5, Complex[0.8660254037844387, 0.49999999999999994]], -1], QGamma[1.5, Complex[0.8660254037844387, 0.49999999999999994]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5], Rule[β, 1.5]} Result: Plus[-3.1415926535897936, NIntegrate[Times[Power[t, -0.5], Power[QPochhammer[Times[-1, t], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], DirectedInfinity[1]], -1], QPochhammer[Times[Complex[-0.8660254037844387, -0.49999999999999994], t], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], DirectedInfinity[1]]] Test Values: {t, 0, DirectedInfinity[1]}]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5], Rule[β, 0.5]} ... skip entries to safe data