Hypergeometric Function - 15.6 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
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15.6.E1 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ ∫ 0 1 t b - 1 ⁒ ( 1 - t ) c - b - 1 ( 1 - z ⁒ t ) a ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 Euler-Gamma 𝑏 Euler-Gamma 𝑐 𝑏 superscript subscript 0 1 superscript 𝑑 𝑏 1 superscript 1 𝑑 𝑐 𝑏 1 superscript 1 𝑧 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma% \left(b\right)\Gamma\left(c-b\right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(% 1-zt)^{a}}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = \frac{1}{\EulerGamma@{b}\EulerGamma@{c-b}}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ c > β„œ ⁑ b , β„œ ⁑ b > 0 , β„œ ⁑ ( c - b ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence 𝑐 𝑏 formulae-sequence 𝑏 0 formulae-sequence 𝑐 𝑏 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>\Re b% ,\Re b>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(GAMMA(b)*GAMMA(c - b))*int(((t)^(b - 1)*(1 - t)^(c - b - 1))/((1 - z*t)^(a)), t = 0..1)

Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,Gamma[b]*Gamma[c - b]]*Integrate[Divide[(t)^(b - 1)*(1 - t)^(c - b - 1),(1 - z*t)^(a)], {t, 0, 1}, GenerateConditions->None]
Failure Successful
Failed [18 / 18]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = 3/2, c = 2, z = 1/2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = 1/2, c = 3/2, z = 1/2}

... skip entries to safe data
 Successful [Tested: 18]
15.6.E2 𝐅 ⁑ ( a , b ; c ; z ) = Ξ“ ⁑ ( 1 + b - c ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( b ) ⁒ ∫ 0 ( 1 + ) t b - 1 ⁒ ( t - 1 ) c - b - 1 ( 1 - z ⁒ t ) a ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 Euler-Gamma 1 𝑏 𝑐 2 πœ‹ imaginary-unit Euler-Gamma 𝑏 superscript subscript 0 limit-from 1 superscript 𝑑 𝑏 1 superscript 𝑑 1 𝑐 𝑏 1 superscript 1 𝑧 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1% +b-c\right)}{2\pi\mathrm{i}\Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t% -1)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = \frac{\EulerGamma@{1+b-c}}{2\pi\iunit\EulerGamma@{b}}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ b > 0 , β„œ ⁑ ( 1 + b - c ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence 𝑏 0 formulae-sequence 1 𝑏 𝑐 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re b>0,% \Re(1+b-c)>0,|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = (GAMMA(1 + b - c))/(2*Pi*I*GAMMA(b))*int(((t)^(b - 1)*(t - 1)^(c - b - 1))/((1 - z*t)^(a)), t = 0..(1 +))

Hypergeometric2F1Regularized[a, b, c, z] == Divide[Gamma[1 + b - c],2*Pi*I*Gamma[b]]*Integrate[Divide[(t)^(b - 1)*(t - 1)^(c - b - 1),(1 - z*t)^(a)], {t, 0, (1 +)}, GenerateConditions->None]
Error Failure - Error
15.6.E3 𝐅 ⁑ ( a , b ; c ; z ) = e - b ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( c - b ) ⁒ ∫ ∞ ( 0 + ) t b - 1 ⁒ ( t + 1 ) a - c ( t - z ⁒ t + 1 ) a ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 𝑒 𝑏 πœ‹ imaginary-unit Euler-Gamma 1 𝑏 2 πœ‹ imaginary-unit Euler-Gamma 𝑐 𝑏 superscript subscript limit-from 0 superscript 𝑑 𝑏 1 superscript 𝑑 1 π‘Ž 𝑐 superscript 𝑑 𝑧 𝑑 1 π‘Ž 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}% \frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{% \infty}^{(0+)}\frac{t^{b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = e^{-b\pi\iunit}\frac{\EulerGamma@{1-b}}{2\pi\iunit\EulerGamma@{c-b}}\int_{\infty}^{(0+)}\frac{t^{b-1}(t+1)^{a-c}}{(t-zt+1)^{a}}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( c - b ) > 0 , β„œ ⁑ ( 1 - b ) > 0 , β„œ ⁑ ( c - b ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence 𝑐 𝑏 0 formulae-sequence 1 𝑏 0 formulae-sequence 𝑐 𝑏 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re\left(c% -b\right)>0,\Re(1-b)>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = exp(- b*Pi*I)*(GAMMA(1 - b))/(2*Pi*I*GAMMA(c - b))*int(((t)^(b - 1)*(t + 1)^(a - c))/((t - z*t + 1)^(a)), t = infinity..(0 +))

Hypergeometric2F1Regularized[a, b, c, z] == Exp[- b*Pi*I]*Divide[Gamma[1 - b],2*Pi*I*Gamma[c - b]]*Integrate[Divide[(t)^(b - 1)*(t + 1)^(a - c),(t - z*t + 1)^(a)], {t, Infinity, (0 +)}, GenerateConditions->None]
Error Failure - Error
15.6.E4 𝐅 ⁑ ( a , b ; c ; z ) = e - b ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - b ) 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( c - b ) ⁒ ∫ 1 ( 0 + ) t b - 1 ⁒ ( 1 - t ) c - b - 1 ( 1 - z ⁒ t ) a ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 𝑒 𝑏 πœ‹ imaginary-unit Euler-Gamma 1 𝑏 2 πœ‹ imaginary-unit Euler-Gamma 𝑐 𝑏 superscript subscript 1 limit-from 0 superscript 𝑑 𝑏 1 superscript 1 𝑑 𝑐 𝑏 1 superscript 1 𝑧 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-b\pi\mathrm{i}}% \frac{\Gamma\left(1-b\right)}{2\pi\mathrm{i}\Gamma\left(c-b\right)}\int_{1}^{(% 0+)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = e^{-b\pi\iunit}\frac{\EulerGamma@{1-b}}{2\pi\iunit\EulerGamma@{c-b}}\int_{1}^{(0+)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( c - b ) > 0 , β„œ ⁑ ( 1 - b ) > 0 , β„œ ⁑ ( c - b ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence 𝑐 𝑏 0 formulae-sequence 1 𝑏 0 formulae-sequence 𝑐 𝑏 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re\left(c% -b\right)>0,\Re(1-b)>0,\Re(c-b)>0,|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = exp(- b*Pi*I)*(GAMMA(1 - b))/(2*Pi*I*GAMMA(c - b))*int(((t)^(b - 1)*(1 - t)^(c - b - 1))/((1 - z*t)^(a)), t = 1..(0 +))

Hypergeometric2F1Regularized[a, b, c, z] == Exp[- b*Pi*I]*Divide[Gamma[1 - b],2*Pi*I*Gamma[c - b]]*Integrate[Divide[(t)^(b - 1)*(1 - t)^(c - b - 1),(1 - z*t)^(a)], {t, 1, (0 +)}, GenerateConditions->None]
Error Failure - Error
15.6.E5 𝐅 ⁑ ( a , b ; c ; z ) = e - c ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( 1 - b ) ⁒ Ξ“ ⁑ ( 1 + b - c ) ⁒ 1 4 ⁒ Ο€ 2 ⁒ ∫ A ( 0 + , 1 + , 0 - , 1 - ) t b - 1 ⁒ ( 1 - t ) c - b - 1 ( 1 - z ⁒ t ) a ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript 𝑒 𝑐 πœ‹ imaginary-unit Euler-Gamma 1 𝑏 Euler-Gamma 1 𝑏 𝑐 1 4 superscript πœ‹ 2 superscript subscript 𝐴 limit-from 0 limit-from 1 limit-from 0 limit-from 1 superscript 𝑑 𝑏 1 superscript 1 𝑑 𝑐 𝑏 1 superscript 1 𝑧 𝑑 π‘Ž 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=e^{-c\pi\mathrm{i}}% \Gamma\left(1-b\right)\Gamma\left(1+b-c\right)\*\frac{1}{4\pi^{2}}\int_{A}^{(0% +,1+,0-,1-)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = e^{-c\pi\iunit}\EulerGamma@{1-b}\EulerGamma@{1+b-c}\*\frac{1}{4\pi^{2}}\int_{A}^{(0+,1+,0-,1-)}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( 1 - b ) > 0 , β„œ ⁑ ( 1 + b - c ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence 1 𝑏 0 formulae-sequence 1 𝑏 𝑐 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re(1-b)>0% ,\Re(1+b-c)>0,|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = exp(- c*Pi*I)*GAMMA(1 - b)*GAMMA(1 + b - c)*(1)/(4*(Pi)^(2))*int(((t)^(b - 1)*(1 - t)^(c - b - 1))/((1 - z*t)^(a)), t = A..(0 + , 1 + , 0 - , 1 -))

Hypergeometric2F1Regularized[a, b, c, z] == Exp[- c*Pi*I]*Gamma[1 - b]*Gamma[1 + b - c]*Divide[1,4*(Pi)^(2)]*Integrate[Divide[(t)^(b - 1)*(1 - t)^(c - b - 1),(1 - z*t)^(a)], {t, A, (0 + , 1 + , 0 - , 1 -)}, GenerateConditions->None]
Error Failure - Error
15.6.E6 𝐅 ⁑ ( a , b ; c ; z ) = 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( a + t ) ⁒ Ξ“ ⁑ ( b + t ) ⁒ Ξ“ ⁑ ( - t ) Ξ“ ⁑ ( c + t ) ⁒ ( - z ) t ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž Euler-Gamma 𝑏 superscript subscript imaginary-unit imaginary-unit Euler-Gamma π‘Ž 𝑑 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑑 Euler-Gamma 𝑐 𝑑 superscript 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)\Gamma\left(b\right)}\int_{-\mathrm{i}\infty}^{% \mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma\left(b+t\right)\Gamma\left% (-t\right)}{\Gamma\left(c+t\right)}(-z)^{t}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = \frac{1}{2\pi\iunit\EulerGamma@{a}\EulerGamma@{b}}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{-t}}{\EulerGamma@{c+t}}(-z)^{t}\diff{t}		 | ph ⁑ ( - z ) | < Ο€ , β„œ ⁑ a > 0 , β„œ ⁑ b > 0 , β„œ ⁑ ( a + t ) > 0 , β„œ ⁑ ( b + t ) > 0 , β„œ ⁑ ( - t ) > 0 , β„œ ⁑ ( c + t ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 𝑧 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 0 formulae-sequence π‘Ž 𝑑 0 formulae-sequence 𝑏 𝑑 0 formulae-sequence 𝑑 0 formulae-sequence 𝑐 𝑑 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi,\Re a>0,\Re b% >0,\Re(a+t)>0,\Re(b+t)>0,\Re(-t)>0,\Re(c+t)>0,|z|<1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(2*Pi*I*GAMMA(a)*GAMMA(b))*int((GAMMA(a + t)*GAMMA(b + t)*GAMMA(- t))/(GAMMA(c + t))*(- z)^(t), t = - I*infinity..I*infinity)

Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,2*Pi*I*Gamma[a]*Gamma[b]]*Integrate[Divide[Gamma[a + t]*Gamma[b + t]*Gamma[- t],Gamma[c + t]]*(- z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Manual Skip! Skipped - Because timed out
15.6.E7 𝐅 ⁑ ( a , b ; c ; z ) = 1 2 ⁒ Ο€ ⁒ i ⁒ Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ Ξ“ ⁑ ( c - a ) ⁒ Ξ“ ⁑ ( c - b ) ⁒ ∫ - i ⁒ ∞ i ⁒ ∞ Ξ“ ⁑ ( a + t ) ⁒ Ξ“ ⁑ ( b + t ) ⁒ Ξ“ ⁑ ( c - a - b - t ) ⁒ Ξ“ ⁑ ( - t ) ⁒ ( 1 - z ) t ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž Euler-Gamma 𝑏 Euler-Gamma 𝑐 π‘Ž Euler-Gamma 𝑐 𝑏 superscript subscript imaginary-unit imaginary-unit Euler-Gamma π‘Ž 𝑑 Euler-Gamma 𝑏 𝑑 Euler-Gamma 𝑐 π‘Ž 𝑏 𝑑 Euler-Gamma 𝑑 superscript 1 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{2\pi% \mathrm{i}\Gamma\left(a\right)\Gamma\left(b\right)\Gamma\left(c-a\right)\Gamma% \left(c-b\right)}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\Gamma\left(a+t% \right)\Gamma\left(b+t\right)\Gamma\left(c-a-b-t\right)\Gamma\left(-t\right)(1% -z)^{t}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = \frac{1}{2\pi\iunit\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c-a}\EulerGamma@{c-b}}\int_{-\iunit\infty}^{\iunit\infty}\EulerGamma@{a+t}\EulerGamma@{b+t}\EulerGamma@{c-a-b-t}\EulerGamma@{-t}(1-z)^{t}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ ( a + t ) > 0 , β„œ ⁑ ( b + t ) > 0 , β„œ ⁑ ( c - a - b - t ) > 0 , β„œ ⁑ ( - t ) > 0 , β„œ ⁑ a > 0 , β„œ ⁑ b > 0 , β„œ ⁑ ( c - a ) > 0 , β„œ ⁑ ( c - b ) > 0 , | z | < 1 , β„œ ⁑ ( c + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence π‘Ž 𝑑 0 formulae-sequence 𝑏 𝑑 0 formulae-sequence 𝑐 π‘Ž 𝑏 𝑑 0 formulae-sequence 𝑑 0 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 0 formulae-sequence 𝑐 π‘Ž 0 formulae-sequence 𝑐 𝑏 0 formulae-sequence 𝑧 1 𝑐 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re(a+t)>0% ,\Re(b+t)>0,\Re(c-a-b-t)>0,\Re(-t)>0,\Re a>0,\Re b>0,\Re(c-a)>0,\Re(c-b)>0,|z|% <1,\Re(c+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(2*Pi*I*GAMMA(a)*GAMMA(b)*GAMMA(c - a)*GAMMA(c - b))*int(GAMMA(a + t)*GAMMA(b + t)*GAMMA(c - a - b - t)*GAMMA(- t)*(1 - z)^(t), t = - I*infinity..I*infinity)

Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,2*Pi*I*Gamma[a]*Gamma[b]*Gamma[c - a]*Gamma[c - b]]*Integrate[Gamma[a + t]*Gamma[b + t]*Gamma[c - a - b - t]*Gamma[- t]*(1 - z)^(t), {t, - I*Infinity, I*Infinity}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
15.6.E8 𝐅 ⁑ ( a , b ; c ; z ) = 1 Ξ“ ⁑ ( c - d ) ⁒ ∫ 0 1 𝐅 ⁑ ( a , b ; d ; z ⁒ t ) ⁒ t d - 1 ⁒ ( 1 - t ) c - d - 1 ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 1 Euler-Gamma 𝑐 𝑑 superscript subscript 0 1 scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑑 𝑧 𝑑 superscript 𝑑 𝑑 1 superscript 1 𝑑 𝑐 𝑑 1 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma% \left(c-d\right)}\int_{0}^{1}\mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-% 1}\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = \frac{1}{\EulerGamma@{c-d}}\int_{0}^{1}\hyperOlverF@{a}{b}{d}{zt}t^{d-1}(1-t)^{c-d-1}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ c > β„œ ⁑ d , β„œ ⁑ d > 0 , β„œ ⁑ ( c - d ) > 0 , | z | < 1 , | ( z ⁒ t ) | < 1 , β„œ ⁑ ( c + s ) > 0 , β„œ ⁑ ( d + s ) > 0 formulae-sequence phase 1 𝑧 formulae-sequence 𝑐 𝑑 formulae-sequence 𝑑 0 formulae-sequence 𝑐 𝑑 0 formulae-sequence 𝑧 1 formulae-sequence 𝑧 𝑑 1 formulae-sequence 𝑐 𝑠 0 𝑑 𝑠 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>\Re d% ,\Re d>0,\Re(c-d)>0,|z|<1,|(zt)|<1,\Re(c+s)>0,\Re(d+s)>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = (1)/(GAMMA(c - d))*int(hypergeom([a, b], [d], z*t)/GAMMA(d)*(t)^(d - 1)*(1 - t)^(c - d - 1), t = 0..1)

Hypergeometric2F1Regularized[a, b, c, z] == Divide[1,Gamma[c - d]]*Integrate[Hypergeometric2F1Regularized[a, b, d, z*t]*(t)^(d - 1)*(1 - t)^(c - d - 1), {t, 0, 1}, GenerateConditions->None]
Failure Successful
Failed [252 / 252]
Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = -3/2, c = 3/2, d = 1/2*3^(1/2)+1/2*I, z = 1/2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {a = -3/2, b = -3/2, c = 3/2, d = 1/2-1/2*I*3^(1/2), z = 1/2}

... skip entries to safe data
 Successful [Tested: 252]
15.6.E9 𝐅 ⁑ ( a , b ; c ; z ) = ∫ 0 1 t d - 1 ⁒ ( 1 - t ) c - d - 1 ( 1 - z ⁒ t ) a + b - Ξ» ⁒ 𝐅 ⁑ ( Ξ» - a , Ξ» - b d ; z ⁒ t ) ⁒ 𝐅 ⁑ ( a + b - Ξ» , Ξ» - d c - d ; ( 1 - t ) ⁒ z 1 - z ⁒ t ) ⁒ d t scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 superscript subscript 0 1 superscript 𝑑 𝑑 1 superscript 1 𝑑 𝑐 𝑑 1 superscript 1 𝑧 𝑑 π‘Ž 𝑏 πœ† scaled-hypergeometric-bold-F πœ† π‘Ž πœ† 𝑏 𝑑 𝑧 𝑑 scaled-hypergeometric-bold-F π‘Ž 𝑏 πœ† πœ† 𝑑 𝑐 𝑑 1 𝑑 𝑧 1 𝑧 𝑑 𝑑 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t% ^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b% \atop d};zt\right)\mathbf{F}\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)% z}{1-zt}\right)\mathrm{d}t}}
\hyperOlverF@{a}{b}{c}{z} = \int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\hyperOlverF@@{\lambda-a}{\lambda-b}{d}{zt}\hyperOlverF@@{a+b-\lambda}{\lambda-d}{c-d}{\frac{(1-t)z}{1-zt}}\diff{t}		 | ph ⁑ ( 1 - z ) | < Ο€ , β„œ ⁑ c > β„œ ⁑ d , β„œ ⁑ d > 0 formulae-sequence phase 1 𝑧 formulae-sequence 𝑐 𝑑 𝑑 0 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi,\Re c>\Re d% ,\Re d>0}} 
hypergeom([a, b], [c], z)/GAMMA(c) = int(((t)^(d - 1)*(1 - t)^(c - d - 1))/((1 - z*t)^(a + b - lambda))*hypergeom([lambda - a, lambda - b], [d], z*t)/GAMMA(d)*hypergeom([a + b - lambda, lambda - d], [c - d], ((1 - t)*z)/(1 - z*t))/GAMMA(c - d), t = 0..1)

Hypergeometric2F1Regularized[a, b, c, z] == Integrate[Divide[(t)^(d - 1)*(1 - t)^(c - d - 1),(1 - z*t)^(a + b - \[Lambda])]*Hypergeometric2F1Regularized[\[Lambda]- a, \[Lambda]- b, d, z*t]*Hypergeometric2F1Regularized[a + b - \[Lambda], \[Lambda]- d, c - d, Divide[(1 - t)*z,1 - z*t]], {t, 0, 1}, GenerateConditions->None]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
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