15.4

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
15.4.E1	${\displaystyle{\displaystyle F\left(1,1;2;z\right)=-z^{-1}\ln\left(1-z\right)}}$ \hyperF@{1}{1}{2}{z} = -z^{-1}\ln@{1-z}		hypergeom([1, 1], [2], z) = - (z)^(- 1)* ln(1 - z)	Hypergeometric2F1[1, 1, 2, z] == - (z)^(- 1)* Log[1 - z]	Successful	Successful	-	Successful [Tested: 7]
15.4.E2	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)=\frac{1}{2z}\ln\left(\frac{1+z}{1-z}\right)}}$ \hyperF@{\tfrac{1}{2}}{1}{\tfrac{3}{2}}{z^{2}} = \frac{1}{2z}\ln@{\frac{1+z}{1-z}}		hypergeom([(1)/(2), 1], [(3)/(2)], (z)^(2)) = (1)/(2*z)*ln((1 + z)/(1 - z))	Hypergeometric2F1[Divide[1,2], 1, Divide[3,2], (z)^(2)] == Divide[1,2*z]*Log[Divide[1 + z,1 - z]]	Failure	Failure	Failed [2 / 7] Result: .1e-9-2.094395103*I Test Values: {z = 3/2} Result: 0.-1.570796327*I Test Values: {z = 2}	Failed [2 / 7] Result: Complex[1.1102230246251565*^-16, -2.0943951023931953] Test Values: {Rule[z, 1.5]} Result: Complex[0.0, -1.5707963267948966] Test Values: {Rule[z, 2]}
15.4.E3	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)=z^{-1}\operatorname{arctan}z}}$ \hyperF@{\tfrac{1}{2}}{1}{\tfrac{3}{2}}{-z^{2}} = z^{-1}\atan@@{z}		hypergeom([(1)/(2), 1], [(3)/(2)], - (z)^(2)) = (z)^(- 1)* arctan(z)	Hypergeometric2F1[Divide[1,2], 1, Divide[3,2], - (z)^(2)] == (z)^(- 1)* ArcTan[z]	Successful	Successful	-	Successful [Tested: 7]
15.4.E4	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}\right)=z^{-1}\operatorname{arcsin}z}}$ \hyperF@{\tfrac{1}{2}}{\tfrac{1}{2}}{\tfrac{3}{2}}{z^{2}} = z^{-1}\asin@@{z}		hypergeom([(1)/(2), (1)/(2)], [(3)/(2)], (z)^(2)) = (z)^(- 1)* arcsin(z)	Hypergeometric2F1[Divide[1,2], Divide[1,2], Divide[3,2], (z)^(2)] == (z)^(- 1)* ArcSin[z]	Successful	Successful	-	Successful [Tested: 7]
15.4.E5	${\displaystyle{\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}\right)=z^{-1}\ln\left(z+\sqrt{1+z^{2}}\right)}}$ \hyperF@{\tfrac{1}{2}}{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = z^{-1}\ln@{z+\sqrt{1+z^{2}}}		hypergeom([(1)/(2), (1)/(2)], [(3)/(2)], - (z)^(2)) = (z)^(- 1)* ln(z +sqrt(1 + (z)^(2)))	Hypergeometric2F1[Divide[1,2], Divide[1,2], Divide[3,2], - (z)^(2)] == (z)^(- 1)* Log[z +Sqrt[1 + (z)^(2)]]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
15.4#Ex1	${\displaystyle{\displaystyle F\left(a,b;a;z\right)=(1-z)^{-b}}}$ \hyperF@{a}{b}{a}{z} = (1-z)^{-b}		hypergeom([a, b], [a], z) = (1 - z)^(- b)	Hypergeometric2F1[a, b, a, z] == (1 - z)^(- b)	Successful	Successful	-	Successful [Tested: 252]
15.4#Ex2	${\displaystyle{\displaystyle F\left(a,b;b;z\right)=(1-z)^{-a}}}$ \hyperF@{a}{b}{b}{z} = (1-z)^{-a}		hypergeom([a, b], [b], z) = (1 - z)^(- a)	Hypergeometric2F1[a, b, b, z] == (1 - z)^(- a)	Successful	Successful	-	Successful [Tested: 252]
15.4.E7	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac{1}{2}\left((1+z)^{-2a}+(1-z)^{-2a}\right)}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{1}{2}}{z^{2}} = \tfrac{1}{2}\left((1+z)^{-2a}+(1-z)^{-2a}\right)		hypergeom([a, (1)/(2)+ a], [(1)/(2)], (z)^(2)) = (1)/(2)*((1 + z)^(- 2*a)+(1 - z)^(- 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[1,2], (z)^(2)] == Divide[1,2]*((1 + z)^(- 2*a)+(1 - z)^(- 2*a))	Successful	Successful	-	Successful [Tested: 42]
15.4.E8	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\cos\left(2az\right)}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{1}{2}}{-\tan^{2}@@{z}} = (\cos@@{z})^{2a}\cos@{2az}		hypergeom([a, (1)/(2)+ a], [(1)/(2)], - (tan(z))^(2)) = (cos(z))^(2*a)* cos(2*a*z)	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[1,2], - (Tan[z])^(2)] == (Cos[z])^(2*a)* Cos[2*a*z]	Failure	Failure	Successful [Tested: 42]	Successful [Tested: 42]
15.4.E9	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{1}{(2-4a)z}\left((1+z)^{1-2a}-(1-z)^{1-2a}\right)}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{3}{2}}{z^{2}} = \frac{1}{(2-4a)z}\left((1+z)^{1-2a}-(1-z)^{1-2a}\right)		hypergeom([a, (1)/(2)+ a], [(3)/(2)], (z)^(2)) = (1)/((2 - 4*a)*z)*((1 + z)^(1 - 2*a)-(1 - z)^(1 - 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2], (z)^(2)] == Divide[1,(2 - 4*a)*z]*((1 + z)^(1 - 2*a)-(1 - z)^(1 - 2*a))	Successful	Successful	-	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E10	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\tan^{2}}z\right)=(\cos z)^{2a}\frac{\sin\left((1-2a)z\right)}{(1-2a)\sin z}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{3}{2}}{-\tan^{2}@@{z}} = (\cos@@{z})^{2a}\frac{\sin@{(1-2a)z}}{(1-2a)\sin@@{z}}		hypergeom([a, (1)/(2)+ a], [(3)/(2)], - (tan(z))^(2)) = (cos(z))^(2*a)*(sin((1 - 2*a)*z))/((1 - 2*a)*sin(z))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2], - (Tan[z])^(2)] == (Cos[z])^(2*a)*Divide[Sin[(1 - 2*a)*z],(1 - 2*a)*Sin[z]]	Failure	Failure	Failed [7 / 42] Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.4.E11	${\displaystyle{\displaystyle F\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right)}}$ \hyperF@{-a}{a}{\tfrac{1}{2}}{-z^{2}} = \tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right)		hypergeom([- a, a], [(1)/(2)], - (z)^(2)) = (1)/(2)*((sqrt(1 + (z)^(2))+ z)^(2*a)+(sqrt(1 + (z)^(2))- z)^(2*a))	Hypergeometric2F1[- a, a, Divide[1,2], - (z)^(2)] == Divide[1,2]*((Sqrt[1 + (z)^(2)]+ z)^(2*a)+(Sqrt[1 + (z)^(2)]- z)^(2*a))	Failure	Failure	Successful [Tested: 42]	Successful [Tested: 42]
15.4.E12	${\displaystyle{\displaystyle F\left(-a,a;\tfrac{1}{2};{\sin^{2}}z\right)=\cos\left(2az\right)}}$ \hyperF@{-a}{a}{\tfrac{1}{2}}{\sin^{2}@@{z}} = \cos@{2az}		hypergeom([- a, a], [(1)/(2)], (sin(z))^(2)) = cos(2*a*z)	Hypergeometric2F1[- a, a, Divide[1,2], (Sin[z])^(2)] == Cos[2*a*z]	Failure	Failure	Failed [4 / 42] Result: -1.920340573 Test Values: {a = -3/2, z = 2} Result: -1.920340573 Test Values: {a = 3/2, z = 2} ... skip entries to safe data	Failed [4 / 42] Result: -1.9203405733007322 Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: -1.9203405733007322 Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.4.E13	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right)}}$ \hyperF@{a}{1-a}{\tfrac{1}{2}}{-z^{2}} = \frac{1}{2\sqrt{1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right)		hypergeom([a, 1 - a], [(1)/(2)], - (z)^(2)) = (1)/(2*sqrt(1 + (z)^(2)))*((sqrt(1 + (z)^(2))+ z)^(2*a - 1)+(sqrt(1 + (z)^(2))- z)^(2*a - 1))	Hypergeometric2F1[a, 1 - a, Divide[1,2], - (z)^(2)] == Divide[1,2*Sqrt[1 + (z)^(2)]]*((Sqrt[1 + (z)^(2)]+ z)^(2*a - 1)+(Sqrt[1 + (z)^(2)]- z)^(2*a - 1))	Successful	Failure	-	Successful [Tested: 42]
15.4.E14	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{1}{2};{\sin^{2}}z\right)=\frac{\cos\left((2a-1)z\right)}{\cos z}}}$ \hyperF@{a}{1-a}{\tfrac{1}{2}}{\sin^{2}@@{z}} = \frac{\cos@{(2a-1)z}}{\cos@@{z}}		hypergeom([a, 1 - a], [(1)/(2)], (sin(z))^(2)) = (cos((2*a - 1)*z))/(cos(z))	Hypergeometric2F1[a, 1 - a, Divide[1,2], (Sin[z])^(2)] == Divide[Cos[(2*a - 1)*z],Cos[z]]	Failure	Failure	Failed [4 / 42] Result: -.6992725697 Test Values: {a = -3/2, z = 2} Result: -3.141408577 Test Values: {a = 3/2, z = 2} ... skip entries to safe data	Failed [4 / 42] Result: -0.6992725693452728 Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: -3.1414085772561924 Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.4.E15	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z}\left(\left(\sqrt{1+z^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right)}}$ \hyperF@{a}{1-a}{\tfrac{3}{2}}{-z^{2}} = \frac{1}{(2-4a)z}\left(\left(\sqrt{1+z^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right)		hypergeom([a, 1 - a], [(3)/(2)], - (z)^(2)) = (1)/((2 - 4*a)*z)*((sqrt(1 + (z)^(2))+ z)^(1 - 2*a)-(sqrt(1 + (z)^(2))- z)^(1 - 2*a))	Hypergeometric2F1[a, 1 - a, Divide[3,2], - (z)^(2)] == Divide[1,(2 - 4*a)*z]*((Sqrt[1 + (z)^(2)]+ z)^(1 - 2*a)-(Sqrt[1 + (z)^(2)]- z)^(1 - 2*a))	Failure	Failure	Failed [7 / 42] Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = 1/2*3^(1/2)+1/2*I} Result: Float(undefined)+Float(undefined)*I Test Values: {a = 1/2, z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [7 / 42] Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.4.E16	${\displaystyle{\displaystyle F\left(a,1-a;\tfrac{3}{2};{\sin^{2}}z\right)=\frac{\sin\left((2a-1)z\right)}{(2a-1)\sin z}}}$ \hyperF@{a}{1-a}{\tfrac{3}{2}}{\sin^{2}@@{z}} = \frac{\sin@{(2a-1)z}}{(2a-1)\sin@@{z}}		hypergeom([a, 1 - a], [(3)/(2)], (sin(z))^(2)) = (sin((2*a - 1)*z))/((2*a - 1)*sin(z))	Hypergeometric2F1[a, 1 - a, Divide[3,2], (Sin[z])^(2)] == Divide[Sin[(2*a - 1)*z],(2*a - 1)*Sin[z]]	Successful	Failure	-	Failed [10 / 42] Result: -0.5440234501032235 Test Values: {Rule[a, -1.5], Rule[z, 2]} Result: 0.8322936730942848 Test Values: {Rule[a, 1.5], Rule[z, 2]} ... skip entries to safe data
15.4.E17	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{-2a}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{1+2a}{z} = \left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{-2a}		hypergeom([a, (1)/(2)+ a], [1 + 2*a], z) = ((1)/(2)+(1)/(2)*sqrt(1 - z))^(- 2*a)	Hypergeometric2F1[a, Divide[1,2]+ a, 1 + 2*a, z] == (Divide[1,2]+Divide[1,2]*Sqrt[1 - z])^(- 2*a)	Successful	Successful	-	Failed [20 / 42] Result: Complex[-0.02099232729741518, 0.019754284780044207] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.009306933376070914, 0.00445671804147707] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E18	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{1-2a}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{2a}{z} = \frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{1-2a}		hypergeom([a, (1)/(2)+ a], [2*a], z) = (1)/(sqrt(1 - z))*((1)/(2)+(1)/(2)*sqrt(1 - z))^(1 - 2*a)	Hypergeometric2F1[a, Divide[1,2]+ a, 2*a, z] == Divide[1,Sqrt[1 - z]]*(Divide[1,2]+Divide[1,2]*Sqrt[1 - z])^(1 - 2*a)	Successful	Successful	-	Failed [17 / 42] Result: Complex[0.009435141098616318, 0.007866769593881467] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.0016763074528445276, -3.2228425612285116*^-4] Test Values: {Rule[a, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E19	${\displaystyle{\displaystyle F\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}}}$ \hyperF@{a+1}{b}{a}{z} = \left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}		hypergeom([a + 1, b], [a], z) = (1 -(1 -((b)/(a)))*z)*(1 - z)^(- 1 - b)	Hypergeometric2F1[a + 1, b, a, z] == (1 -(1 -(Divide[b,a]))*z)*(1 - z)^(- 1 - b)	Successful	Successful	-	Failed [34 / 252] Result: Complex[-0.041984654594830306, 0.03950856956008836] Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.018613866752141828, 0.008913436082954251] Test Values: {Rule[a, -2], Rule[b, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
15.4.E20	${\displaystyle{\displaystyle F\left(a,b;c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}}}$ \hyperF@{a}{b}{c}{1} = \frac{\EulerGamma@{c}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}	${\displaystyle{\displaystyle\Re c>0,\Re(c-a-b)>0,\Re(c-a)>0,\Re(c-b)>0}}$	hypergeom([a, b], [c], 1) = (GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))	Hypergeometric2F1[a, b, c, 1] == Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]	Failure	Failure	Successful [Tested: 47]	Successful [Tested: 47]
15.4.E21	${\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln\left(1-z\right)}=\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$ \lim_{z\to 1-}\frac{\hyperF@{a}{b}{a+b}{z}}{-\ln@{1-z}} = \frac{\EulerGamma@{a+b}}{\EulerGamma@{a}\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re(a+b)>0,\Re a>0,\Re b>0}}$	limit((hypergeom([a, b], [a + b], z))/(- ln(1 - z)), z = 1, left) = (GAMMA(a + b))/(GAMMA(a)*GAMMA(b))	Limit[Divide[Hypergeometric2F1[a, b, a + b, z],- Log[1 - z]], z -> 1, Direction -> "FromBelow", GenerateConditions->None] == Divide[Gamma[a + b],Gamma[a]*Gamma[b]]	Successful	Successful	-	Successful [Tested: 9]
15.4.E22	${\displaystyle{\displaystyle\lim_{z\to 1-}(1-z)^{a+b-c}\left(F\left(a,b;c;z\right)-\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\right)=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$ \lim_{z\to 1-}(1-z)^{a+b-c}\left(\hyperF@{a}{b}{c}{z}-\frac{\EulerGamma@{c}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}\right) = \frac{\EulerGamma@{c}\EulerGamma@{a+b-c}}{\EulerGamma@{a}\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re c>0,\Re(c-a-b)>0,\Re(c-a)>0,\Re(c-b)>0,\Re(a+b-c)>0,\Re a>0,\Re b>0}}$	limit((1 - z)^(a + b - c)*(hypergeom([a, b], [c], z)-(GAMMA(c)*GAMMA(c - a - b))/(GAMMA(c - a)*GAMMA(c - b))), z = 1, left) = (GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))	Limit[(1 - z)^(a + b - c)*(Hypergeometric2F1[a, b, c, z]-Divide[Gamma[c]*Gamma[c - a - b],Gamma[c - a]*Gamma[c - b]]), z -> 1, Direction -> "FromBelow", GenerateConditions->None] == Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]	Failure	Aborted	Error	Skipped - Because timed out
15.4.E23	${\displaystyle{\displaystyle\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{c-a-b}}=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}}}$ \lim_{z\to 1-}\frac{\hyperF@{a}{b}{c}{z}}{(1-z)^{c-a-b}} = \frac{\EulerGamma@{c}\EulerGamma@{a+b-c}}{\EulerGamma@{a}\EulerGamma@{b}}	${\displaystyle{\displaystyle\Re c>0,\Re(a+b-c)>0,\Re a>0,\Re b>0}}$	limit((hypergeom([a, b], [c], z))/((1 - z)^(c - a - b)), z = 1, left) = (GAMMA(c)*GAMMA(a + b - c))/(GAMMA(a)*GAMMA(b))	Limit[Divide[Hypergeometric2F1[a, b, c, z],(1 - z)^(c - a - b)], z -> 1, Direction -> "FromBelow", GenerateConditions->None] == Divide[Gamma[c]*Gamma[a + b - c],Gamma[a]*Gamma[b]]	Failure	Failure	Manual Skip!	Successful [Tested: 23]
15.4.E24	${\displaystyle{\displaystyle F\left(-n,b;c;1\right)=\frac{{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}}}$ \hyperF@{-n}{b}{c}{1} = \frac{\Pochhammersym{c-b}{n}}{\Pochhammersym{c}{n}}		hypergeom([- n, b], [c], 1) = (pochhammer(c - b, n))/(pochhammer(c, n))	Hypergeometric2F1[- n, b, c, 1] == Divide[Pochhammer[c - b, n],Pochhammer[c, n]]	Failure	Failure	Error	Failed [6 / 36] Result: Indeterminate Test Values: {Rule[b, -1.5], Rule[c, -2], Rule[n, 3]} Result: Indeterminate Test Values: {Rule[b, 1.5], Rule[c, -2], Rule[n, 3]} ... skip entries to safe data
15.4.E25	${\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{\Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a\right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)}}}$ \sum_{n=-\infty}^{\infty}\frac{\EulerGamma@{a+n}\EulerGamma@{b+n}}{\EulerGamma@{c+n}\EulerGamma@{d+n}} = \frac{\pi^{2}}{\sin@{\pi a}\sin@{\pi b}}\*\frac{\EulerGamma@{c+d-a-b-1}}{\EulerGamma@{c-a}\EulerGamma@{d-a}\EulerGamma@{c-b}\EulerGamma@{d-b}}	${\displaystyle{\displaystyle\Re(a+n)>0,\Re(b+n)>0,\Re(c+n)>0,\Re(d+n)>0,\Re(c+d-a-b-1)>0,\Re(c-a)>0,\Re(d-a)>0,\Re(c-b)>0,\Re(d-b)>0}}$	sum((GAMMA(a + n)*GAMMA(b + n))/(GAMMA(c + n)*GAMMA(d + n)), n = - infinity..infinity) = ((Pi)^(2))/(sin(Pi*a)*sin(Pi*b))*(GAMMA(c + d - a - b - 1))/(GAMMA(c - a)*GAMMA(d - a)*GAMMA(c - b)*GAMMA(d - b))	Sum[Divide[Gamma[a + n]*Gamma[b + n],Gamma[c + n]*Gamma[d + n]], {n, - Infinity, Infinity}, GenerateConditions->None] == Divide[(Pi)^(2),Sin[Pi*a]*Sin[Pi*b]]*Divide[Gamma[c + d - a - b - 1],Gamma[c - a]*Gamma[d - a]*Gamma[c - b]*Gamma[d - b]]	Failure	Aborted	Manual Skip!	Failed [160 / 281] Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[c, 1.5], Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Indeterminate Test Values: {Rule[a, -1.5], Rule[b, -2], Rule[c, 1.5], Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]} ... skip entries to safe data
15.4.E26	${\displaystyle{\displaystyle F\left(a,b;a-b+1;-1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}a+1\right)}{\Gamma\left(a+1\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}}}$ \hyperF@{a}{b}{a-b+1}{-1} = \frac{\EulerGamma@{a-b+1}\EulerGamma@{\tfrac{1}{2}a+1}}{\EulerGamma@{a+1}\EulerGamma@{\tfrac{1}{2}a-b+1}}	${\displaystyle{\displaystyle\Re(a-b+1)>0,\Re(\tfrac{1}{2}a+1)>0,\Re(a+1)>0,\Re(\tfrac{1}{2}a-b+1)>0}}$	hypergeom([a, b], [a - b + 1], - 1) = (GAMMA(a - b + 1)*GAMMA((1)/(2)*a + 1))/(GAMMA(a + 1)*GAMMA((1)/(2)*a - b + 1))	Hypergeometric2F1[a, b, a - b + 1, - 1] == Divide[Gamma[a - b + 1]*Gamma[Divide[1,2]*a + 1],Gamma[a + 1]*Gamma[Divide[1,2]*a - b + 1]]	Successful	Successful	-	Successful [Tested: 17]
15.4.E27	${\displaystyle{\displaystyle F\left(1,a;a+1;-1\right)=\tfrac{1}{2}a\left(\psi\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)-\psi\left(\tfrac{1}{2}a\right)\right)}}$ \hyperF@{1}{a}{a+1}{-1} = \tfrac{1}{2}a\left(\digamma@{\tfrac{1}{2}a+\tfrac{1}{2}}-\digamma@{\tfrac{1}{2}a}\right)		hypergeom([1, a], [a + 1], - 1) = (1)/(2)*a*(Psi((1)/(2)*a +(1)/(2))- Psi((1)/(2)*a))	Hypergeometric2F1[1, a, a + 1, - 1] == Divide[1,2]*a*(PolyGamma[Divide[1,2]*a +Divide[1,2]]- PolyGamma[Divide[1,2]*a])	Successful	Successful	-	Successful [Tested: 6]
15.4.E28	${\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};\tfrac{1}{2}\right)=\sqrt{\pi}\frac{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}}}$ \hyperF@{a}{b}{\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}}{\tfrac{1}{2}} = \sqrt{\pi}\frac{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}b+\tfrac{1}{2}}}	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}b+\tfrac{1}{2})>0}}$	hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b +(1)/(2)], (1)/(2)) = sqrt(Pi)*(GAMMA((1)/(2)*a +(1)/(2)*b +(1)/(2)))/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b +(1)/(2)))	Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b +Divide[1,2], Divide[1,2]] == Sqrt[Pi]*Divide[Gamma[Divide[1,2]*a +Divide[1,2]*b +Divide[1,2]],Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b +Divide[1,2]]]	Successful	Successful	-	Successful [Tested: 15]
15.4.E29	${\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}\right)=\frac{2\sqrt{\pi}}{a-b}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}-\frac{1}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b\right)}\right)}}$ \hyperF@{a}{b}{\tfrac{1}{2}a+\tfrac{1}{2}b+1}{\tfrac{1}{2}} = \frac{2\sqrt{\pi}}{a-b}\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b+1}\*\left(\frac{1}{\EulerGamma@{\tfrac{1}{2}a}\EulerGamma@{\tfrac{1}{2}b+\tfrac{1}{2}}}-\frac{1}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}}\EulerGamma@{\tfrac{1}{2}b}}\right)	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}a+\tfrac{1}{2}b+1)>0,\Re(\tfrac{1}{2}a)>0,\Re(\tfrac{1}{2}b+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2})>0,\Re(\tfrac{1}{2}b)>0}}$	hypergeom([a, b], [(1)/(2)*a +(1)/(2)*b + 1], (1)/(2)) = (2*sqrt(Pi))/(a - b)*GAMMA((1)/(2)*a +(1)/(2)*b + 1)*((1)/(GAMMA((1)/(2)*a)*GAMMA((1)/(2)*b +(1)/(2)))-(1)/(GAMMA((1)/(2)*a +(1)/(2))*GAMMA((1)/(2)*b)))	Hypergeometric2F1[a, b, Divide[1,2]*a +Divide[1,2]*b + 1, Divide[1,2]] == Divide[2*Sqrt[Pi],a - b]*Gamma[Divide[1,2]*a +Divide[1,2]*b + 1]*(Divide[1,Gamma[Divide[1,2]*a]*Gamma[Divide[1,2]*b +Divide[1,2]]]-Divide[1,Gamma[Divide[1,2]*a +Divide[1,2]]*Gamma[Divide[1,2]*b]])	Failure	Failure	Failed [3 / 9] Result: Float(-infinity) Test Values: {a = 3/2, b = 3/2} Result: Float(undefined) Test Values: {a = 1/2, b = 1/2} ... skip entries to safe data	Failed [3 / 9] Result: Indeterminate Test Values: {Rule[a, 1.5], Rule[b, 1.5]} Result: Indeterminate Test Values: {Rule[a, 0.5], Rule[b, 0.5]} ... skip entries to safe data
15.4.E30	${\displaystyle{\displaystyle F\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{1-b}\sqrt{\pi}\Gamma\left(b\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\Gamma\left(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}\right)}}}$ \hyperF@{a}{1-a}{b}{\tfrac{1}{2}} = \frac{2^{1-b}\sqrt{\pi}\EulerGamma@{b}}{\EulerGamma@{\tfrac{1}{2}a+\tfrac{1}{2}b}\EulerGamma@{\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}}}	${\displaystyle{\displaystyle\Re b>0,\Re(\tfrac{1}{2}a+\tfrac{1}{2}b)>0,\Re(\tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2})>0}}$	hypergeom([a, 1 - a], [b], (1)/(2)) = ((2)^(1 - b)*sqrt(Pi)*GAMMA(b))/(GAMMA((1)/(2)*a +(1)/(2)*b)*GAMMA((1)/(2)*b -(1)/(2)*a +(1)/(2)))	Hypergeometric2F1[a, 1 - a, b, Divide[1,2]] == Divide[(2)^(1 - b)*Sqrt[Pi]*Gamma[b],Gamma[Divide[1,2]*a +Divide[1,2]*b]*Gamma[Divide[1,2]*b -Divide[1,2]*a +Divide[1,2]]]	Successful	Failure	-	Successful [Tested: 10]
15.4.E31	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}\right)^{-2a}\frac{\Gamma\left(\tfrac{4}{3}\right)\Gamma\left(\tfrac{3}{2}-2a\right)}{\Gamma\left(\tfrac{3}{2}\right)\Gamma\left(\tfrac{4}{3}-2a\right)}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{3}{2}-2a}{-\tfrac{1}{3}} = \left(\frac{8}{9}\right)^{-2a}\frac{\EulerGamma@{\tfrac{4}{3}}\EulerGamma@{\tfrac{3}{2}-2a}}{\EulerGamma@{\tfrac{3}{2}}\EulerGamma@{\tfrac{4}{3}-2a}}	${\displaystyle{\displaystyle\Re(\tfrac{3}{2}-2a)>0,\Re(\tfrac{4}{3}-2a)>0}}$	hypergeom([a, (1)/(2)+ a], [(3)/(2)- 2*a], -(1)/(3)) = ((8)/(9))^(- 2*a)*(GAMMA((4)/(3))*GAMMA((3)/(2)- 2*a))/(GAMMA((3)/(2))*GAMMA((4)/(3)- 2*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[3,2]- 2*a, -Divide[1,3]] == (Divide[8,9])^(- 2*a)*Divide[Gamma[Divide[4,3]]*Gamma[Divide[3,2]- 2*a],Gamma[Divide[3,2]]*Gamma[Divide[4,3]- 2*a]]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 4]
15.4.E32	${\displaystyle{\displaystyle F\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;\tfrac{1}{9}\right)=\sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\Gamma\left(\tfrac{5}{6}+\tfrac{2}{3}a\right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\Gamma\left(\tfrac{5}{6}+\tfrac{1}{3}a\right)}}}$ \hyperF@{a}{\tfrac{1}{2}+a}{\tfrac{5}{6}+\tfrac{2}{3}a}{\tfrac{1}{9}} = \sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\EulerGamma@{\tfrac{5}{6}+\tfrac{2}{3}a}}{\EulerGamma@{\tfrac{1}{2}+\tfrac{1}{3}a}\EulerGamma@{\tfrac{5}{6}+\tfrac{1}{3}a}}	${\displaystyle{\displaystyle\Re(\tfrac{5}{6}+\tfrac{2}{3}a)>0,\Re(\tfrac{1}{2}+\tfrac{1}{3}a)>0,\Re(\tfrac{5}{6}+\tfrac{1}{3}a)>0}}$	hypergeom([a, (1)/(2)+ a], [(5)/(6)+(2)/(3)*a], (1)/(9)) = sqrt(Pi)*((3)/(4))^(a)*(GAMMA((5)/(6)+(2)/(3)*a))/(GAMMA((1)/(2)+(1)/(3)*a)*GAMMA((5)/(6)+(1)/(3)*a))	Hypergeometric2F1[a, Divide[1,2]+ a, Divide[5,6]+Divide[2,3]*a, Divide[1,9]] == Sqrt[Pi]*(Divide[3,4])^(a)*Divide[Gamma[Divide[5,6]+Divide[2,3]*a],Gamma[Divide[1,2]+Divide[1,3]*a]*Gamma[Divide[5,6]+Divide[1,3]*a]]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 4]
15.4.E33	${\displaystyle{\displaystyle F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{\ifrac{\mathrm{i}\pi}{3}}\right)=\sqrt{\pi}e^{\ifrac{\mathrm{i}\pi a}{2}}\left(\frac{16}{27}\right)^{(3a+1)/6}\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma\left(\frac{2}{3}+a\right)\Gamma\left(\frac{2}{3}\right)}}}$ \hyperF@{3a}{\tfrac{1}{3}+a}{\tfrac{2}{3}+2a}{e^{\ifrac{\iunit\pi}{3}}} = \sqrt{\pi}e^{\ifrac{\iunit\pi a}{2}}\left(\frac{16}{27}\right)^{(3a+1)/6}\frac{\EulerGamma@{\frac{5}{6}+a}}{\EulerGamma@{\frac{2}{3}+a}\EulerGamma@{\frac{2}{3}}}	${\displaystyle{\displaystyle\Re(\frac{5}{6}+a)>0,\Re(\frac{2}{3}+a)>0}}$	hypergeom([3*a, (1)/(3)+ a], [(2)/(3)+ 2*a], exp((I*Pi)/(3))) = sqrt(Pi)*exp((I*Pi*a)/(2))*((16)/(27))^((3*a + 1)/6)*(GAMMA((5)/(6)+ a))/(GAMMA((2)/(3)+ a)*GAMMA((2)/(3)))	Hypergeometric2F1[3*a, Divide[1,3]+ a, Divide[2,3]+ 2*a, Exp[Divide[I*Pi,3]]] == Sqrt[Pi]*Exp[Divide[I*Pi*a,2]]*(Divide[16,27])^((3*a + 1)/6)*Divide[Gamma[Divide[5,6]+ a],Gamma[Divide[2,3]+ a]*Gamma[Divide[2,3]]]	Failure	Failure	Successful [Tested: 4]	Successful [Tested: 4]