7.11

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.11.E1	${\displaystyle{\displaystyle\operatorname{erf}z=\frac{1}{\sqrt{\pi}}\gamma\left(\tfrac{1}{2},z^{2}\right)}}$ \erf@@{z} = \frac{1}{\sqrt{\pi}}\incgamma@{\tfrac{1}{2}}{z^{2}}		erf(z) = (1)/(sqrt(Pi))*GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2))	Erf[z] == Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], 0, (z)^(2)]	Failure	Failure	Failed [7 / 7] Result: .756123263e-1-.1955582163*I Test Values: {z = 1/2*3^(1/2)+1/2*I} Result: -1.938247417+2.376161732*I Test Values: {z = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [2 / 7] Result: Complex[-1.955452759718527, 1.7141217559576072] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[-1.8042282364091204, -0.5063298374329108] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
7.11.E2	${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}\Gamma\left(\tfrac{1}{2},z^{2}\right)}}$ \erfc@@{z} = \frac{1}{\sqrt{\pi}}\incGamma@{\tfrac{1}{2}}{z^{2}}		erfc(z) = (1)/(sqrt(Pi))*GAMMA((1)/(2), (z)^(2))	Erfc[z] == Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], (z)^(2)]	Failure	Failure	Failed [2 / 7] Result: 1.955452760-1.714121756*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: 1.804228236+.5063298372*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[1.9554527597185267, -1.7141217559576072] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.8042282364091202, 0.5063298374329108] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
7.11.E3	${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{z}{\sqrt{\pi}}E_{\frac{1}{2}}\left(z^{2}\right)}}$ \erfc@@{z} = \frac{z}{\sqrt{\pi}}\genexpintE{\frac{1}{2}}@{z^{2}}		erfc(z) = (z)/(sqrt(Pi))*Ei((1)/(2), (z)^(2))	Erfc[z] == Divide[z,Sqrt[Pi]]*ExpIntegralE[Divide[1,2], (z)^(2)]	Failure	Failure	Failed [2 / 7] Result: 2.000000000+.1e-9*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: 2.000000000+.1e-9*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[2.0000000000000004, -7.771561172376096*^-16] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.9999999999999998, -5.551115123125783*^-17] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
7.11.E4	${\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)}}$ \erf@@{z} = \frac{2z}{\sqrt{\pi}}\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}}		erf(z) = (2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2))	Erf[z] == Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
7.11.E4	${\displaystyle{\displaystyle\frac{2z}{\sqrt{\pi}}M\left(\tfrac{1}{2},\tfrac{3}{2},-z^{2}\right)=\frac{2z}{\sqrt{\pi}}e^{-z^{2}}M\left(1,\tfrac{3}{2},z^{2}\right)}}$ \frac{2z}{\sqrt{\pi}}\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = \frac{2z}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperM@{1}{\tfrac{3}{2}}{z^{2}}		(2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2)) = (2*z)/(sqrt(Pi))*exp(- (z)^(2))*KummerM(1, (3)/(2), (z)^(2))	Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)] == Divide[2*z,Sqrt[Pi]]*Exp[- (z)^(2)]*Hypergeometric1F1[1, Divide[3,2], (z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
7.11.E5	${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)}}$ \erfc@@{z} = \frac{1}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}}		erfc(z) = (1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2))	Erfc[z] == Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)]	Failure	Failure	Failed [2 / 7] Result: 1.955452760-1.714121756*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: 1.804228236+.5063298372*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[1.9554527597185267, -1.7141217559576072] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[1.8042282364091202, 0.5063298374329108] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
7.11.E5	${\displaystyle{\displaystyle\frac{1}{\sqrt{\pi}}e^{-z^{2}}U\left(\tfrac{1}{2},\tfrac{1}{2},z^{2}\right)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}U\left(1,\tfrac{3}{2},z^{2}\right)}}$ \frac{1}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}} = \frac{z}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperU@{1}{\tfrac{3}{2}}{z^{2}}		(1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2)) = (z)/(sqrt(Pi))*exp(- (z)^(2))*KummerU(1, (3)/(2), (z)^(2))	Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)] == Divide[z,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[1, Divide[3,2], (z)^(2)]	Failure	Failure	Failed [2 / 7] Result: .4454723945e-1+1.714121756*I Test Values: {z = -1/2+1/2*I*3^(1/2)} Result: .1957717634-.5063298372*I Test Values: {z = -1/2*3^(1/2)-1/2*I}	Failed [2 / 7] Result: Complex[0.04454724028147337, 1.7141217559576065] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.19577176359087947, -0.5063298374329108] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}
7.11.E6	${\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}\right)}}$ \Fresnelcosint@{z}+i\Fresnelsinint@{z} = z\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}\pi iz^{2}}		FresnelC(z)+ I*FresnelS(z) = z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2))	FresnelC[z]+ I*FresnelS[z] == z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)]	Failure	Successful	Successful [Tested: 7]	Successful [Tested: 7]
7.11.E6	${\displaystyle{\displaystyle zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^{2}\right)}}$ z\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}\pi iz^{2}} = ze^{\pi iz^{2}/2}\KummerconfhyperM@{1}{\tfrac{3}{2}}{-\tfrac{1}{2}\pi iz^{2}}		z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2)) = z*exp(Pi*I*(z)^(2)/2)*KummerM(1, (3)/(2), -(1)/(2)*Pi*I*(z)^(2))	z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)] == z*Exp[Pi*I*(z)^(2)/2]*Hypergeometric1F1[1, Divide[3,2], -Divide[1,2]*Pi*I*(z)^(2)]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 7]
7.11.E7	${\displaystyle{\displaystyle C\left(z\right)=z{{}_{1}F_{2}}\left(\tfrac{1}{4};\tfrac{5}{4},\tfrac{1}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right)}}$ \Fresnelcosint@{z} = z\genhyperF{1}{2}@{\tfrac{1}{4}}{\tfrac{5}{4},\tfrac{1}{2}}{-\tfrac{1}{16}\pi^{2}z^{4}}		FresnelC(z) = z*hypergeom([(1)/(4)], [(5)/(4),(1)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4))	FresnelC[z] == z*HypergeometricPFQ[{Divide[1,4]}, {Divide[5,4],Divide[1,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)]	Successful	Successful	-	Successful [Tested: 7]
7.11.E8	${\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{6}\pi z^{3}{{}_{1}F_{2}}\left(\tfrac{3}{4};\tfrac{7}{4},\tfrac{3}{2};-\tfrac{1}{16}\pi^{2}z^{4}\right)}}$ \Fresnelsinint@{z} = \tfrac{1}{6}\pi z^{3}\genhyperF{1}{2}@{\tfrac{3}{4}}{\tfrac{7}{4},\tfrac{3}{2}}{-\tfrac{1}{16}\pi^{2}z^{4}}		FresnelS(z) = (1)/(6)*Pi*(z)^(3)* hypergeom([(3)/(4)], [(7)/(4),(3)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4))	FresnelS[z] == Divide[1,6]*Pi*(z)^(3)* HypergeometricPFQ[{Divide[3,4]}, {Divide[7,4],Divide[3,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)]	Successful	Successful	-	Successful [Tested: 7]