Error Functions, Dawson’s and Fresnel Integrals - 7.11 Relations to Other Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
7.11.E1 erf ⁡ z = 1 π ⁢ γ ⁡ ( 1 2 , z 2 ) error-function 𝑧 1 𝜋 incomplete-gamma 1 2 superscript 𝑧 2 {\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 erfc ⁡ z = 1 π ⁢ Γ ⁡ ( 1 2 , z 2 ) complementary-error-function 𝑧 1 𝜋 incomplete-Gamma 1 2 superscript 𝑧 2 {\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 erfc ⁡ z = z π ⁢ E 1 2 ⁡ ( z 2 ) complementary-error-function 𝑧 𝑧 𝜋 exponential-integral-En 1 2 superscript 𝑧 2 {\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 erf ⁡ z = 2 ⁢ z π ⁢ M ⁡ ( 1 2 , 3 2 , - z 2 ) error-function 𝑧 2 𝑧 𝜋 Kummer-confluent-hypergeometric-M 1 2 3 2 superscript 𝑧 2 {\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 2 ⁢ z π ⁢ M ⁡ ( 1 2 , 3 2 , - z 2 ) = 2 ⁢ z π ⁢ e - z 2 ⁢ M ⁡ ( 1 , 3 2 , z 2 ) 2 𝑧 𝜋 Kummer-confluent-hypergeometric-M 1 2 3 2 superscript 𝑧 2 2 𝑧 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-M 1 3 2 superscript 𝑧 2 {\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 erfc ⁡ z = 1 π ⁢ e - z 2 ⁢ U ⁡ ( 1 2 , 1 2 , z 2 ) complementary-error-function 𝑧 1 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 1 2 superscript 𝑧 2 {\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 1 π ⁢ e - z 2 ⁢ U ⁡ ( 1 2 , 1 2 , z 2 ) = z π ⁢ e - z 2 ⁢ U ⁡ ( 1 , 3 2 , z 2 ) 1 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 2 1 2 superscript 𝑧 2 𝑧 𝜋 superscript 𝑒 superscript 𝑧 2 Kummer-confluent-hypergeometric-U 1 3 2 superscript 𝑧 2 {\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 C ⁡ ( z ) + i ⁢ S ⁡ ( z ) = z ⁢ M ⁡ ( 1 2 , 3 2 , 1 2 ⁢ π ⁢ i ⁢ z 2 ) Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 𝑧 Kummer-confluent-hypergeometric-M 1 2 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 {\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 z ⁢ M ⁡ ( 1 2 , 3 2 , 1 2 ⁢ π ⁢ i ⁢ z 2 ) = z ⁢ e π ⁢ i ⁢ z 2 / 2 ⁢ M ⁡ ( 1 , 3 2 , - 1 2 ⁢ π ⁢ i ⁢ z 2 ) 𝑧 Kummer-confluent-hypergeometric-M 1 2 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 𝑧 superscript 𝑒 𝜋 𝑖 superscript 𝑧 2 2 Kummer-confluent-hypergeometric-M 1 3 2 1 2 𝜋 𝑖 superscript 𝑧 2 {\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 C ⁡ ( z ) = z ⁢ F 2 1 ⁡ ( 1 4 ; 5 4 , 1 2 ; - 1 16 ⁢ π 2 ⁢ z 4 ) Fresnel-cosine-integral 𝑧 𝑧 Gauss-hypergeometric-pFq 1 2 1 4 5 4 1 2 1 16 superscript 𝜋 2 superscript 𝑧 4 {\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 S ⁡ ( z ) = 1 6 ⁢ π ⁢ z 3 ⁢ F 2 1 ⁡ ( 3 4 ; 7 4 , 3 2 ; - 1 16 ⁢ π 2 ⁢ z 4 ) Fresnel-sine-integral 𝑧 1 6 𝜋 superscript 𝑧 3 Gauss-hypergeometric-pFq 1 2 3 4 7 4 3 2 1 16 superscript 𝜋 2 superscript 𝑧 4 {\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]
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