Error Functions, Dawson’s and Fresnel Integrals - 7.5 Interrelations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
7.5.E2 C ⁡ ( z ) + i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 + i ) - ℱ ⁡ ( z ) Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 1 𝑖 Fresnel-integral 𝑧 {\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i% )-\mathcal{F}\left(z\right)}}
\Fresnelcosint@{z}+i\Fresnelsinint@{z} = \tfrac{1}{2}(1+i)-\FresnelintF@{z}		 

Error

FresnelC[z]+ I*FresnelS[z] == Divide[1,2]*(1 + I)- (1+I)/2-FresnelC[z]-I*FresnelS[z]
Missing Macro Error Failure -
Failed [7 / 7]
Result: Complex[1.0249430142401041, 0.8677085978643018]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5229723981935741, 3.2881446840443265]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.5.E3 C ⁡ ( z ) = 1 2 + f ⁡ ( z ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) - g ⁡ ( z ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-cosine-integral 𝑧 1 2 Fresnel-auxilliary-function-f 𝑧 1 2 𝜋 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z% \right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\cos% \left(\tfrac{1}{2}\pi z^{2}\right)}}
\Fresnelcosint@{z} = \tfrac{1}{2}+\auxFresnelf@{z}\sin@{\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}\cos@{\tfrac{1}{2}\pi z^{2}}		 

FresnelC(z) = (1)/(2)+ Fresnelf(z)*sin((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*cos((1)/(2)*Pi*(z)^(2))

FresnelC[z] == Divide[1,2]+ FresnelF[z]*Sin[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Cos[Divide[1,2]*Pi*(z)^(2)]
Successful Failure - Successful [Tested: 7]
7.5.E4 S ⁡ ( z ) = 1 2 - f ⁡ ( z ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) - g ⁡ ( z ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-sine-integral 𝑧 1 2 Fresnel-auxilliary-function-f 𝑧 1 2 𝜋 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{2}-\mathrm{f}\left(z% \right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right)}}
\Fresnelsinint@{z} = \tfrac{1}{2}-\auxFresnelf@{z}\cos@{\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}\sin@{\tfrac{1}{2}\pi z^{2}}		 

FresnelS(z) = (1)/(2)- Fresnelf(z)*cos((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*sin((1)/(2)*Pi*(z)^(2))

FresnelS[z] == Divide[1,2]- FresnelF[z]*Cos[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Sin[Divide[1,2]*Pi*(z)^(2)]
Successful Failure - Successful [Tested: 7]
7.5.E5 e - 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ℱ ⁡ ( z ) = g ⁡ ( z ) + i ⁢ f ⁡ ( z ) superscript 𝑒 1 2 𝜋 𝑖 superscript 𝑧 2 Fresnel-integral 𝑧 Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 {\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z% \right)=\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)}}
e^{-\frac{1}{2}\pi iz^{2}}\FresnelintF@{z} = \auxFresnelg@{z}+i\auxFresnelf@{z}		 

Error

Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(1+I)/2-FresnelC[z]-I*FresnelS[z] == FresnelG[z]+ I*FresnelF[z]
Missing Macro Error Failure -
Failed [6 / 7]
Result: Complex[2.0955908860316255, -0.6505223669676224]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.08422623998042833, -1.3932392044453867]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.5.E6 e + 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ( g ⁡ ( z ) + i ⁢ f ⁡ ( z ) ) = 1 2 ⁢ ( 1 + i ) - ( C ⁡ ( z ) + i ⁢ S ⁡ ( z ) ) superscript 𝑒 1 2 𝜋 𝑖 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle e^{+\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)+i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1+i)-(C\left(z\right)+iS\left(% z\right))}}
e^{+\frac{1}{2}\pi iz^{2}}(\auxFresnelg@{z}+ i\auxFresnelf@{z}) = \tfrac{1}{2}(1+ i)-(\Fresnelcosint@{z}+ i\Fresnelsinint@{z})		 

exp(+(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)+ I*Fresnelf(z)) = (1)/(2)*(1 + I)-(FresnelC(z)+ I*FresnelS(z))

Exp[+Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]+ I*FresnelF[z]) == Divide[1,2]*(1 + I)-(FresnelC[z]+ I*FresnelS[z])
Successful Failure - Successful [Tested: 7]
7.5.E6 e - 1 2 ⁢ π ⁢ i ⁢ z 2 ⁢ ( g ⁡ ( z ) - i ⁢ f ⁡ ( z ) ) = 1 2 ⁢ ( 1 - i ) - ( C ⁡ ( z ) - i ⁢ S ⁡ ( z ) ) superscript 𝑒 1 2 𝜋 𝑖 superscript 𝑧 2 Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 {\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z% \right)-i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1-i)-(C\left(z\right)-iS\left(% z\right))}}
e^{-\frac{1}{2}\pi iz^{2}}(\auxFresnelg@{z}- i\auxFresnelf@{z}) = \tfrac{1}{2}(1- i)-(\Fresnelcosint@{z}- i\Fresnelsinint@{z})		 

exp(-(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)- I*Fresnelf(z)) = (1)/(2)*(1 - I)-(FresnelC(z)- I*FresnelS(z))

Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]- I*FresnelF[z]) == Divide[1,2]*(1 - I)-(FresnelC[z]- I*FresnelS[z])
Successful Failure - Successful [Tested: 7]
7.5.E8 C ⁡ ( z ) + i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 + i ) ⁢ erf ⁡ ζ Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 1 𝑖 error-function 𝜁 {\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i% )\operatorname{erf}\zeta}}
\Fresnelcosint@{z}+ i\Fresnelsinint@{z} = \tfrac{1}{2}(1+ i)\erf@@{\zeta}		 

FresnelC(z)+ I*FresnelS(z) = (1)/(2)*(1 + I)*erf((1)/(2)*sqrt(Pi)*(1 - I)*z)

FresnelC[z]+ I*FresnelS[z] == Divide[1,2]*(1 + I)*Erf[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 7]
7.5.E8 C ⁡ ( z ) - i ⁢ S ⁡ ( z ) = 1 2 ⁢ ( 1 - i ) ⁢ erf ⁡ ζ Fresnel-cosine-integral 𝑧 𝑖 Fresnel-sine-integral 𝑧 1 2 1 𝑖 error-function 𝜁 {\displaystyle{\displaystyle C\left(z\right)-iS\left(z\right)=\tfrac{1}{2}(1-i% )\operatorname{erf}\zeta}}
\Fresnelcosint@{z}- i\Fresnelsinint@{z} = \tfrac{1}{2}(1- i)\erf@@{\zeta}		 

FresnelC(z)- I*FresnelS(z) = (1)/(2)*(1 - I)*erf((1)/(2)*sqrt(Pi)*(1 - I)*z)

FresnelC[z]- I*FresnelS[z] == Divide[1,2]*(1 - I)*Erf[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
Failure Failure
Failed [7 / 7]
Result: 1.210218044+.7739577054*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -2.077926642+.2509853077*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[1.210218043090013, 0.7739577062168396]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.0779266409543133, 0.2509853080232649]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.5.E10 g ⁡ ( z ) + i ⁢ f ⁡ ( z ) = 1 2 ⁢ ( 1 + i ) ⁢ e ζ 2 ⁢ erfc ⁡ ζ Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 superscript 𝑒 superscript 𝜁 2 complementary-error-function 𝜁 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)% =\tfrac{1}{2}(1+i)e^{\zeta^{2}}\operatorname{erfc}\zeta}}
\auxFresnelg@{z}+ i\auxFresnelf@{z} = \tfrac{1}{2}(1+ i)e^{\zeta^{2}}\erfc@@{\zeta}		 

Fresnelg(z)+ I*Fresnelf(z) = (1)/(2)*(1 + I)*exp(((1)/(2)*sqrt(Pi)*(1 - I)*z)^(2))*erfc((1)/(2)*sqrt(Pi)*(1 - I)*z)

FresnelG[z]+ I*FresnelF[z] == Divide[1,2]*(1 + I)*Exp[(Divide[1,2]*Sqrt[Pi]*(1 - I)*z)^(2)]*Erfc[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
Successful Failure Skip - symbolical successful subtest Successful [Tested: 7]
7.5.E10 g ⁡ ( z ) - i ⁢ f ⁡ ( z ) = 1 2 ⁢ ( 1 - i ) ⁢ e ζ 2 ⁢ erfc ⁡ ζ Fresnel-auxilliary-function-g 𝑧 𝑖 Fresnel-auxilliary-function-f 𝑧 1 2 1 𝑖 superscript 𝑒 superscript 𝜁 2 complementary-error-function 𝜁 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)-i\mathrm{f}\left(z\right)% =\tfrac{1}{2}(1-i)e^{\zeta^{2}}\operatorname{erfc}\zeta}}
\auxFresnelg@{z}- i\auxFresnelf@{z} = \tfrac{1}{2}(1- i)e^{\zeta^{2}}\erfc@@{\zeta}		 

Fresnelg(z)- I*Fresnelf(z) = (1)/(2)*(1 - I)*exp(((1)/(2)*sqrt(Pi)*(1 - I)*z)^(2))*erfc((1)/(2)*sqrt(Pi)*(1 - I)*z)

FresnelG[z]- I*FresnelF[z] == Divide[1,2]*(1 - I)*Exp[(Divide[1,2]*Sqrt[Pi]*(1 - I)*z)^(2)]*Erfc[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
Failure Failure
Failed [7 / 7]
Result: -.2860780540-.1977870141*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -.1472580850-5.018337775*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[-0.2860780524436176, -0.19778701442673574]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.14725808362732817, -5.018337771876615]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.5.E11 | ℱ ⁡ ( x ) | 2 = f 2 ⁡ ( x ) + g 2 ⁡ ( x ) superscript Fresnel-integral 𝑥 2 Fresnel-auxilliary-function-f 2 𝑥 Fresnel-auxilliary-function-g 2 𝑥 {\displaystyle{\displaystyle|\mathcal{F}\left(x\right)|^{2}={\mathrm{f}^{2}}% \left(x\right)+{\mathrm{g}^{2}}\left(x\right)}}
|\FresnelintF@{x}|^{2} = \auxFresnelf^{2}@{x}+\auxFresnelg^{2}@{x}		 x ≥ 0 𝑥 0 {\displaystyle{\displaystyle x\geq 0}} 
Error

(Abs[(1+I)/2-FresnelC[x]-I*FresnelS[x]])^(2) == (FresnelF[x])^(2)+ (FresnelG[x])^(2)
Missing Macro Error Failure - Successful [Tested: 3]
7.5.E12 | ℱ ⁡ ( x ) | 2 = 2 + f 2 ⁡ ( - x ) + g 2 ⁡ ( - x ) - 2 ⁢ 2 ⁢ cos ⁡ ( 1 4 ⁢ π + 1 2 ⁢ π ⁢ x 2 ) ⁢ f ⁡ ( - x ) - 2 ⁢ 2 ⁢ cos ⁡ ( 1 4 ⁢ π - 1 2 ⁢ π ⁢ x 2 ) ⁢ g ⁡ ( - x ) superscript Fresnel-integral 𝑥 2 2 Fresnel-auxilliary-function-f 2 𝑥 Fresnel-auxilliary-function-g 2 𝑥 2 2 1 4 𝜋 1 2 𝜋 superscript 𝑥 2 Fresnel-auxilliary-function-f 𝑥 2 2 1 4 𝜋 1 2 𝜋 superscript 𝑥 2 Fresnel-auxilliary-function-g 𝑥 {\displaystyle{\displaystyle|\mathcal{F}\left(x\right)|^{2}=2+{\mathrm{f}^{2}}% \left(-x\right)+{\mathrm{g}^{2}}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4% }\pi+\tfrac{1}{2}\pi x^{2}\right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos\left(% \tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}\right)\mathrm{g}\left(-x\right)}}
|\FresnelintF@{x}|^{2} = 2+\auxFresnelf^{2}@{-x}+\auxFresnelg^{2}@{-x}-2\sqrt{2}\cos@{\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}}\auxFresnelf@{-x}-2\sqrt{2}\cos@{\tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}}\auxFresnelg@{-x}		 x ≤ 0 𝑥 0 {\displaystyle{\displaystyle x\leq 0}} 
Error

(Abs[(1+I)/2-FresnelC[x]-I*FresnelS[x]])^(2) == 2 + (FresnelF[- x])^(2)+ (FresnelG[- x])^(2)- 2*Sqrt[2]*Cos[Divide[1,4]*Pi +Divide[1,2]*Pi*(x)^(2)]*FresnelF[- x]- 2*Sqrt[2]*Cos[Divide[1,4]*Pi -Divide[1,2]*Pi*(x)^(2)]*FresnelG[- x]
Missing Macro Error Failure - Skip - No test values generated
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