7.5

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.5.E2	${\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i)-\mathcal{F}\left(z\right)}}$ \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	${\displaystyle{\displaystyle C\left(z\right)=\tfrac{1}{2}+\mathrm{f}\left(z\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)}}$ \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	${\displaystyle{\displaystyle S\left(z\right)=\tfrac{1}{2}-\mathrm{f}\left(z\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}$ \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	${\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}\mathcal{F}\left(z\right)=\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)}}$ 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	${\displaystyle{\displaystyle e^{+\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1+i)-(C\left(z\right)+iS\left(z\right))}}$ 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	${\displaystyle{\displaystyle e^{-\frac{1}{2}\pi iz^{2}}(\mathrm{g}\left(z\right)-i\mathrm{f}\left(z\right))=\tfrac{1}{2}(1-i)-(C\left(z\right)-iS\left(z\right))}}$ 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	${\displaystyle{\displaystyle C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i)\operatorname{erf}\zeta}}$ \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	${\displaystyle{\displaystyle C\left(z\right)-iS\left(z\right)=\tfrac{1}{2}(1-i)\operatorname{erf}\zeta}}$ \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	${\displaystyle{\displaystyle\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=\tfrac{1}{2}(1+i)e^{\zeta^{2}}\operatorname{erfc}\zeta}}$ \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	${\displaystyle{\displaystyle\mathrm{g}\left(z\right)-i\mathrm{f}\left(z\right)=\tfrac{1}{2}(1-i)e^{\zeta^{2}}\operatorname{erfc}\zeta}}$ \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	${\displaystyle{\displaystyle|\mathcal{F}\left(x\right)|^{2}={\mathrm{f}^{2}}\left(x\right)+{\mathrm{g}^{2}}\left(x\right)}}$ |\FresnelintF@{x}|^{2} = \auxFresnelf^{2}@{x}+\auxFresnelg^{2}@{x}	${\displaystyle{\displaystyle x\geq 0}}$	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	${\displaystyle{\displaystyle|\mathcal{F}\left(x\right)|^{2}=2+{\mathrm{f}^{2}}\left(-x\right)+{\mathrm{g}^{2}}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}\right)\mathrm{f}\left(-x\right)-2\sqrt{2}\cos\left(\tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}\right)\mathrm{g}\left(-x\right)}}$ |\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}	${\displaystyle{\displaystyle x\leq 0}}$	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