Exponential, Logarithmic, Sine, and Cosine Integrals - 7.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
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Mathematica
7.2.E1 erf ⁡ z = 2 π ⁢ ∫ 0 z e - t 2 ⁢ d t error-function 𝑧 2 𝜋 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{% z}e^{-t^{2}}\mathrm{d}t}}
\erf@@{z} = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\diff{t}		 

erf(z) = (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = 0..z)

Erf[z] == Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.2.E2 erfc ⁡ z = 2 π ⁢ ∫ z ∞ e - t 2 ⁢ d t complementary-error-function 𝑧 2 𝜋 superscript subscript 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^% {\infty}e^{-t^{2}}\mathrm{d}t}}
\erfc@@{z} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t}		 

erfc(z) = (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity)

Erfc[z] == Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.2.E2 2 π ⁢ ∫ z ∞ e - t 2 ⁢ d t = 1 - erf ⁡ z 2 𝜋 superscript subscript 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 1 error-function 𝑧 {\displaystyle{\displaystyle\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}% \mathrm{d}t=1-\operatorname{erf}z}}
\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t} = 1-\erf@@{z}		 

(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity) = 1 - erf(z)

Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None] == 1 - Erf[z]
Successful Successful - Successful [Tested: 7]
7.2.E3 e - z 2 ⁢ ( 1 + 2 ⁢ i π ⁢ ∫ 0 z e t 2 ⁢ d t ) = e - z 2 ⁢ erfc ⁡ ( - i ⁢ z ) superscript 𝑒 superscript 𝑧 2 1 2 𝑖 𝜋 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 superscript 𝑒 superscript 𝑧 2 complementary-error-function 𝑖 𝑧 {\displaystyle{\displaystyle e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{% z}e^{t^{2}}\mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right)}}
e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\diff{t}\right) = e^{-z^{2}}\erfc@{-iz}		 

exp(- (z)^(2))*(1 +(2*I)/(sqrt(Pi))*int(exp((t)^(2)), t = 0..z)) = exp(- (z)^(2))*erfc(- I*z)

Exp[- (z)^(2)]*(1 +Divide[2*I,Sqrt[Pi]]*Integrate[Exp[(t)^(2)], {t, 0, z}, GenerateConditions->None]) == Exp[- (z)^(2)]*Erfc[- I*z]
Successful Successful - Successful [Tested: 7]
7.2#Ex1 lim z → ∞ ⁡ erf ⁡ z = 1 subscript → 𝑧 error-function 𝑧 1 {\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erf}z=1}}
\lim_{z\to\infty}\erf@@{z} = 1		 

limit(erf(z), z = infinity) = 1

Limit[Erf[z], z -> Infinity, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 1]
7.2#Ex2 lim z → ∞ ⁡ erfc ⁡ z = 0 subscript → 𝑧 complementary-error-function 𝑧 0 {\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erfc}z=0}}
\lim_{z\to\infty}\erfc@@{z} = 0		 | ph ⁡ z | ≤ 1 4 ⁢ π - δ ( < 1 4 ⁢ π ) phase 𝑧 annotated 1 4 𝜋 𝛿 absent 1 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<% \tfrac{1}{4}\pi)}} 
limit(erfc(z), z = infinity) = 0

Limit[Erfc[z], z -> Infinity, GenerateConditions->None] == 0
Successful Successful - Successful [Tested: 1]
7.2.E5 F ⁡ ( z ) = e - z 2 ⁢ ∫ 0 z e t 2 ⁢ d t Dawsons-integral 𝑧 superscript 𝑒 superscript 𝑧 2 superscript subscript 0 𝑧 superscript 𝑒 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}% \mathrm{d}t}}
\DawsonsintF@{z} = e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\diff{t}		 

dawson(z) = exp(- (z)^(2))*int(exp((t)^(2)), t = 0..z)

DawsonF[z] == Exp[- (z)^(2)]*Integrate[Exp[(t)^(2)], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.2.E6 ℱ ⁡ ( z ) = ∫ z ∞ e 1 2 ⁢ π ⁢ i ⁢ t 2 ⁢ d t Fresnel-integral 𝑧 superscript subscript 𝑧 superscript 𝑒 1 2 𝜋 imaginary-unit superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle\mathcal{F}\left(z\right)=\int_{z}^{\infty}e^{% \tfrac{1}{2}\pi\mathrm{i}t^{2}}\mathrm{d}t}}
\FresnelintF@{z} = \int_{z}^{\infty}e^{\tfrac{1}{2}\pi\iunit t^{2}}\diff{t}		 

Error

(1+I)/2-FresnelC[z]-I*FresnelS[z] == Integrate[Exp[Divide[1,2]*Pi*I*(t)^(2)], {t, z, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [2 / 7]
Result: Complex[-0.17236809983536389, -1.1316008349021112]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.17236809983536283, 1.1316008349021118]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

7.2.E7 C ⁡ ( z ) = ∫ 0 z cos ⁡ ( 1 2 ⁢ π ⁢ t 2 ) ⁢ d t Fresnel-cosine-integral 𝑧 superscript subscript 0 𝑧 1 2 𝜋 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2% }\pi t^{2}\right)\mathrm{d}t}}
\Fresnelcosint@{z} = \int_{0}^{z}\cos@{\tfrac{1}{2}\pi t^{2}}\diff{t}		 

FresnelC(z) = int(cos((1)/(2)*Pi*(t)^(2)), t = 0..z)

FresnelC[z] == Integrate[Cos[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.2.E8 S ⁡ ( z ) = ∫ 0 z sin ⁡ ( 1 2 ⁢ π ⁢ t 2 ) ⁢ d t Fresnel-sine-integral 𝑧 superscript subscript 0 𝑧 1 2 𝜋 superscript 𝑡 2 𝑡 {\displaystyle{\displaystyle S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2% }\pi t^{2}\right)\mathrm{d}t}}
\Fresnelsinint@{z} = \int_{0}^{z}\sin@{\tfrac{1}{2}\pi t^{2}}\diff{t}		 

FresnelS(z) = int(sin((1)/(2)*Pi*(t)^(2)), t = 0..z)

FresnelS[z] == Integrate[Sin[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.2#Ex3 lim x → ∞ ⁡ C ⁡ ( x ) = 1 2 subscript → 𝑥 Fresnel-cosine-integral 𝑥 1 2 {\displaystyle{\displaystyle\lim_{x\to\infty}C\left(x\right)=\tfrac{1}{2}}}
\lim_{x\to\infty}\Fresnelcosint@{x} = \tfrac{1}{2}		 

limit(FresnelC(x), x = infinity) = (1)/(2)

Limit[FresnelC[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]
Successful Successful - Successful [Tested: 1]
7.2#Ex4 lim x → ∞ ⁡ S ⁡ ( x ) = 1 2 subscript → 𝑥 Fresnel-sine-integral 𝑥 1 2 {\displaystyle{\displaystyle\lim_{x\to\infty}S\left(x\right)=\tfrac{1}{2}}}
\lim_{x\to\infty}\Fresnelsinint@{x} = \tfrac{1}{2}		 

limit(FresnelS(x), x = infinity) = (1)/(2)

Limit[FresnelS[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]
Successful Successful - Successful [Tested: 1]
7.2.E10 f ⁡ ( z ) = ( 1 2 - S ⁡ ( z ) ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) - ( 1 2 - C ⁡ ( z ) ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-auxilliary-function-f 𝑧 1 2 Fresnel-sine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 1 2 Fresnel-cosine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left% (z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C% \left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}
\auxFresnelf@{z} = \left(\tfrac{1}{2}-\Fresnelsinint@{z}\right)\cos@{\tfrac{1}{2}\pi z^{2}}-\left(\tfrac{1}{2}-\Fresnelcosint@{z}\right)\sin@{\tfrac{1}{2}\pi z^{2}}		 

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

FresnelF[z] == (Divide[1,2]- FresnelS[z])*Cos[Divide[1,2]*Pi*(z)^(2)]-(Divide[1,2]- FresnelC[z])*Sin[Divide[1,2]*Pi*(z)^(2)]
Successful Successful - Successful [Tested: 7]
7.2.E11 g ⁡ ( z ) = ( 1 2 - C ⁡ ( z ) ) ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) + ( 1 2 - S ⁡ ( z ) ) ⁢ sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-auxilliary-function-g 𝑧 1 2 Fresnel-cosine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 1 2 Fresnel-sine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left% (z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S% \left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}
\auxFresnelg@{z} = \left(\tfrac{1}{2}-\Fresnelcosint@{z}\right)\cos@{\tfrac{1}{2}\pi z^{2}}+\left(\tfrac{1}{2}-\Fresnelsinint@{z}\right)\sin@{\tfrac{1}{2}\pi z^{2}}		 

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

FresnelG[z] == (Divide[1,2]- FresnelC[z])*Cos[Divide[1,2]*Pi*(z)^(2)]+(Divide[1,2]- FresnelS[z])*Sin[Divide[1,2]*Pi*(z)^(2)]
Successful Successful - Successful [Tested: 7]
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