Error Functions, Dawson’s and Fresnel Integrals - 7.6 Series Expansions

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7.6.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}}
\erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
erf(z) = (2)/(sqrt(Pi))*sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(n)*(2*n + 1)), n = 0..infinity)

Erf[z] == Divide[2,Sqrt[Pi]]*Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(n)!*(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.6.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2}{\sqrt{\pi}}e^{-z^{2}}\sum_{n=0}^{\infty}\frac{2^{n}z^{2n+1}}{1\cdot 3\cdots(2n+1)}}
\erf@@{z} = \frac{2}{\sqrt{\pi}}e^{-z^{2}}\sum_{n=0}^{\infty}\frac{2^{n}z^{2n+1}}{1\cdot 3\cdots(2n+1)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
erf(z) = (2)/(sqrt(Pi))*exp(- (z)^(2))*sum(((2)^(n)* (z)^(2*n + 1))/(1 * 3*(2*n + 1)), n = 0..infinity)

Erf[z] == Divide[2,Sqrt[Pi]]*Exp[- (z)^(2)]*Sum[Divide[(2)^(n)* (z)^(2*n + 1),1 * 3*(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [7 / 7]
Result: .7078919422+.2093474075*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -.5779386350+.6643773058*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.7078919419896831, 0.20934740753145048]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.5779386346997313, 0.6643773053985802]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.6.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}}
\Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
FresnelC(z) = sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n))/(factorial(2*n)*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity)

FresnelC[z] == Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n),(2*n)!*(4*n + 1)]*(z)^(4*n + 1), {n, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.6.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = \cos@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}+\sin@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}}
\Fresnelcosint@{z} = \cos@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}+\sin@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
FresnelC(z) = cos((1)/(2)*Pi*(z)^(2))*sum(((- 1)^(n)* (Pi)^(2*n))/(1 * 3*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity)+ sin((1)/(2)*Pi*(z)^(2))*sum(((- 1)^(n)* (Pi)^(2*n + 1))/(1 * 3*(4*n + 3))*(z)^(4*n + 3), n = 0..infinity)

FresnelC[z] == Cos[Divide[1,2]*Pi*(z)^(2)]*Sum[Divide[(- 1)^(n)* (Pi)^(2*n),1 * 3*(4*n + 1)]*(z)^(4*n + 1), {n, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[1,2]*Pi*(z)^(2)]*Sum[Divide[(- 1)^(n)* (Pi)^(2*n + 1),1 * 3*(4*n + 3)]*(z)^(4*n + 3), {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [7 / 7]
Result: .6549946728+.3747413995*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -.3747413995+.6549946728*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.6549946726974499, 0.37474139987534255]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.37474139987534216, 0.6549946726974494]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.6.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}}
\Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
FresnelS(z) = sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n + 1))/(factorial(2*n + 1)*(4*n + 3))*(z)^(4*n + 3), n = 0..infinity)

FresnelS[z] == Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n + 1),(2*n + 1)!*(4*n + 3)]*(z)^(4*n + 3), {n, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 7]
7.6.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = -\cos@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}+\sin@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}}
\Fresnelsinint@{z} = -\cos@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}+\sin@{\tfrac{1}{2}\pi z^{2}}\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
FresnelS(z) = - cos((1)/(2)*Pi*(z)^(2))*sum(((- 1)^(n)* (Pi)^(2*n + 1))/(1 * 3*(4*n + 3))*(z)^(4*n + 3), n = 0..infinity)+ sin((1)/(2)*Pi*(z)^(2))*sum(((- 1)^(n)* (Pi)^(2*n))/(1 * 3*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity)

FresnelS[z] == - Cos[Divide[1,2]*Pi*(z)^(2)]*Sum[Divide[(- 1)^(n)* (Pi)^(2*n + 1),1 * 3*(4*n + 3)]*(z)^(4*n + 3), {n, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[1,2]*Pi*(z)^(2)]*Sum[Divide[(- 1)^(n)* (Pi)^(2*n),1 * 3*(4*n + 1)]*(z)^(4*n + 1), {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [7 / 7]
Result: .306970168e-1+.2085514294*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: .2085514294-.306970168e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.030697016764588636, 0.2085514288007122]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.2085514288007118, -0.030697016764589136]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.6.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left(\modsphBesseli{1}{2n}@{z^{2}}-\modsphBesseli{1}{2n+1}@{z^{2}}\right)}
\erf@@{z} = \frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left(\modsphBesseli{1}{2n}@{z^{2}}-\modsphBesseli{1}{2n+1}@{z^{2}}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

Erf[z] == Divide[2*z,Sqrt[Pi]]*Sum[(- 1)^(n)*(Sqrt[Divide[Pi, (z)^(2)]/2] BesselI[(-1)^(1-1)*(2*n + 1/2), 2*n]- Sqrt[Divide[Pi, (z)^(2)]/2] BesselI[(-1)^(1-1)*(2*n + 1 + 1/2), 2*n + 1]), {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [7 / 7]
Result: Plus[Complex[0.90211411820456, 0.25316491871645536], Times[Complex[-0.9772050238058398, -0.5641895835477562], NSum[Times[Power[-1, n], Plus[Times[Power[Power[E, Times[Complex[0, Rational[-1, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[2, n]], Times[2, n]]], Times[-1, Power[Power[E, Times[Complex[0, Rational[-1, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[3, 2], Times[2, n]], Plus[1, Times[2, n]]]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.9777263798592635, 0.8570608779788039], Times[Complex[0.5641895835477561, -0.9772050238058398], NSum[Times[Power[-1, n], Plus[Times[Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[2, n]], Times[2, n]]], Times[-1, Power[Power[E, Times[Complex[0, Rational[2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[3, 2], Times[2, n]], Plus[1, Times[2, n]]]]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
7.6.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@{az} = \frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}\ChebyshevpolyT{2n+1}@{a}\modsphBesseli{1}{n}@{\tfrac{1}{2}z^{2}}}
\erf@{az} = \frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}\ChebyshevpolyT{2n+1}@{a}\modsphBesseli{1}{n}@{\tfrac{1}{2}z^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -1 \leq a, a \leq 1} 
Error

Erf[a*z] == Divide[2*z,Sqrt[Pi]]*Exp[(Divide[1,2]- (a)^(2))*(z)^(2)]*Sum[ChebyshevT[2*n + 1, a]*Sqrt[Divide[Pi, Divide[1,2]*(z)^(2)]/2] BesselI[(-1)^(1-1)*(n + 1/2), n], {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
7.6.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}}
\Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

FresnelC[z] == z*Sum[SphericalBesselJ[2*n, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Failure - Skipped - Because timed out
7.6.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}}
\Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

FresnelS[z] == z*Sum[SphericalBesselJ[2*n + 1, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Failure - Skipped - Because timed out
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