7.6: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/7.6.E1 7.6.E1] || [[Item:Q2360|<math>\erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>erf(z) = (2)/(sqrt(Pi))*sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(n)*(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Erf[z] == Divide[2,Sqrt[Pi]]*Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(n)!*(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
| [https://dlmf.nist.gov/7.6.E1 7.6.E1] || <math qid="Q2360">\erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>erf(z) = (2)/(sqrt(Pi))*sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(n)*(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Erf[z] == Divide[2,Sqrt[Pi]]*Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(n)!*(2*n + 1)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.6.E2 7.6.E2] || [[Item:Q2361|<math>\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)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>erf(z) = (2)/(sqrt(Pi))*exp(- (z)^(2))*sum(((2)^(n)* (z)^(2*n + 1))/(1 * 3*(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7078919422+.2093474075*I
| [https://dlmf.nist.gov/7.6.E2 7.6.E2] || <math qid="Q2361">\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)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>erf(z) = (2)/(sqrt(Pi))*exp(- (z)^(2))*sum(((2)^(n)* (z)^(2*n + 1))/(1 * 3*(2*n + 1)), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .7078919422+.2093474075*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5779386350+.6643773058*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.5779386350+.6643773058*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7078919419896831, 0.20934740753145048]
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.7078919419896831, 0.20934740753145048]
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/7.6.E4 7.6.E4] || [[Item:Q2363|<math>\Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>FresnelC(z) = sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n))/(factorial(2*n)*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
| [https://dlmf.nist.gov/7.6.E4 7.6.E4] || <math qid="Q2363">\Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>FresnelC(z) = sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n))/(factorial(2*n)*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.6.E5 7.6.E5] || [[Item:Q2364|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6549946728+.3747413995*I
| [https://dlmf.nist.gov/7.6.E5 7.6.E5] || <math qid="Q2364">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .6549946728+.3747413995*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.3747413995+.6549946728*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.3747413995+.6549946728*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.6549946726974499, 0.37474139987534255]
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.6549946726974499, 0.37474139987534255]
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/7.6.E6 7.6.E6] || [[Item:Q2365|<math>\Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
| [https://dlmf.nist.gov/7.6.E6 7.6.E6] || <math qid="Q2365">\Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.6.E7 7.6.E7] || [[Item:Q2366|<math>\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}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .306970168e-1+.2085514294*I
| [https://dlmf.nist.gov/7.6.E7 7.6.E7] || <math qid="Q2366">\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}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>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)</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .306970168e-1+.2085514294*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2085514294-.306970168e-1*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2085514294-.306970168e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.030697016764588636, 0.2085514288007122]
Test Values: {z = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.030697016764588636, 0.2085514288007122]
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Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/7.6.E8 7.6.E8] || [[Item:Q2367|<math>\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)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]]]]
| [https://dlmf.nist.gov/7.6.E8 7.6.E8] || <math qid="Q2367">\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)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [7 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>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]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|-  
|-  
| [https://dlmf.nist.gov/7.6.E9 7.6.E9] || [[Item:Q2368|<math>\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}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math>-1 \leq a, a \leq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
| [https://dlmf.nist.gov/7.6.E9 7.6.E9] || <math qid="Q2368">\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}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math>-1 \leq a, a \leq 1</math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>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]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
|-  
|-  
| [https://dlmf.nist.gov/7.6.E10 7.6.E10] || [[Item:Q2369|<math>\Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>FresnelC[z] == z*Sum[SphericalBesselJ[2*n, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
| [https://dlmf.nist.gov/7.6.E10 7.6.E10] || <math qid="Q2369">\Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>FresnelC[z] == z*Sum[SphericalBesselJ[2*n, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
|-  
|-  
| [https://dlmf.nist.gov/7.6.E11 7.6.E11] || [[Item:Q2370|<math>\Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>FresnelS[z] == z*Sum[SphericalBesselJ[2*n + 1, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
| [https://dlmf.nist.gov/7.6.E11 7.6.E11] || <math qid="Q2370">\Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>FresnelS[z] == z*Sum[SphericalBesselJ[2*n + 1, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Skipped - Because timed out
|}
|}
</div>
</div>

Latest revision as of 11:15, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
7.6.E1 erf ⁡ z = 2 π ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ z 2 ⁢ n + 1 n ! ⁢ ( 2 ⁢ n + 1 ) error-function 𝑧 2 𝜋 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑧 2 𝑛 1 𝑛 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{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)}		 

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 erf ⁡ z = 2 π ⁢ e - z 2 ⁢ ∑ n = 0 ∞ 2 n ⁢ z 2 ⁢ n + 1 1 ⋅ 3 ⁢ ⋯ ⁢ ( 2 ⁢ n + 1 ) error-function 𝑧 2 𝜋 superscript 𝑒 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 2 𝑛 superscript 𝑧 2 𝑛 1 ⋅ 1 3 ⋯ 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{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)}		 

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 C ⁡ ( z ) = ∑ n = 0 ∞ ( - 1 ) n ⁢ ( 1 2 ⁢ π ) 2 ⁢ n ( 2 ⁢ n ) ! ⁢ ( 4 ⁢ n + 1 ) ⁢ z 4 ⁢ n + 1 Fresnel-cosine-integral 𝑧 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 1 2 𝜋 2 𝑛 2 𝑛 4 𝑛 1 superscript 𝑧 4 𝑛 1 {\displaystyle{\displaystyle C\left(z\right)=\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}		 

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 C ⁡ ( z ) = cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ π 2 ⁢ n 1 ⋅ 3 ⁢ ⋯ ⁢ ( 4 ⁢ n + 1 ) ⁢ z 4 ⁢ n + 1 + sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ π 2 ⁢ n + 1 1 ⋅ 3 ⁢ ⋯ ⁢ ( 4 ⁢ n + 3 ) ⁢ z 4 ⁢ n + 3 Fresnel-cosine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝜋 2 𝑛 ⋅ 1 3 ⋯ 4 𝑛 1 superscript 𝑧 4 𝑛 1 1 2 𝜋 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝜋 2 𝑛 1 ⋅ 1 3 ⋯ 4 𝑛 3 superscript 𝑧 4 𝑛 3 {\displaystyle{\displaystyle C\left(z\right)=\cos\left(\tfrac{1}{2}\pi z^{2}% \right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1% }+\sin\left(\tfrac{1}{2}\pi z^{2}\right)\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}		 

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 S ⁡ ( z ) = ∑ n = 0 ∞ ( - 1 ) n ⁢ ( 1 2 ⁢ π ) 2 ⁢ n + 1 ( 2 ⁢ n + 1 ) ! ⁢ ( 4 ⁢ n + 3 ) ⁢ z 4 ⁢ n + 3 Fresnel-sine-integral 𝑧 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 1 2 𝜋 2 𝑛 1 2 𝑛 1 4 𝑛 3 superscript 𝑧 4 𝑛 3 {\displaystyle{\displaystyle S\left(z\right)=\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}		 

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 S ⁡ ( z ) = - cos ⁡ ( 1 2 ⁢ π ⁢ z 2 ) ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ π 2 ⁢ n + 1 1 ⋅ 3 ⁢ ⋯ ⁢ ( 4 ⁢ n + 3 ) ⁢ z 4 ⁢ n + 3 + sin ⁡ ( 1 2 ⁢ π ⁢ z 2 ) ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ π 2 ⁢ n 1 ⋅ 3 ⁢ ⋯ ⁢ ( 4 ⁢ n + 1 ) ⁢ z 4 ⁢ n + 1 Fresnel-sine-integral 𝑧 1 2 𝜋 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝜋 2 𝑛 1 ⋅ 1 3 ⋯ 4 𝑛 3 superscript 𝑧 4 𝑛 3 1 2 𝜋 superscript 𝑧 2 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝜋 2 𝑛 ⋅ 1 3 ⋯ 4 𝑛 1 superscript 𝑧 4 𝑛 1 {\displaystyle{\displaystyle S\left(z\right)=-\cos\left(\tfrac{1}{2}\pi z^{2}% \right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n% +3}+\sin\left(\tfrac{1}{2}\pi z^{2}\right)\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}		 

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 erf ⁡ z = 2 ⁢ z π ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ ( 𝗂 2 ⁢ n ( 1 ) ⁡ ( z 2 ) - 𝗂 2 ⁢ n + 1 ( 1 ) ⁡ ( z 2 ) ) error-function 𝑧 2 𝑧 𝜋 superscript subscript 𝑛 0 superscript 1 𝑛 spherical-Bessel-I-1 2 𝑛 superscript 𝑧 2 spherical-Bessel-I-1 2 𝑛 1 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0% }^{\infty}(-1)^{n}\left({\mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^% {(1)}_{2n+1}}\left(z^{2}\right)\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)		 

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 erf ⁡ ( a ⁢ z ) = 2 ⁢ z π ⁢ e ( 1 2 - a 2 ) ⁢ z 2 ⁢ ∑ n = 0 ∞ T 2 ⁢ n + 1 ⁡ ( a ) ⁢ 𝗂 n ( 1 ) ⁡ ( 1 2 ⁢ z 2 ) error-function 𝑎 𝑧 2 𝑧 𝜋 superscript 𝑒 1 2 superscript 𝑎 2 superscript 𝑧 2 superscript subscript 𝑛 0 Chebyshev-polynomial-first-kind-T 2 𝑛 1 𝑎 spherical-Bessel-I-1 𝑛 1 2 superscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{% \pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){% \mathsf{i}^{(1)}_{n}}\left(\tfrac{1}{2}z^{2}\right)}}
\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}}		 - 1 ≤ a , a ≤ 1 formulae-sequence 1 𝑎 𝑎 1 {\displaystyle{\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 C ⁡ ( z ) = z ⁢ ∑ n = 0 ∞ 𝗃 2 ⁢ n ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-cosine-integral 𝑧 𝑧 superscript subscript 𝑛 0 spherical-Bessel-J 2 𝑛 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2% n}\left(\tfrac{1}{2}\pi z^{2}\right)}}
\Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}		 

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 S ⁡ ( z ) = z ⁢ ∑ n = 0 ∞ 𝗃 2 ⁢ n + 1 ⁡ ( 1 2 ⁢ π ⁢ z 2 ) Fresnel-sine-integral 𝑧 𝑧 superscript subscript 𝑛 0 spherical-Bessel-J 2 𝑛 1 1 2 𝜋 superscript 𝑧 2 {\displaystyle{\displaystyle S\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2% n+1}\left(\tfrac{1}{2}\pi z^{2}\right)}}
\Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}		 

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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