Error Functions, Dawson’s and Fresnel Integrals - 7.12 Asymptotic Expansions
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 7.12.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelf@{z} = \frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{\Pochhammersym{\tfrac{1}{2}}{2m}}{(\pi z^{2}/2)^{2m}}+R_{n}^{(\auxFresnelf)}(z)}
\auxFresnelf@{z} = \frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{\Pochhammersym{\tfrac{1}{2}}{2m}}{(\pi z^{2}/2)^{2m}}+R_{n}^{(\auxFresnelf)}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Fresnelf(z) = (1)/(Pi*z)*sum((- 1)^(m)*(pochhammer((1)/(2), 2*m))/((Pi*(z)^(2)/2)^(2*m)), m = 0..n - 1)+(((- 1)^(n))/(Pi*sqrt(2))*int((exp(- Pi*(z)^(2)* t/2)*(t)^(2*n -(1/2)))/((t)^(2)+ 1), t = 0..infinity))
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FresnelF[z] == Divide[1,Pi*z]*Sum[(- 1)^(m)*Divide[Pochhammer[Divide[1,2], 2*m],(Pi*(z)^(2)/2)^(2*m)], {m, 0, n - 1}, GenerateConditions->None]+(Divide[(- 1)^(n),Pi*Sqrt[2]]*Integrate[Divide[Exp[- Pi*(z)^(2)* t/2]*(t)^(2*n -(1/2)),(t)^(2)+ 1], {t, 0, Infinity}, GenerateConditions->None])
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Failure | Aborted | Failed [6 / 21] Result: 2.675539142-.1105161248e-1*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 1}
Result: 2.578784539+.1565322760*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2}
... skip entries to safe data |
Failed [12 / 21]
Result: Complex[2.6755391417586893, -0.011051611690896284]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[2.578784538459091, 0.1565322770901798]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 7.12.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelg@{z} = \frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{\Pochhammersym{\tfrac{1}{2}}{2m+1}}{(\pi z^{2}/2)^{2m+1}},+R_{n}^{(\auxFresnelg)}(z)}
\auxFresnelg@{z} = \frac{1}{\pi z}\sum_{m=0}^{n-1}(-1)^{m}\frac{\Pochhammersym{\tfrac{1}{2}}{2m+1}}{(\pi z^{2}/2)^{2m+1}},+R_{n}^{(\auxFresnelg)}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Fresnelg(z) = (1)/(Pi*z)*(- 1)^(m)*(pochhammer((1)/(2), 2*m + 1))/((Pi*(z)^(2)/2)^(2*m + 1)); sum(+, m = 0..n - 1)(((- 1)^(n))/(Pi*sqrt(2))*int((exp(- Pi*(z)^(2)* t/2)*(t)^(2*n +(1/2)))/((t)^(2)+ 1), t = 0..infinity))
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FresnelG[z] == Divide[1,Pi*z]*(- 1)^(m)*Divide[Pochhammer[Divide[1,2], 2*m + 1],(Pi*(z)^(2)/2)^(2*m + 1)]
Sum[+, {m, 0, n - 1}, GenerateConditions->None](Divide[(- 1)^(n),Pi*Sqrt[2]]*Integrate[Divide[Exp[- Pi*(z)^(2)* t/2]*(t)^(2*n +(1/2)),(t)^(2)+ 1], {t, 0, Infinity}, GenerateConditions->None])
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Failure | Failure | Error | Error |