Error Functions, Dawson’s and Fresnel Integrals - 7.20 Mathematical Applications
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 7.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}\diff{t} = \frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}}}
\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}\diff{t} = \frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (1)/(sigma*sqrt(2*Pi))*int(exp(-(t - m)^(2)/(2*(sigma)^(2))), t = - infinity..x) = (1)/(2)*erfc((m - x)/(sigma*sqrt(2)))
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Divide[1,\[Sigma]*Sqrt[2*Pi]]*Integrate[Exp[-(t - m)^(2)/(2*\[Sigma]^(2))], {t, - Infinity, x}, GenerateConditions->None] == Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]]
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Failure | Failure | Failed [54 / 90] Result: Float(undefined)+Float(undefined)*I
Test Values: {sigma = -1/2+1/2*I*3^(1/2), x = 1.5, m = 1}
Result: Float(undefined)+Float(undefined)*I
Test Values: {sigma = -1/2+1/2*I*3^(1/2), x = 1.5, m = 2}
... skip entries to safe data |
Failed [45 / 90]
Result: Complex[-1.0, -1.942890293094024*^-16]
Test Values: {Rule[m, 1], Rule[x, 1.5], Rule[σ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
Result: Complex[-1.0, -1.6653345369377348*^-16]
Test Values: {Rule[m, 2], Rule[x, 1.5], Rule[σ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 7.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}} = Q\left(\frac{m-x}{\sigma}\right)}
\frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}} = Q\left(\frac{m-x}{\sigma}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (1)/(2)*erfc((m - x)/(sigma*sqrt(2))) = Q((m - x)/(sigma))
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Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]] == Q[Divide[m - x,\[Sigma]]]
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Failure | Failure | Failed [300 / 300] Result: 1.172485186-.9158452425e-1*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 1}
Result: -.1724851867+.9158452425e-1*I
Test Values: {Q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 2}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[1.1724851867610806, -0.09158452430796671]
Test Values: {Rule[m, 1], Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.1724851867610806, 0.09158452430796671]
Test Values: {Rule[m, 2], Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 7.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle Q\left(\frac{m-x}{\sigma}\right) = P\left(\frac{x-m}{\sigma}\right)}
Q\left(\frac{m-x}{\sigma}\right) = P\left(\frac{x-m}{\sigma}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Q((m - x)/(sigma)) = P((x - m)/(sigma))
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Q[Divide[m - x,\[Sigma]]] == P[Divide[x - m,\[Sigma]]]
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Failure | Failure | Failed [240 / 300] Result: -1.0
Test Values: {P = 1/2*3^(1/2)+1/2*I, Q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 1}
Result: 1.0
Test Values: {P = 1/2*3^(1/2)+1/2*I, Q = 1/2*3^(1/2)+1/2*I, sigma = 1/2*3^(1/2)+1/2*I, x = 1.5, m = 2}
... skip entries to safe data |
Failed [240 / 300]
Result: Complex[-1.0, 0.0]
Test Values: {Rule[m, 1], Rule[P, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[1.0, 0.0]
Test Values: {Rule[m, 2], Rule[P, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[σ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |