Error Functions, Dawson’s and Fresnel Integrals - 7.20 Mathematical Applications

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DLMF Formula Constraints Maple Mathematica Symbolic
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7.20.E1 1 σ 2 π - x e - ( t - m ) 2 / ( 2 σ 2 ) d t = 1 2 erfc ( m - x σ 2 ) 1 𝜎 2 𝜋 superscript subscript 𝑥 superscript 𝑒 superscript 𝑡 𝑚 2 2 superscript 𝜎 2 𝑡 1 2 complementary-error-function 𝑚 𝑥 𝜎 2 {\displaystyle{\displaystyle\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(% t-m)^{2}/(2\sigma^{2})}\mathrm{d}t=\frac{1}{2}\operatorname{erfc}\left(\frac{m% -x}{\sigma\sqrt{2}}\right)}}
\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}}}

(1)/(sigma*sqrt(2*Pi))*int(exp(-(t - m)^(2)/(2*(sigma)^(2))), t = - infinity..x) = (1)/(2)*erfc((m - x)/(sigma*sqrt(2)))
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]]]
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 1 2 erfc ( m - x σ 2 ) = Q ( m - x σ ) 1 2 complementary-error-function 𝑚 𝑥 𝜎 2 𝑄 𝑚 𝑥 𝜎 {\displaystyle{\displaystyle\frac{1}{2}\operatorname{erfc}\left(\frac{m-x}{% \sigma\sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma}\right)}}
\frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}} = Q\left(\frac{m-x}{\sigma}\right)

(1)/(2)*erfc((m - x)/(sigma*sqrt(2))) = Q((m - x)/(sigma))
Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]] == Q[Divide[m - x,\[Sigma]]]
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 Q ( m - x σ ) = P ( x - m σ ) 𝑄 𝑚 𝑥 𝜎 𝑃 𝑥 𝑚 𝜎 {\displaystyle{\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)

Q((m - x)/(sigma)) = P((x - m)/(sigma))
Q[Divide[m - x,\[Sigma]]] == P[Divide[x - m,\[Sigma]]]
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