Gold 34

Voigt profile

Gold ID: 34
Link: https://sigir21.wmflabs.org/wiki/Voigt_profile#math.84.31
Formula: ${\frac {\partial ^{2}}{\partial x^{2}}}V(x;\sigma ,\gamma )={\frac {x^{2}-\gamma ^{2}-\sigma ^{2}}{\sigma ^{4}}}{\frac {\operatorname {Re} [w(z)]}{\sigma {\sqrt {2\pi }}}}-{\frac {2x\gamma }{\sigma ^{4}}}{\frac {\operatorname {Im} [w(z)]}{\sigma {\sqrt {2\pi }}}}+{\frac {\gamma }{\sigma ^{4}}}{\frac {1}{\pi }}$
TeX Source: \frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations
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Translation: \deriv [2]{ }{x} V(x ; \sigma , \gamma) = \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\realpart@@{[\Faddeevaw@{z}]}}{\sigma \sqrt{2 \cpi}} - \frac{2 x \gamma}{\sigma^4} \frac{\imagpart [\Faddeevaw@{z}]}{\sigma \sqrt{2 \cpi}} + \frac{\gamma}{\sigma^4} \frac{1}{\cpi}
Expected (Gold Entry): \deriv[2]{}{x} V(x ; \sigma , \gamma) = \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\realpart [\Faddeevaw@{z}]}{\sigma \sqrt{2 \cpi}} - \frac{2 x \gamma}{\sigma^4} \frac{\imagpart [\Faddeevaw@{z}]}{\sigma \sqrt{2 \cpi}} + \frac{\gamma}{\sigma^4} \frac{1}{\cpi}

Translation
Expected (Gold Entry): D[PDF[VoigtDistribution[\[Gamma], \[Sigma]], x], {x, 2}] == Divide[x^2 - \[Gamma]^2 - \[Sigma]^2, \[Sigma]^4] * Divide[ Re[ Exp[-(Divide[x+I*y,\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\[Sigma]*Sqrt[2]])] ], \[Sigma]*Sqrt[2*Pi]] - Divide[2*x*y, \[Sigma]^4] * Divide[Im[Exp[-(Divide[x+I*y,\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\[Sigma]*Sqrt[2]])]], \[Sigma]*Sqrt[2*Pi]] + Divide[\[Gamma],\[Sigma]^4]*Divide[1,Pi]