Gold 11

Error function

Gold ID

11

Link

https://sigir21.wmflabs.org/wiki/Error_function#math.61.27

Formula

$\operatorname {erf} ^{(k)}(z)={\frac {2(-1)^{k-1}}{\sqrt {\pi }}}{\mathit {H}}_{k-1}(z)e^{-z^{2}}={\frac {2}{\sqrt {\pi }}}{\frac {d^{k-1}}{dz^{k-1}}}\left(e^{-z^{2}}\right),\qquad k=1,2,\dots$

TeX Source

\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Semantic LaTeX

Translation

\erf@@{(z)}^{(k)} = \frac{2 (-1)^{k-1}}{\sqrt{\cpi}} \HermitepolyH{k-1}@{z} \expe^{-z^2} = \frac{2}{\sqrt{\cpi}} \deriv [{k-1}]{ }{z}(\expe^{-z^2}) , \qquad k = 1 , 2 , \dots

Expected (Gold Entry)

\erf@@{(z)}^{(k)} = \frac{2 (-1)^{k-1}}{\sqrt{\cpi}} \HermitepolyH{k-1}@{z} \expe^{-z^2} = \frac{2}{\sqrt{\cpi}} \deriv [{k-1}]{ }{z}(\expe^{-z^2}) , \qquad k = 1 , 2 , \dots

Mathematica

Translation

(Erf[z])^(k) == Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] == Divide[2,Sqrt[Pi]]*(D[(Exp[- (temp)^(2)]), {temp, k - 1}]/.temp-> z)

Expected (Gold Entry)

D[Erf[z], {z, k}] == Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] == Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]

Maple

Translation

(erf(z))^(k) = (2*(- 1)^(k - 1))/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)/(sqrt(Pi))*subs( temp=z, diff( (exp(- (temp)^(2))), temp$(k - 1) ) )

Expected (Gold Entry)

diff(erf(z), [z$\$$k]) = (2*(- 1)^(k - 1))/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$\$$(k - 1)])