# §7.10 Derivatives

 7.10.1 $\frac{{\mathrm{d}}^{n+1}\mathop{\mathrm{erf}\/}\nolimits z}{{\mathrm{d}z}^{n+1% }}=(-1)^{n}\frac{2}{\sqrt{\pi}}\mathop{H_{n}\/}\nolimits\!\left(z\right)e^{-z^% {2}},$ $n=0,1,2,\dots$.

For the Hermite polynomial $\mathop{H_{n}\/}\nolimits\!\left(z\right)$ see §18.3.

 7.10.2 $\mathop{w\/}\nolimits'\!\left(z\right)=-2z\mathop{w\/}\nolimits\!\left(z\right% )+(2i/\sqrt{\pi}),$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{w\/}\nolimits\!\left(\NVar{z}\right)$: complementary error function and $z$: complex variable A&S Ref: 7.1.20 Permalink: http://dlmf.nist.gov/7.10.E2 Encodings: TeX, pMML, png See also: Annotations for 7.10
 7.10.3 ${{\mathop{w\/}\nolimits^{(n+2)}}\!\left(z\right)+2z{\mathop{w\/}\nolimits^{(n+% 1)}}\!\left(z\right)+2(n+1){\mathop{w\/}\nolimits^{(n)}}\!\left(z\right)=0},$ $n=0,1,2,\dots$. Symbols: $\mathop{w\/}\nolimits\!\left(\NVar{z}\right)$: complementary error function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.1.20 Permalink: http://dlmf.nist.gov/7.10.E3 Encodings: TeX, pMML, png See also: Annotations for 7.10
 7.10.4 $\displaystyle\frac{\mathrm{d}\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)}{% \mathrm{d}z}$ $\displaystyle=-\pi z\mathop{\mathrm{g}\/}\nolimits\!\left(z\right),$ $\displaystyle\frac{\mathrm{d}\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)}{% \mathrm{d}z}$ $\displaystyle=\pi z\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)-1.$