7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.9 Continued Fractions 7.11 Relations to Other Functions

§7.10 Derivatives

Notes:: These results may be verified by differentiation of the definitions given in §7.2.
Keywords:: Hermite polynomials, auxiliary functions for Fresnel integrals, derivatives, error functions
Permalink:: http://dlmf.nist.gov/7.10

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

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\frac{df}{dx}$ : derivative of with respect to , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : base of exponential function, : complex variable and : nonnegative integer
A&S Ref:: 7.1.19
Permalink:: http://dlmf.nist.gov/7.10.E1
Encodings:: TeX, pMML, png

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

7.10.2 ${\mathop{w\/}\nolimits^{{\prime}}}\!\left(z\right)=-2z\mathop{w\/}\nolimits\!% \left(z\right)+(2i/\sqrt{\pi}),$

Symbols:: $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function and : complex variable
A&S Ref:: 7.1.20
Permalink:: http://dlmf.nist.gov/7.10.E2
Encodings:: TeX, pMML, png

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:: $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, : complex variable and : nonnegative integer
A&S Ref:: 7.1.20
Permalink:: http://dlmf.nist.gov/7.10.E3
Encodings:: TeX, pMML, png

7.10.4

$\frac{d\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)}{dz}=-\pi z\mathop{% \mathrm{g}\/}\nolimits\!\left(z\right),$

$\frac{d\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)}{dz}=\pi z\mathop{% \mathrm{f}\/}\nolimits\!\left(z\right)-1.$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\frac{df}{dx}$ : derivative of with respect to and : complex variable
A&S Ref:: 7.3.21 (in slightly different form)
Permalink:: http://dlmf.nist.gov/7.10.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

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