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

§7.10 Derivatives

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Keywords:: Hermite polynomials, auxiliary functions for Fresnel integrals, derivatives, derivatives of the error function, error functions, relations to other functions
Notes:: These results may be verified by differentiation of the definitions given in §7.2.
Permalink:: http://dlmf.nist.gov/7.10
See also:: Annotations for Ch.7

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

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Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, : complex variable and : nonnegative integer
A&S Ref:: 7.1.19
Permalink:: http://dlmf.nist.gov/7.10.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §7.10 and Ch.7

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

7.10.2 $w'\left(z\right)=-2zw\left(z\right)+(2i/\sqrt{\pi}),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, $\mathrm{i}$ : imaginary unit and : 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 and Ch.7

7.10.3 ${{w}^{(n+2)}\left(z\right)+2z{w}^{(n+1)}\left(z\right)+2(n+1){w}^{(n)}\left(z% \right)=0},$ $n=0,1,2,\dots$ .

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Symbols:: $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, : complex variable and : 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 and Ch.7

7.10.4

$\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}=-\pi z\mathrm{g}\left(z% \right),$

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

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Symbols:: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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
See also:: Annotations for §7.10 and Ch.7

7.9 Continued Fractions 7.11 Relations to Other Functions

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