7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.5 Interrelations7.7 Integral Representations

§7.6 Series Expansions

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http://dlmf.nist.gov/7.6

See also:

Annotations for Ch.7

Contents

1. §7.6(i) Power Series
2. §7.6(ii) Expansions in Series of Spherical Bessel Functions

§7.6(i) Power Series

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Keywords:

Fresnel integrals, error functions, power-series expansions

Notes:

For (7.6.1)–(7.6.3) see van der Laan and Temme (1984, pp. 185–186). (7.6.4) and (7.6.6) follow from (7.2.7) and (7.2.8). (7.6.5) and (7.6.7) follow from (7.5.8) and (7.6.2).

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http://dlmf.nist.gov/7.6.i

See also:

Annotations for §7.6 and Ch.7

7.6.1		$\mathrm{erf}z$	$={\displaystyle \frac{2}{\sqrt{\pi }}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n{z}^{2n+1}}{n!(2n+1)}},$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.1.5 Referenced by: §7.6(i) Permalink: http://dlmf.nist.gov/7.6.E1 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7
7.6.2		$\mathrm{erf}z$	$={\displaystyle \frac{2}{\sqrt{\pi }}}{\mathrm{e}}^{-z^2}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{2^n{z}^{2n+1}}{1\cdot 3\mathrm{\cdots }(2n+1)}},$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.1.6 Referenced by: §7.6(i), §7.6(ii) Permalink: http://dlmf.nist.gov/7.6.E2 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7
7.6.3		$w\left(z\right)$	$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(\mathrm{i}z)}^n}{\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}}.$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.1.8 Referenced by: §7.6(i) Permalink: http://dlmf.nist.gov/7.6.E3 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7
7.6.4		$C\left(z\right)$	$={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n}}{(2n)!(4n+1)}}{z}^{4n+1},$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.3.11 Referenced by: §7.6(i) Permalink: http://dlmf.nist.gov/7.6.E4 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7

7.6.5		$$C\left(z\right)=\begin{array}{l}\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}\\ +\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}.\end{array}$$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.3.12 Referenced by: §11.10(vi), §7.6(i) Permalink: http://dlmf.nist.gov/7.6.E5 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7

7.6.6		$$S\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n+1}}{(2n+1)!(4n+3)}{z}^{4n+3},$$
ⓘ Symbols: $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.3.13 Referenced by: §7.6(i) Permalink: http://dlmf.nist.gov/7.6.E6 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7

7.6.7		$$S\left(z\right)=\begin{array}{l}-\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}\\ +\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}.\end{array}$$
ⓘ Symbols: $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.3.14 Referenced by: §11.10(vi), §7.6(i) Permalink: http://dlmf.nist.gov/7.6.E7 Encodings: TeX, pMML, png See also: Annotations for §7.6(i), §7.6 and Ch.7

The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$.

§7.6(ii) Expansions in Series of Spherical Bessel Functions

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Keywords:

Fresnel integrals, error functions, expansions in spherical Bessel functions

Notes:

For (7.6.8) differentiate: use (7.2.1) for the left-hand side, and (10.47.7) and the second of (10.29.1) for the right-hand side. The same method can be used for (7.6.10) and (7.6.11). For (7.6.9) write the coefficients in the Chebyshev-series expansion of $\exp \left(a^2{z}^2\right)\mathrm{erf}\left(az\right)$ as integrals (§3.11(ii)), then apply (5.12.5), (7.6.2), and (13.6.9).

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.ii

See also:

Annotations for §7.6 and Ch.7

For the notation see §§10.47(ii) and 18.3.

7.6.8		$$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n\left({𝗂}_{2n}^{(1)}\left(z^2\right)-{𝗂}_{2n+1}^{(1)}\left(z^2\right)\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 7.1.7 (in different form) Referenced by: §7.6(ii) Permalink: http://dlmf.nist.gov/7.6.E8 Encodings: TeX, pMML, png See also: Annotations for §7.6(ii), §7.6 and Ch.7

7.6.9		$$\mathrm{erf}\left(az\right)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{(\frac{1}{2}-a^2)z^2}\sum _{n=0}^{\mathrm{\infty }}{T}_{2n+1}\left(a\right){𝗂}_n^{(1)}\left(\frac{1}{2}{z}^2\right),$$
$-1\le a\le 1$.
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer Referenced by: §7.6(ii) Permalink: http://dlmf.nist.gov/7.6.E9 Encodings: TeX, pMML, png See also: Annotations for §7.6(ii), §7.6 and Ch.7

7.6.10		$$C\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n}\left(\frac{1}{2}\pi z^2\right),$$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer Referenced by: §7.6(ii) Permalink: http://dlmf.nist.gov/7.6.E10 Encodings: TeX, pMML, png See also: Annotations for §7.6(ii), §7.6 and Ch.7

7.6.11		$$S\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n+1}\left(\frac{1}{2}\pi z^2\right).$$
ⓘ Symbols: $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer Referenced by: §7.6(ii) Permalink: http://dlmf.nist.gov/7.6.E11 Encodings: TeX, pMML, png See also: Annotations for §7.6(ii), §7.6 and Ch.7

For further results see Luke (1969b, pp. 57–58).

7.5 Interrelations7.7 Integral Representations

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