7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.5 Interrelations 7.7 Integral Representations

§7.6 Series Expansions

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Permalink:: http://dlmf.nist.gov/7.6
See also:: Annotations for Ch.7

§7.6(i) Power Series
§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).
Permalink:: http://dlmf.nist.gov/7.6.i
See also:: Annotations for §7.6 and Ch.7

7.6.1	$\displaystyle\operatorname{erf}z$	$\displaystyle=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n% !(2n+1)},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : 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	$\displaystyle\operatorname{erf}z$	$\displaystyle=\frac{2}{\sqrt{\pi}}e^{-z^{2}}\sum_{n=0}^{\infty}\frac{2^{n}z^{2% n+1}}{1\cdot 3\cdots(2n+1)},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : 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	$\displaystyle w\left(z\right)$	$\displaystyle=\sum_{n=0}^{\infty}\frac{(iz)^{n}}{\Gamma\left(\frac{1}{2}n+1% \right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $w\left(\NVar{z}\right)$ : 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	$\displaystyle C\left(z\right)$	$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n% +1)}z^{4n+1},$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : 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		$C\left(z\right)=\cos\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}\frac% {(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}+\sin\left(\tfrac{1}{2}\pi z^{% 2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{% 4n+3}.$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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		$S\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+% 1)!(4n+3)}z^{4n+3},$
ⓘ Symbols: $S\left(\NVar{z}\right)$ : 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		$S\left(z\right)=-\cos\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}% \frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}+\sin\left(\tfrac{1}{2}% \pi z^{2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1% )}z^{4n+1}.$
ⓘ Symbols: $S\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : 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 $\exp\left(a^{2}z^{2}\right)\operatorname{erf}\left(az\right)$ 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		$\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left({% \mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^{(1)}_{2n+1}}\left(z^{2}% \right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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		$\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z% ^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){\mathsf{i}^{(1)}_{n}}\left(% \tfrac{1}{2}z^{2}\right),$
$-1\leq a\leq 1$ .
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, ${\mathsf{i}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : 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		$C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n}\left(\tfrac{1}{2}\pi z^{2}% \right),$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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		$S\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n+1}\left(\tfrac{1}{2}\pi z^{% 2}\right).$
ⓘ Symbols: $S\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : 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 Interrelations 7.7 Integral Representations

NIST

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§7.6 Series Expansions

Contents

§7.6(i) Power Series

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