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

§7.6 Series Expansions

ⓘ

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

ⓘ

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

ⓘ

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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§7.6 Series Expansions

Contents

§7.6(i) Power Series

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