# §7.6 Series Expansions

## §7.6(i) Power Series

 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 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 exponential function, $z$: 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 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, $w\left(\NVar{z}\right)$: complementary error function, $z$: 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 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 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 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 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 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

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 7
 7.6.9 $\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z% ^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){\mathsf{i}^{(1)}_{n}}\left(% \tfrac{1}{2}z^{2}\right),$ $-1\leq a\leq 1$.
 7.6.10 $C\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n}\left(\tfrac{1}{2}\pi z^{2}% \right),$
 7.6.11 $S\left(z\right)=z\sum_{n=0}^{\infty}\mathsf{j}_{2n+1}\left(\tfrac{1}{2}\pi z^{% 2}\right).$

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