§7.6 Series Expansions

§7.6(i) Power Series

 7.6.1 $\displaystyle\mathop{\mathrm{erf}\/}\nolimits z$ $\displaystyle=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n% !(2n+1)},$ 7.6.2 $\displaystyle\mathop{\mathrm{erf}\/}\nolimits z$ $\displaystyle=\frac{2}{\sqrt{\pi}}e^{-z^{2}}\sum_{n=0}^{\infty}\frac{2^{n}z^{2% n+1}}{1\cdot 3\cdots(2n+1)},$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{erf}\/}\nolimits\NVar{z}$: 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) 7.6.3 $\displaystyle\mathop{w\/}\nolimits\!\left(z\right)$ $\displaystyle=\sum_{n=0}^{\infty}\frac{(iz)^{n}}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}n+1\right)}.$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{w\/}\nolimits\!\left(\NVar{z}\right)$: 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) 7.6.4 $\displaystyle\mathop{C\/}\nolimits\!\left(z\right)$ $\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n% +1)}z^{4n+1},$
 7.6.5 $\mathop{C\/}\nolimits\!\left(z\right)=\mathop{\cos\/}\nolimits\!\left(\tfrac{1% }{2}\pi z^{2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(% 4n+1)}z^{4n+1}+\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right)% \sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}.$
 7.6.6 $\mathop{S\/}\nolimits\!\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{% 1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3},$
 7.6.7 $\mathop{S\/}\nolimits\!\left(z\right)=-\mathop{\cos\/}\nolimits\!\left(\tfrac{% 1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3% \cdots(4n+3)}z^{4n+3}+\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}% \right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}.$

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 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1% )^{n}\left(\mathop{{\mathsf{i}^{(1)}_{2n}}\/}\nolimits\!\left(z^{2}\right)-% \mathop{{\mathsf{i}^{(1)}_{2n+1}}\/}\nolimits\!\left(z^{2}\right)\right),$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{erf}\/}\nolimits\NVar{z}$: error function, $\mathop{{\mathsf{i}^{(1)}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 7.6.9 $\mathop{\mathrm{erf}\/}\nolimits\!\left(az\right)=\frac{2z}{\sqrt{\pi}}e^{(% \frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}\mathop{T_{2n+1}\/}\nolimits\!\left% (a\right)\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(\tfrac{1}{2}z^{2}% \right),$ $-1\leq a\leq 1$.
 7.6.10 $\mathop{C\/}\nolimits\!\left(z\right)=z\sum_{n=0}^{\infty}\mathop{\mathsf{j}_{% 2n}\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right),$
 7.6.11 $\mathop{S\/}\nolimits\!\left(z\right)=z\sum_{n=0}^{\infty}\mathop{\mathsf{j}_{% 2n+1}\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right).$

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