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 $\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n% +1}}{n!(2n+1)},$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erf}\NVar{z}$ : error function, : factorial (as in ), : complex variable and : 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 $\operatorname{erf}z=\frac{2}{\sqrt{\pi}}e^{-z^{2}}\sum_{n=0}^{\infty}\frac{2^{% n}z^{2n+1}}{1\cdot 3\cdots(2n+1)},$

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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, : complex variable and : 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 $w\left(z\right)=\sum_{n=0}^{\infty}\frac{(iz)^{n}}{\Gamma\left(\frac{1}{2}n+1% \right)}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, $\mathrm{i}$ : imaginary unit, : complex variable and : 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 $C\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(% 4n+1)}z^{4n+1},$

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Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : complex variable and : 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 $C\left(z\right)=\cos\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}\frac% {(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1)}z^{4n+1}+\sin\left(\tfrac{1}{2}\pi z^{% 2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{% 4n+3}.$

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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, : complex variable and : 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 $S\left(z\right)=\sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+% 1)!(4n+3)}z^{4n+3},$

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Symbols:: $S\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : complex variable and : 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 $S\left(z\right)=-\cos\left(\tfrac{1}{2}\pi z^{2}\right)\sum_{n=0}^{\infty}% \frac{(-1)^{n}\pi^{2n+1}}{1\cdot 3\cdots(4n+3)}z^{4n+3}+\sin\left(\tfrac{1}{2}% \pi z^{2}\right)\sum_{n=0}^{\infty}\frac{(-1)^{n}\pi^{2n}}{1\cdot 3\cdots(4n+1% )}z^{4n+1}.$

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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, : complex variable and : 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 .

§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 $\operatorname{erf}z=\frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left({% \mathsf{i}^{(1)}_{2n}}\left(z^{2}\right)-{\mathsf{i}^{(1)}_{2n+1}}\left(z^{2}% \right)\right),$

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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, : complex variable and : 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 $\operatorname{erf}\left(az\right)=\frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z% ^{2}}\sum_{n=0}^{\infty}T_{2n+1}\left(a\right){\mathsf{i}^{(1)}_{n}}\left(% \tfrac{1}{2}z^{2}\right),$ $-1\leq a\leq 1$ .

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