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

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

§7.6(i) Power Series

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).
Keywords:: Fresnel integrals, error functions
Permalink:: http://dlmf.nist.gov/7.6.i

7.6.1 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{2}{\sqrt{\pi}}\sum_{{n=0}}^{\infty}% \frac{(-1)^{n}z^{{2n+1}}}{n!(2n+1)},$

Symbols:: $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : : factorial, : 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

7.6.2 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{2}{\sqrt{\pi}}e^{{-z^{2}}}\sum_{{n=0}% }^{\infty}\frac{2^{n}z^{{2n+1}}}{1\cdot 3\cdots(2n+1)},$

Symbols:: $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : base of exponential function, : 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

7.6.3 $\mathop{w\/}\nolimits\!\left(z\right)=\sum_{{n=0}}^{\infty}\frac{(iz)^{n}}{% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}n+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, : 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

7.6.4 $\mathop{C\/}\nolimits\!\left(z\right)=\sum_{{n=0}}^{\infty}\frac{(-1)^{n}(% \frac{1}{2}\pi)^{{2n}}}{(2n)!(4n+1)}z^{{4n+1}},$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, : : factorial, : 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

7.6.5 $\mathop{C\/}\nolimits\!\left(z\right)=\mathop{\cos\/}\nolimits\!\left(\tfrac{1% }{2}\pi z^{2}\right)\sum_{{n=0}}^{\infty}\frac{(-1)^{n}\pi^{{2n}}}{1\cdot 3% \cdots(4n+1)}z^{{4n+1}}+\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}% \right)\sum_{{n=0}}^{\infty}\frac{(-1)^{n}\pi^{{2n+1}}}{1\cdot 3\cdots(4n+3)}z% ^{{4n+3}}.$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

7.6.6 $\mathop{S\/}\nolimits\!\left(z\right)=\sum_{{n=0}}^{\infty}\frac{(-1)^{n}(% \frac{1}{2}\pi)^{{2n+1}}}{(2n+1)!(4n+3)}z^{{4n+3}},$

Symbols:: $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, : : factorial, : 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

7.6.7 $\mathop{S\/}\nolimits\!\left(z\right)=-\mathop{\cos\/}\nolimits\!\left(\tfrac{% 1}{2}\pi z^{2}\right)\sum_{{n=0}}^{\infty}\frac{(-1)^{n}\pi^{{2n+1}}}{1\cdot 3% \cdots(4n+3)}z^{{4n+3}}+\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}% \right)\sum_{{n=0}}^{\infty}\frac{(-1)^{n}\pi^{{2n}}}{1\cdot 3\cdots(4n+1)}z^{% {4n+1}}.$

Symbols:: $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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

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

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 $\mathop{\exp\/}\nolimits\!\left(a^{2}z^{2}\right)\mathop{\mathrm{erf}\/}% \nolimits\!\left(az\right)$ as integrals (§3.11(ii)), then apply (5.12.5), (7.6.2), and (13.6.9).
Keywords:: Fresnel integrals, error functions
Referenced by:: §7.6(i)
Permalink:: http://dlmf.nist.gov/7.6.ii

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:: $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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

7.6.9 $\mathop{\mathrm{erf}\/}\nolimits\!\left(az\right)=\frac{2z}{\sqrt{\pi}}e^{{(% \frac{1}{2}-a^{2})z^{2}}}\sum_{{n=0}}^{\infty}\mathop{T_{{2n+1}}\/}\nolimits\!% \left(a\right)\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(\tfrac{1}{% 2}z^{2}\right),$ $-1\leq a\leq 1$ .

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, : base of exponential function, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : complex variable and : nonnegative integer
Referenced by:: §7.6(ii)
Permalink:: http://dlmf.nist.gov/7.6.E9
Encodings:: TeX, pMML, png

7.6.10 $\mathop{C\/}\nolimits\!\left(z\right)=z\sum_{{n=0}}^{\infty}\mathop{\mathsf{j}% _{{2n}}\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right),$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : complex variable and : nonnegative integer
Referenced by:: §7.6(ii)
Permalink:: http://dlmf.nist.gov/7.6.E10
Encodings:: TeX, pMML, png

7.6.11 $\mathop{S\/}\nolimits\!\left(z\right)=z\sum_{{n=0}}^{\infty}\mathop{\mathsf{j}% _{{2n+1}}\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right).$

Symbols:: $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : complex variable and : nonnegative integer
Referenced by:: §7.6(ii)
Permalink:: http://dlmf.nist.gov/7.6.E11
Encodings:: TeX, pMML, png

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

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§7.6 Series Expansions

Contents

§7.6(i) Power Series

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