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7 Error Functions, Dawson’s and Fresnel IntegralsProperties

§7.6 Series Expansions

  1. §7.6(i) Power Series
  2. §7.6(ii) Expansions in Series of Spherical Bessel Functions

§7.6(i) Power Series

7.6.1 erfz =2πn=0(1)nz2n+1n!(2n+1),
7.6.2 erfz =2πez2n=02nz2n+113(2n+1),
7.6.3 w(z) =n=0(iz)nΓ(12n+1).
7.6.4 C(z) =n=0(1)n(12π)2n(2n)!(4n+1)z4n+1,
7.6.5 C(z)=cos(12πz2)n=0(1)nπ2n13(4n+1)z4n+1+sin(12πz2)n=0(1)nπ2n+113(4n+3)z4n+3.
7.6.6 S(z)=n=0(1)n(12π)2n+1(2n+1)!(4n+3)z4n+3,
7.6.7 S(z)=cos(12πz2)n=0(1)nπ2n+113(4n+3)z4n+3+sin(12πz2)n=0(1)nπ2n13(4n+1)z4n+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 erfz=2zπn=0(1)n(𝗂2n(1)(z2)𝗂2n+1(1)(z2)),
7.6.9 erf(az)=2zπe(12a2)z2n=0T2n+1(a)𝗂n(1)(12z2),
7.6.10 C(z)=zn=0𝗃2n(12πz2),
7.6.11 S(z)=zn=0𝗃2n+1(12πz2).

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