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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.18 Relations to Other Functions


§13.18(i) Elementary Functions

13.18.1 M0,12(2z)=2sinhz,
13.18.2 Mκ,κ-12(z)=Wκ,κ-12(z)=Wκ,-κ+12(z)=e-12zzκ,
13.18.3 Mκ,-κ-12(z)=e12zz-κ.

§13.18(ii) Incomplete Gamma Functions

For the notation see §§6.2(i), 7.2(i), and 8.2(i). When 12-κ±μ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). For example,

13.18.4 Mμ-12,μ(z)=2μe12zz12-μγ(2μ,z),
13.18.5 Wμ-12,μ(z)=e12zz12-μΓ(2μ,z).

Special cases are the error functions

13.18.6 M-14,14(z2)=12e12z2πzerf(z),
13.18.7 W-14,±14(z2)=e12z2πzerfc(z).

§13.18(iii) Modified Bessel Functions

When κ=0 the Whittaker functions can be expressed as modified Bessel functions. For the notation see §§10.25(ii) and 9.2(i).

13.18.8 M0,ν(2z)=22ν+12Γ(1+ν)zIν(z),
13.18.9 W0,ν(2z)=2z/πKν(z),
13.18.10 W0,13(43z32)=2πz14Ai(z).

§13.18(iv) Parabolic Cylinder Functions

For the notation see §12.2.

13.18.11 W-12a,±14(12z2) =212azU(a,z),
13.18.12 M-12a,-14(12z2) =212a-1Γ(12a+34)z/π(U(a,z)+U(a,-z)),
13.18.13 M-12a,14(12z2) =212a-2Γ(12a+14)z/π(U(a,-z)-U(a,z)).

§13.18(v) Orthogonal Polynomials

Special cases of §13.18(iv) are as follows. For the notation see §18.3.

Hermite Polynomials

13.18.14 M14+n,-14(z2)=(-1)nn!(2n)!e-12z2zH2n(z),
13.18.15 M34+n,14(z2)=(-1)nn!(2n+1)!e-12z2z2H2n+1(z),
13.18.16 W14+12n,14(z2)=2-ne-12z2zHn(z).

Laguerre Polynomials

§13.18(vi) Coulomb Functions

For representations of Coulomb functions in terms of Whittaker functions see (33.2.3), (33.2.7), (33.14.4) and (33.14.7)