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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.18 Relations to Other Functions

  1. §13.18(i) Elementary Functions
  2. §13.18(ii) Incomplete Gamma Functions
  3. §13.18(iii) Modified Bessel Functions
  4. §13.18(iv) Parabolic Cylinder Functions
  5. §13.18(v) Orthogonal Polynomials
  6. §13.18(vi) Coulomb Functions

§13.18(i) Elementary Functions

13.18.1 M0,12(2z)=2sinhz,
13.18.2 Mκ,κ12(z)=Wκ,κ12(z)=Wκ,κ+12(z)=e12zzκ,
13.18.3 Mκ,κ12(z)=e12zzκ.

§13.18(ii) Incomplete Gamma Functions

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

13.18.4 Mμ12,μ(z)=2μe12zz12μγ(2μ,z),
13.18.5 Wμ12,μ(z)=e12zz12μΓ(2μ,z).

Special cases are the error functions

13.18.6 M14,14(z2)=12e12z2πzerf(z),
13.18.7 W14,±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 W12a,±14(12z2) =212azU(a,z),
13.18.12 M12a,14(12z2) =212a1Γ(12a+34)z/π(U(a,z)+U(a,z)),
13.18.13 M12a,14(12z2) =212a2Γ(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)!e12z2zH2n(z),
13.18.15 M34+n,14(z2)=(1)nn!(2n+1)!e12z2z2H2n+1(z),
13.18.16 W14+12n,14(z2)=2ne12z2zHn(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)