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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.14 Definitions and Basic Properties


§13.14(i) Differential Equation

Whittaker’s Equation

13.14.1 d2Wdz2+(-14+κz+14-μ2z2)W=0.

This equation is obtained from Kummer’s equation (13.2.1) via the substitutions W=e-12zz12+μw, κ=12b-a, and μ=12b-12. It has a regular singularity at the origin with indices 12±μ, and an irregular singularity at infinity of rank one.

Standard Solutions

Standard solutions are:

13.14.2 Mκ,μ(z)=e-12zz12+μM(12+μ-κ,1+2μ,z),
13.14.3 Wκ,μ(z)=e-12zz12+μU(12+μ-κ,1+2μ,z),

except that Mκ,μ(z) does not exist when 2μ=-1,-2,-3,.

The series

13.14.6 Mκ,μ(z)=e-12zz12+μs=0(12+μ-κ)s(1+2μ)ss!zs=z12+μn=0F12(-n,12+μ-κ1+2μ;2)(-12z)nn!,

converge for all z.

In general Mκ,μ(z) and Wκ,μ(z) are many-valued functions of z with branch points at z=0 and z=. The principal branches correspond to the principal branches of the functions z12+μ and U(12+μ-κ,1+2μ,z) on the right-hand sides of the equations (13.14.2) and (13.14.3); compare §4.2(i).

Although Mκ,μ(z) does not exist when 2μ=-1,-2,-3,, many formulas containing Mκ,μ(z) continue to apply in their limiting form. For example, if n=0,1,2,, then

13.14.7 lim2μ-n-1Mκ,μ(z)Γ(2μ+1)=(-12n-κ)n+1(n+1)!Mκ,12(n+1)(z)=e-12zz-12ns=n+1(-12n-κ)sΓ(s-n)s!zs.

If 2μ=±n, where n=0,1,2,, then

13.14.8 Wκ,±12n(z)=(-1)ne-12zz12n+12n!Γ(12-12n-κ)(k=1nn!(k-1)!(n-k)!(κ+12-12n)kz-k-k=0(12n+12-κ)k(n+1)kk!zk(lnz+ψ(12n+12-κ+k)-ψ(1+k)-ψ(n+1+k))),


13.14.9 Wκ,±12n(z)=(-1)κ-12n-12e-12zz12n+12×k=0κ-12n-12(κ-12n-12k)(n+1+k)κ-k-12n-12(-z)k,

§13.14(ii) Analytic Continuation

In (13.14.11)–(13.14.13) m is any integer.

13.14.11 Mκ,μ(ze2mπi)=(-1)me2mμπiMκ,μ(z).
13.14.12 Wκ,μ(ze2mπi)=(-1)m+12πisin(2πμm)Γ(12-μ-κ)Γ(1+2μ)sin(2πμ)Mκ,μ(z)+(-1)me-2mμπiWκ,μ(z).
13.14.13 (-1)mWκ,μ(ze2mπi)=-e2κπisin(2mμπ)+sin((2m-2)μπ)sin(2μπ)Wκ,μ(z)-sin(2mμπ)2πieκπisin(2μπ)Γ(12+μ-κ)Γ(12-μ-κ)W-κ,μ(zeπi).

Except when z=0, each branch of the functions Mκ,μ(z)/Γ(2μ+1) and Wκ,μ(z) is entire in κ and μ. Also, unless specified otherwise Mκ,μ(z) and Wκ,μ(z) are assumed to have their principal values.

§13.14(iii) Limiting Forms as z0

13.14.14 Mκ,μ(z)=zμ+12(1+O(z)),

In cases when 12-κ±μ=-n, where n is a nonnegative integer,

13.14.15 W12±μ+n,μ(z)=(-1)n(1±2μ)nz12±μ+O(z32±μ).

In all other cases

13.14.16 Wκ,μ(z)=Γ(2μ)Γ(12+μ-κ)z12-μ+O(z32-μ),
μ12, μ12,
13.14.17 Wκ,12(z)=1Γ(1-κ)+O(zlnz),
13.14.18 Wκ,μ(z)=Γ(2μ)Γ(12+μ-κ)z12-μ+Γ(-2μ)Γ(12-μ-κ)z12+μ+O(z32-μ),
0μ<12, μ0,
13.14.19 Wκ,0(z)=-zΓ(12-κ)(lnz+ψ(12-κ)+2γ)+O(z3/2lnz).

For Wκ,μ(z) with μ<0 use (13.14.31).

§13.14(iv) Limiting Forms as z

Except when μ-κ=-12,-32, (polynomial cases),

13.14.20 Mκ,μ(z)Γ(1+2μ)e12zz-κ/Γ(12+μ-κ),

where δ is an arbitrary small positive constant. Also,

13.14.21 Wκ,μ(z)e-12zzκ,

§13.14(v) Numerically Satisfactory Solutions

Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

13.14.22 Wκ,μ(z),
13.14.23 Wκ,μ(z),

A fundamental pair of solutions that is numerically satisfactory in the sector |phz|π near the origin is

13.14.24 Mκ,μ(z),

When 2μ is an integer we may use the results of §13.2(v) with the substitutions b=2μ+1, a=μ-κ+12, and W=e-12zz12+μw, where W is the solution of (13.14.1) corresponding to the solution w of (13.2.1).

§13.14(vi) Wronskians

13.14.25 𝒲{Mκ,μ(z),Mκ,-μ(z)}=-2μ,
13.14.26 𝒲{Mκ,μ(z),Wκ,μ(z)}=-Γ(1+2μ)Γ(12+μ-κ),
13.14.27 𝒲{Mκ,μ(z),W-κ,μ(e±πiz)}=Γ(1+2μ)Γ(12+μ+κ)e(12+μ)πi,
13.14.28 𝒲{Mκ,-μ(z),Wκ,μ(z)}=-Γ(1-2μ)Γ(12-μ-κ),
13.14.29 𝒲{Mκ,-μ(z),W-κ,μ(e±πiz)}=Γ(1-2μ)Γ(12-μ+κ)e(12-μ)πi,
13.14.30 𝒲{Wκ,μ(z),W-κ,μ(e±πiz)}=eκπi.

§13.14(vii) Connection Formulas

13.14.31 Wκ,μ(z)=Wκ,-μ(z).
13.14.32 1Γ(1+2μ)Mκ,μ(z)=e±(κ-μ-12)πiΓ(12+μ+κ)Wκ,μ(z)+e±κπiΓ(12+μ-κ)W-κ,μ(e±πiz).

When 2μ is not an integer

13.14.33 Wκ,μ(z)=Γ(-2μ)Γ(12-μ-κ)Mκ,μ(z)+Γ(2μ)Γ(12+μ-κ)Mκ,-μ(z).