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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.14 Definitions and Basic Properties

  1. §13.14(i) Differential Equation
  2. §13.14(ii) Analytic Continuation
  3. §13.14(iii) Limiting Forms as z0
  4. §13.14(iv) Limiting Forms as z
  5. §13.14(v) Numerically Satisfactory Solutions
  6. §13.14(vi) Wronskians
  7. §13.14(vii) Connection Formulas

§13.14(i) Differential Equation

Whittaker’s Equation

13.14.1 d2Wdz2+(14+κz+14μ2z2)W=0.

This equation is obtained from Kummer’s equation (13.2.1) via the substitutions W=e12zz12+μw, κ=12ba, and μ=12b12. 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)=e12zz12+μM(12+μκ,1+2μ,z),
13.14.3 Wκ,μ(z)=e12zz12+μU(12+μκ,1+2μ,z),

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

The series

13.14.6 Mκ,μ(z)=e12zz12+μs=0(12+μκ)s(1+2μ)ss!zs=z12+μn=0F12(n,12+μκ1+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+μκ,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 lim2μn1Mκ,μ(z)Γ(2μ+1)=(12nκ)n+1(n+1)!Mκ,12(n+1)(z)=e12zz12ns=n+1(12nκ)sΓ(sn)s!zs.

If 2μ=±n, where n=0,1,2,, then

13.14.8 Wκ,±12n(z)=(1)ne12zz12n+12n!Γ(1212nκ)(k=1nn!(k1)!(nk)!(κ+1212n)kzkk=0(12n+12κ)k(n+1)kk!zk(lnz+ψ(12n+12κ+k)ψ(1+k)ψ(n+1+k))),


13.14.9 Wκ,±12n(z)=(1)κ12n12e12zz12n+12×k=0κ12n12(κ12n12k)(n+1+k)κk12n12(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(12μκ)Γ(1+2μ)sin(2πμ)Mκ,μ(z)+(1)me2mμπiWκ,μ(z).
13.14.13 (1)mWκ,μ(ze2mπi)=e2κπisin(2mμπ)+sin((2m2)μπ)sin(2μπ)Wκ,μ(z)sin(2mμπ)2πieκπisin(2μπ)Γ(12+μκ)Γ(12μκ)Wκ,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 12κ±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+μκ)z12μ+O(z32μ),
μ12, μ12,
13.14.17 Wκ,12(z)=1Γ(1κ)+O(zlnz),
13.14.18 Wκ,μ(z)=Γ(2μ)Γ(12+μκ)z12μ+Γ(2μ)Γ(12μκ)z12+μ+O(z32μ),
0μ<12, μ0,
13.14.19 Wκ,0(z)=zΓ(12κ)(lnz+ψ(12κ)+2γ)+O(z3/2lnz).

For Wκ,μ(z) with μ<0 use (13.14.31).

§13.14(iv) Limiting Forms as z

Except when μκ=12,32, (polynomial cases),

13.14.20 Mκ,μ(z)Γ(1+2μ)e12zzκ/Γ(12+μκ),

where δ is an arbitrary small positive constant. Also,

13.14.21 Wκ,μ(z)e12zzκ,

§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=μκ+12, and W=e12zz12+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+μκ),
13.14.27 𝒲{Mκ,μ(z),Wκ,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(12μ)Γ(12μκ),
13.14.29 𝒲{Mκ,μ(z),Wκ,μ(e±πiz)}=Γ(12μ)Γ(12μ+κ)e(12μ)πi,
13.14.30 𝒲{Wκ,μ(z),Wκ,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±(κμ12)πiΓ(12+μ+κ)Wκ,μ(z)+e±κπiΓ(12+μκ)Wκ,μ(e±πiz).

When 2μ is not an integer

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