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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.16 Integral Representations

Contents

§13.16(i) Integrals Along the Real Line

In this subsection see §§10.2(ii), 10.25(ii) for the functions J2μ, I2μ, and K2μ, and §§15.1, 15.2(i) for F12.

13.16.1 Mκ,μ(z) =Γ(1+2μ)zμ+122-2μΓ(12+μ-κ)Γ(12+μ+κ)-1112zt(1+t)μ-12-κ(1-t)μ-12+κt,
μ+12>|κ|,
13.16.2 Mκ,μ(z) =Γ(1+2μ)zλΓ(1+2μ-2λ)Γ(2λ)01Mκ-λ,μ-λ(zt)12z(t-1)tμ-λ-12(1-t)2λ-1t,
μ+12>λ>0,
13.16.3 1Γ(1+2μ)Mκ,μ(z) =z12zΓ(12+μ+κ)0-ttκ-12J2μ(2zt)t,
(κ+μ)+12>0,
13.16.4 1Γ(1+2μ)Mκ,μ(z) =z-12zΓ(12+μ-κ)0-tt-κ-12I2μ(2zt)t,
(κ-μ)-12<0.
13.16.5 Wκ,μ(z)=zμ+122-2μΓ(12+μ-κ)1-12zt(t-1)μ-12-κ(t+1)μ-12+κt,
μ+12>κ, |phz|<12π,
13.16.6 Wκ,μ(z)=-12zzκ+1Γ(12+μ-κ)Γ(12-μ-κ)0W-κ,μ(t)-12tt-κ-1t+zt,
|phz|<π, (12+μ-κ)>max(2μ,0),
13.16.7 Wκ,μ(z)=(-1)n-12zz12-μ-nΓ(1+2μ)Γ(12-μ-κ)0M-κ,μ(t)-12ttn+μ-12t+zt,
|phz|<π, n=0,1,2,, -(1+2μ)<n<|μ|+κ<12,
13.16.8 Wκ,μ(z)=2z-12zΓ(12+μ-κ)Γ(12-μ-κ)0-tt-κ-12K2μ(2zt)t,
(μ-κ)+12>0,
13.16.9 Wκ,μ(z)=-12zzκ+c0-zttc-1F12(12+μ-κ,12-μ-κc;-t)t,
|phz|<12π,

where c is arbitrary, c>0.

§13.16(ii) Contour Integrals

For contour integral representations combine (13.14.2) and (13.14.3) with §13.4(ii). See Buchholz (1969, §2.3), Erdélyi et al. (1953a, §6.11.3), and Slater (1960, Chapter 3). See also §13.16(iii).

§13.16(iii) Mellin–Barnes Integrals

If 12+μ-κ0,-1,-2,, then

13.16.10 1Γ(1+2μ)Mκ,μ(±πz)=12z±(12+μ)π2πΓ(12+μ-κ)-Γ(t-κ)Γ(12+μ-t)Γ(12+μ+t)ztt,
|phz|<12π,

where the contour of integration separates the poles of Γ(t-κ) from those of Γ(12+μ-t).

If 12±μ-κ0,-1,-2,, then

13.16.11 Wκ,μ(z)=-12z2π-Γ(12+μ+t)Γ(12-μ+t)Γ(-κ-t)Γ(12+μ-κ)Γ(12-μ-κ)z-tt,
|phz|<32π,

where the contour of integration separates the poles of Γ(12+μ+t)Γ(12-μ+t) from those of Γ(-κ-t).

13.16.12 Wκ,μ(z)=12z2π-Γ(12+μ+t)Γ(12-μ+t)Γ(1-κ+t)z-tt,
|phz|<12π,

where the contour of integration passes all the poles of Γ(12+μ+t)Γ(12-μ+t) on the right-hand side.