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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.16 Integral Representations

  1. §13.16(i) Integrals Along the Real Line
  2. §13.16(ii) Contour Integrals
  3. §13.16(iii) Mellin–Barnes Integrals

§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+1222μΓ(12+μκ)Γ(12+μ+κ)11e12zt(1+t)μ12κ(1t)μ12+κdt,
13.16.2 Mκ,μ(z) =Γ(1+2μ)zλΓ(1+2μ2λ)Γ(2λ)01Mκλ,μλ(zt)e12z(t1)tμλ12(1t)2λ1dt,
13.16.3 1Γ(1+2μ)Mκ,μ(z) =ze12zΓ(12+μ+κ)0ettκ12J2μ(2zt)dt,
13.16.4 1Γ(1+2μ)Mκ,μ(z) =ze12zΓ(12+μκ)0ettκ12I2μ(2zt)dt,
13.16.5 Wκ,μ(z)=zμ+1222μΓ(12+μκ)1e12zt(t1)μ12κ(t+1)μ12+κdt,
μ+12>κ, |phz|<12π,
13.16.6 Wκ,μ(z)=e12zzκ+1Γ(12+μκ)Γ(12μκ)0Wκ,μ(t)e12ttκ1t+zdt,
|phz|<π, (12+μκ)>max(2μ,0),
13.16.7 Wκ,μ(z)=(1)ne12zz12μnΓ(1+2μ)Γ(12μκ)0Mκ,μ(t)e12ttn+μ12t+zdt,
|phz|<π, n=0,1,2,, (1+2μ)<n<|μ|+κ<12,
13.16.8 Wκ,μ(z)=2ze12zΓ(12+μκ)Γ(12μκ)0ettκ12K2μ(2zt)dt,
13.16.9 Wκ,μ(z)=e12zzκ+c0ezttc1𝐅12(12+μκ,12μκc;t)dt,

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+μκ0,1,2,, then

13.16.10 1Γ(1+2μ)Mκ,μ(e±πiz)=e12z±(12+μ)πi2πiΓ(12+μκ)iiΓ(tκ)Γ(12+μt)Γ(12+μ+t)ztdt,

where the contour of integration separates the poles of Γ(tκ) from those of Γ(12+μt).

If 12±μκ0,1,2,, then

13.16.11 Wκ,μ(z)=e12z2πiiiΓ(12+μ+t)Γ(12μ+t)Γ(κt)Γ(12+μκ)Γ(12μκ)ztdt,

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

13.16.12 Wκ,μ(z)=e12z2πiiiΓ(12+μ+t)Γ(12μ+t)Γ(1κ+t)ztdt,

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