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13 Confluent Hypergeometric FunctionsKummer Functions

§13.4 Integral Representations

  1. §13.4(i) Integrals Along the Real Line
  2. §13.4(ii) Contour Integrals
  3. §13.4(iii) Mellin–Barnes Integrals

§13.4(i) Integrals Along the Real Line

13.4.1 𝐌(a,b,z)=1Γ(a)Γ(ba)01eztta1(1t)ba1dt,
13.4.2 𝐌(a,b,z)=1Γ(bc)01𝐌(a,c,zt)tc1(1t)bc1dt,
13.4.3 𝐌(a,b,z)=z1212bΓ(a)0etta12b12Jb1(2zt)dt,

For the function Jb1 see §10.2(ii).

13.4.4 U(a,b,z)=1Γ(a)0eztta1(1+t)ba1dt,
a>0, |phz|<12π,
13.4.5 U(a,b,z)=z1aΓ(a)Γ(1+ab)0U(ba,b,t)etta1t+zdt,
|phz|<π, a>max(b1,0),
13.4.6 U(a,b,z)=(1)nz1bnΓ(1+ab)0𝐌(ba,b,t)ettb+n1t+zdt,
|phz|<π, n=0,1,2,, b<n<1+(ab),
13.4.7 U(a,b,z)=2z1212bΓ(a)Γ(ab+1)0etta12b12Kb1(2zt)dt,
13.4.8 U(a,b,z)=zca0ezttc1𝐅12(a,ab+1;c;t)dt,

where c is arbitrary, c>0. For the functions Kb1 and 𝐅12 see §10.25(ii) and §§15.1, 15.2(i).

§13.4(ii) Contour Integrals

13.4.9 𝐌(a,b,z)=Γ(1+ab)2πiΓ(a)0(1+)eztta1(t1)ba1dt,
ba1,2,3,, a>0.
13.4.10 𝐌(a,b,z)=eaπiΓ(1a)2πiΓ(ba)1(0+)eztta1(1t)ba1dt,
a1,2,3,, (ba)>0.
See accompanying text
Figure 13.4.1: Contour of integration in (13.4.11). (Compare Figure 5.12.3.) Magnify
13.4.11 𝐌(a,b,z)=ebπiΓ(1a)Γ(1+ab)14π2×α(0+,1+,0,1)eztta1(1t)ba1dt,

The contour of integration starts and terminates at a point α on the real axis between 0 and 1. It encircles t=0 and t=1 once in the positive sense, and then once in the negative sense. See Figure 13.4.1. The fractional powers are continuous and assume their principal values at t=α. Similar conventions also apply to the remaining integrals in this subsection.

13.4.12 𝐌(a,c,z)=Γ(b)2πiz1b(0+,1+)ezttb𝐅12(a,b;c;1/t)dt,
b0,1,2,, |phz|<12π.

At the point where the contour crosses the interval (1,), tb and the 𝐅12 function assume their principal values; compare §§15.1 and 15.2(i). A special case is

13.4.13 𝐌(a,b,z)=z1b2πi(0+,1+)ezttb(11t)adt,
13.4.14 U(a,b,z)=eaπiΓ(1a)2πi(0+)eztta1(1+t)ba1dt,
a1,2,3,, |phz|<12π.

The contour cuts the real axis between 1 and 0. At this point the fractional powers are determined by pht=π and ph(1+t)=0.

Again, tc and the 𝐅12 function assume their principal values where the contour intersects the positive real axis.

§13.4(iii) Mellin–Barnes Integrals

If a0,1,2,, then

13.4.16 𝐌(a,b,z)=12πiΓ(a)iiΓ(a+t)Γ(t)Γ(b+t)ztdt,

where the contour of integration separates the poles of Γ(a+t) from those of Γ(t).

If a and ab+10,1,2,, then

13.4.17 U(a,b,z)=za2πiiiΓ(a+t)Γ(1+ab+t)Γ(t)Γ(a)Γ(1+ab)ztdt,

where the contour of integration separates the poles of Γ(a+t)Γ(1+ab+t) from those of Γ(t).

13.4.18 U(a,b,z)=z1bez2πiiiΓ(b1+t)Γ(t)Γ(a+t)ztdt,

where the contour of integration passes all the poles of Γ(b1+t)Γ(t) on the right-hand side.