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13 Confluent Hypergeometric FunctionsKummer Functions

§13.4 Integral Representations

Contents
  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,
b>a>0,
13.4.2 M(a,b,z)=1Γ(bc)01M(a,c,zt)tc1(1t)bc1dt,
b>c>0,
13.4.3 M(a,b,z)=z1212bΓ(a)0etta12b12Jb1(2zt)dt,
a>0.

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,
a>max(b1,0),
13.4.8 U(a,b,z)=zca0ezttc1F12(a,ab+1;c;t)dt,
|phz|<12π,

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,
a,ba1,2,3,.

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,
|phz|<12π.
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,
|phz|<12π,

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,
|phz|<32π,

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,
|phz|<12π,

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