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13 Confluent Hypergeometric FunctionsKummer Functions

§13.4 Integral Representations

Contents

§13.4(i) Integrals Along the Real Line

13.4.1 M(a,b,z)=1Γ(a)Γ(b-a)01eztta-1(1-t)b-a-1dt,
b>a>0,
13.4.2 M(a,b,z)=1Γ(b-c)01M(a,c,zt)tc-1(1-t)b-c-1dt,
b>c>0,
13.4.3 M(a,b,-z)=z12-12bΓ(a)0e-tta-12b-12Jb-1(2zt)dt,
a>0.

For the function Jb-1 see §10.2(ii).

13.4.4 U(a,b,z)=1Γ(a)0e-ztta-1(1+t)b-a-1dt,
a>0, |phz|<12π,
13.4.5 U(a,b,z)=z1-aΓ(a)Γ(1+a-b)0U(b-a,b,t)e-tta-1t+zdt,
|phz|<π, a>max(b-1,0),
13.4.6 U(a,b,z)=(-1)nz1-b-nΓ(1+a-b)0M(b-a,b,t)e-ttb+n-1t+zdt,
|phz|<π, n=0,1,2,, -b<n<1+(a-b),
13.4.7 U(a,b,z)=2z12-12bΓ(a)Γ(a-b+1)0e-tta-12b-12Kb-1(2zt)dt,
a>max(b-1,0),
13.4.8 U(a,b,z)=zc-a0e-zttc-1F12(a,a-b+1;c;-t)dt,
|phz|<12π,

where c is arbitrary, c>0. For the functions Kb-1 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+a-b)2πiΓ(a)0(1+)eztta-1(t-1)b-a-1dt,
b-a1,2,3,, a>0.
13.4.10 M(a,b,z)=e-aπiΓ(1-a)2πiΓ(b-a)1(0+)eztta-1(1-t)b-a-1dt,
a1,2,3,, (b-a)>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)=e-bπiΓ(1-a)Γ(1+a-b)14π2α(0+,1+,0-,1-)eztta-1(1-t)b-a-1dt,
a,b-a1,2,3,.

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)2πiz1-b-(0+,1+)eztt-bF12(a,b;c;1/t)dt,
b0,-1,-2,, |phz|<12π.

At the point where the contour crosses the interval (1,), t-b 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)=z1-b2πi-(0+,1+)eztt-b(1-1t)-adt,
|phz|<12π.
13.4.14 U(a,b,z)=e-aπiΓ(1-a)2πi(0+)e-ztta-1(1+t)b-a-1dt,
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.

13.4.15 U(a,b,z)Γ(c)Γ(c-b+1)=z1-c2πi-(0+)eztt-cF12(a,c;a+c-b+1;1-1t)dt,
|phz|<12π.

Again, t-c 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 a-b+10,-1,-2,, then

13.4.17 U(a,b,z)=z-a2πi-iiΓ(a+t)Γ(1+a-b+t)Γ(-t)Γ(a)Γ(1+a-b)z-tdt,
|phz|<32π,

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

13.4.18 U(a,b,z)=z1-bez2πi-iiΓ(b-1+t)Γ(t)Γ(a+t)z-tdt,
|phz|<12π,

where the contour of integration passes all the poles of Γ(b-1+t)Γ(t) on the right-hand side.