Next: Integral representations of the Up: No Title Previous: A guide to the

# An integral for the reciprocal Gamma function

As a preliminary to finding an integral representation of the Bessel function we shall first find an integral for the reciprocal Gamma function. Consider the integral

 (137)

where Rez>0 and where the contour C starts slightly above the real axis at , runs down to t=0, where it goes around counter-clockwise in a small circle and returns to just below the real axis. The t-plane is cut from 0 to . We define the logarithm such that is real on the negative axis. Thus, on the negative real axis arg(-t)=0. On the contour C we have , such that arg(-t)=0 on the negative real t-axis. Therefore, just above the positive real t-axis we have arg, whereas just below we have arg, the angle being measured counter-clockwise. It then follows that

 (138)

just above the positive real axis, and

 (139)

just below the positive real axis. On the small circle enclosing t=0 we have . Then
 I = (140)

It was used that the integral over the small circle vanishes as . The integral on the right hand side is the usual representation for the Gamma function, so , or

 (141)

This is Hankel's formula for the Gamma function, valid for all . Using the formula

 (142)

derived in the book in eqs. (3-49)-(3-50), we get the following integral representation for the reciprocal Gamma function,

 (143)

Here we have written for , meaning thereby a path starting at infinity on the real axis, encircling 0 in a positive sense, and returning to infinity along the real axis, respecting the cut along the positive real axis.

Next: Integral representations of the Up: No Title Previous: A guide to the
Mette Lund
2000-04-27