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 Re

(138) |

just above the positive real axis, and

(139) |

just below the positive real axis. On the small circle enclosing

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

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

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.