In section 6 we found the power series (47) for the Bessel
function .
By means of the Hankel representation (143) in
the last section we can now easily find an integral representation for the
Bessel function,

= | |||

= | (144) |

Often the contour is turned , so that it runs from to a small circel enclosing zero, and then returning to . Then

This can be brought to another form by taking

(146) |

We now deform the contour so that it runs from to -1, encircles 0 through the unit circle, and then returns from -1 to . Thus

(147) |

In the first integral on the right we have , in the second and in the third . Thus

(148) |

In the last integral we substitute and obtain the standard integral representation

If is an integer equal to

(150) |

From the integral representation (149) we can obtain an asymptotic
formula
for
by means of the saddle point method. First, let
us notice that the second integral on the right hand side does
not have a saddle, because if we differentiate the exponent we get
,
which has no solution in the range of integration from
0 to infinity. The first integral on the right hand side of (149)
can be written as

(151) |

and has two saddle points. Differentiating the exponent we find them at , corresponding to . The values for sin are therefore . Using the formulas (136) with the function

(152) |

Here it was used that the second derivative of , namely , becomes for . This was important in (136) for the sign of the term in the exponent. Thus, summing over the saddle points we have the asymptotic expansion

showing that the Bessel function have damped oscillations for large values of the argument

From the integral representation (145) it is easy to derive
some important recursion relations by differentiation. The reader
should have no difficulty in showing that

and hence

From (154) and

= | |||

= | (156) |

or

(157) |

we can easily derive

(158) |

which are useful relations.

We can trivially rewrite (154) as

(159) |

By induction it is then simple to show