Approximating rapidly divergent integrals

A while back I ran across a paper [1] giving a trick for evaluating integrals of the form where M is large and f is an increasing function.

For large M, the integral is asymptotically That is, the ratio of A(M) to I(M) goes to 1 as M goes to infinity.

This looks like a strange variation on Laplace’s approximation.

And although Laplace’s method is often useful in practice, no applications of the approximation above come to mind.

Any ideas? I have a vague feeling I could have used something like this before.

There is one more requirement on the function f.

In addition to being an increasing function, it must also satisfy In [1] the author gives several examples, including using f(x) = x².

If we wanted to approximate the method above gives exp(10000)/200 = 4.

4034 × 104340 whereas the correct value to five significant figures is 4.

4036 × 104340.

Even getting an estimate of the order of magnitude for such a large integral could be useful, and the approximation does better than that.

[1] Ira Rosenholtz.

Estimating Large Integrals: The Bigger They Are, The Harder They Fall.

The College Mathematics Journal, Vol.

32, No.

5 (Nov.

, 2001), pp.


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