# Asymptotic expansion of complex integrals?

1. Mar 18, 2006

### eljose

Let,s suppose we have the next integral:

$$\int_{c-i\infty}^{c+i\infty}dsF(s)e^{sx}$$

with c a real number..of course F(s) is so complicated that we can not evaluate it... first of all we make the change of variable s=c+iu so we would get the "new" integral:

$$\int_{-\infty}^{\infty}iduF(c+iu)e^{iux+cx}$$ being this a Fourier transform..my question is supposing x---->oo (infinity) ¿How could we evaluate it by a "Saddle point" method or other form similar to the one Laplace did to calculate the factorial?..thanks in advance.

2. Mar 18, 2006

### Clausius2

What about using Stationary Phase Method?. I think it works.

3. Mar 18, 2006

### Tide

Eljose asked about the "saddle point" method so I think he is doing stationary phase.

4. Mar 19, 2006

### eljose

Ummm..thanks for replying my problem is that for example when we have the integral:

$$\int_{-\infty}^{\infty}dxe^{-sM(x)}$$

where a is a big parameter (real) s------>oo, then we could use Laplace method to solve it, but my integral proposed has a complex integrand and is itself complex, i think that Cauchy developed a formula to deal with those integrals but i don,t know the results i have looked into the wikipedia or mathworld but have not found any information when the integral is complex.

5. Mar 19, 2006

### shmoe

Try looking in some textbooks, anything dealing with exponential sums will have some methods to deal with exponential integrals (e.g. titchmarsh or ivic's zeta books).

A complex integral is just two real integrals.