Asymptotic expansion of complex integrals?

[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.

 

[Clausius2]

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.

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

 

[Tide]

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

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

 

[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.

 

[shmoe]

...i have looked into the wikipedia or mathworld but have not found any information when the integral is complex.

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.