Question on the Method of Steepest Decent

[Edwin]

Is the following integral of the "Laplace Type?"

If so, is it possible to create an asymptotic expansion for contour integrals of the following form?

Contour Integral around y of e^[K*h(z)]/[f(z)*g(c/z) - epsilon],

where epsilon is a very small real valued positive constant,

C is an integer; h, f, and g are holomorphic in the region containing the contour y, y not containing the origin, and y containing a simple pole of the integral on the complex plane. The parameter K is a parameter that makes the integral convergent.

Inquisitively,

Edwin

 

[eljose]

I asked the same months ago in fact with a change of variable s=c+ix you get integrals of the form:

$$\int_{-\infty}^{\infty}dxF(c+ix)e^{ixt}$$ and a term outside the integral proportional to exp(ct) ow the problem is how do you get an asymptotic expansion for a Fourier Integral (transform) above,...in fact several Number-theoretic function have the form of an inverse Laplace transform evaluated at t=logx

 

[Edwin]

the problem is how do you get an asymptotic expansion for a Fourier Integral (transform) above,...

There are a couple of papers I am currently reading up on related to methods of generating asymptotic series for multi-point Taylor, Laurent, and Taylor/Laurent expansions. The particular paper in question generalizes 2 point Taylor expansions to m point Taylor expansions and allows one to "expand in Laurent series at some points, and Taylor series at others."

See link below for further details:

http://ftp.cwi.nl/CWIreports/MAS/MAS-E0402.pdf#search='asymptotic%20expansion%20multipoint%20taylor%20expansions'


I'm not sure whether this benefits me on my problem however; because, the particular problem I seem to have is that no combination of contours that run through saddle-points tend to enclose the desired simple pole. So, for my problem, this makes using the "method of steepest decent" less attractive, but not necessarily impossible. However, it may benefit you in constructing an asymptotic expansion for the Fourier Integral transform listed above.

Best Regards,

Edwin