# Circle method for Laplace transform.

1. Apr 2, 2007

### tpm

If we have the Laplace transform:

$$\int_{0}^{\infty}dtf(t)exp(-st) = \sum_{n=0}^{\infty}(f(n)-f(n-1))\frac{exp(-sn)}{s}=g(s)$$

(We have used Abel sum formula on the right hand)

then making Z=exp(s) we find the Z-transform:

$$\sum_{n=0}^{\infty}(f(n)-f(n-1))Z^{-n}$$

which can be inverted to get:

$$2\pi i (f(n)-f(n-1))=\oint g(lnZ) (lnZ)Z^{n-1}$$

my quetion is if using 'Circle method' you can get an asymptotic expansion for: $$f(n)-f(n-1)$$ n big, also another question if we have that:

$$f(n)-f(n-1) \sim h(n)$$ then ??? $$f(n) \sim \int h(n)dn$$

Last edited: Apr 2, 2007