Editing Laplace's method (section)

==Complex integrals==
For complex integrals in the form:

:<math>\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(s)e^{st} \,ds</math>

with <math>t \gg 1,</math> we make the substitution ''t'' = ''iu'' and the change of variable <math>s=c+ix</math> to get the bilateral Laplace transform:

:<math>\frac{1}{2 \pi}\int_{-\infty}^\infty g(c+ix)e^{-ux}e^{icu} \, dx.</math>

We then split ''g''(''c'' + ''ix'') in its real and complex part, after which we recover ''u'' = ''t''/''i''. This is useful for [[inverse Laplace transform]]s, the [[Perron formula]] and complex integration.