Integration by parts (IBP)

See examples in notes, and problems.

One has to choose the right functions. Nice because it gives error term explicitly, and can often be bounded.

Trick of separating integral domain.

Failure of integration by parts

General rule: Integration by parts will not work if the contribution from one of the limits of integration is much larger than the size of the integral.

It can still fail in other cases, if for some reason the terms in the expansion can't be generated by the IBP.

Laplace-type integrals

$I(x) = \int_a^b f(t) e^{x\phi(t)} dt$   as $x\rightarrow \infty$

Laplace method

For $\phi(t)$ real. Contributions near global maxima of $\phi(t)$.

Method of stationary phase

For $\phi(t)$ imaginary. Contributions regions of stationary phase $\psi(t)$ (where $\phi(t)=i\psi(t)$).

Method of steepest descents

Most general and powerful. For $\phi(t)$ generally complex, and the integral being along a complex contour in general too.

Splitting range of integration

splitting the range of integration and using different approximations in each range.

See examples

Bounding integrals

Trick I use similar to IBP