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A detailed tutorial on saddle-point approximation. Step by step tutorial including several examples of saddle-point approximation for reference.

Saddle-point approximation is also referred to as the method of steepest descent and Laplace’s method. It is a way of approximating integrals in the form . f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b have a possibilty of being infinite.

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A detailed tutorial of solving Dirichlet problems. Step by step tutorial including several examples of how to solve Dirichlet problems for reference.

A Dirichlet problem is a problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region. It was originally supposed to be used for Laplace’s equation, although other equations can use it as well. The Dirichlet problem can be stated as: given a function f  that has values everywhere on the boundary of a region in R^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary? A mathematical solution can be expressed as: