9/8/12: Basic Stuff
The Gaussian Integral: An useful integral that can be used to evaluate many definite integrals occurring in different areas of physics.
Extending the concept of Gaussian integral we arrive at a very useful relation: The Hubbard-Stratonovich Transformation, that allows formulating a problem involving a system of interacting particles through two-body potentials into a problem involving a system of independent particles
interacting with a fluctuating field.
For further reading:
Michael Cross demonstrates how the Hubbard-Stratonovich transformation can be used to obtain the partition function for the Ising model.

Evaluating the same integrand as the Gaussian integral gives us other functions, such as the error function erf(x).

A closely related function is the Gamma function.

These are just a sampler from the large variety of special functions that one comes across. Of course, in the majority of cases we will be working with trigonometric functions and their close relatives, the hyperbolic functions.

Physical interpretation of hyperbolic functions: The motion of particles away from an unstable equilibrium, the catenary curve and catenary arch (see e.g., the Wikipedia entry on catenary.)

An useful technique one should learn is how to draw the graph of arbitrary functions.