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.