complex analysis

For instance, with real variables the integral of a function from z₁ to z₂ is well defined along a unique path. But on the complex plane, there are infinitely many paths from one number to another, and the integral may depend on the choice of the path. Similar complications exist with complex derivatives. There's a good reason why these numbers are called complex ;-).

On the plus side, complex analysis has powerful results which can simplify some calculations enormously. For example:

• The residue theorem tells how the integral of a function over a closed path only depends on the behaviour of the function in certain points (the singular points) inside the loop.
• Any well-behaved (analytic) function has a physical interpretation in terms of potentials and fields; both its real and imaginary parts separately fulfill Laplace's equation (see also Cauchy-Riemann equations). Moreover, if one part represents equipotential lines, the other describes field lines.

Among other uses, complex analysis provides powerful tools for physics in a two-dimensional system, examples being fluid dynamics and electromagnetism. For a higher number of dimensions, there are similar techniques in Geometric algebra.