The calculus (i.e. analysis) of complex numbers. It is quite advanced compared to ordinary calculus which uses real numbers, mainly because complex numbers lie on a two-dimensional plane, instead of a line of real numbers.

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:

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.

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