Principle of permanence


In mathematics, the principle of permanence states that a complex function, suitably well-behaved, which is 0 on a set containing a non-isolated point is 0 everywhere. There are various statements of the principle, depending on the type of function or equation considered.

For a complex function of one variable

For one variable, the principle of permanence states that if f is an analytic function defined on an open connected subset U of the complex numbers C, and there exists a convergent sequence having a limit L which is in U, such that f = 0 for all n, then f is uniformly zero on U.

Applications

One of the main uses of the principle of permanence is to show that a functional equation that holds for the real numbers also holds for the complex numbers.
As an example, the function es+t − eset = 0 on the real numbers. By the principle of permanence for functions of two variables, this implies that es+t − eset = 0 for all complex numbers as well, thus proving one of the laws of exponents for complex exponents.