In complex analysis, Morera's theorem states that if the integral of a continuous complex-valued function f of a complex variable along every simple closed curve within an open set is zero, that is, if Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

for C any simple closed curve, then f is differentiable at every point in that open set.

Morera's theorem can be used to show the analyticity of functions defined by sums or integrals, such as the Riemann zeta function In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...

or the Gamma function The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ...

It also leads to a quick proof of the general result that if a sequence This is a page about mathematics. ...

f_n(z),

of analytic functions on a given open set D of complex numbers, converges to a function In mathematics, an analytic function is one that is locally given by a convergent power series. ...

f(z)

uniformly on every compact subset K, then f is analytic. The condition can easily be reduced to K being a closed disk. In mathematical analysis, a sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. ... In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has...