Morera’s Theorem
stands as an interesting counterpoint to the Cauchy-Goursat Theorem. The latter theorem posits that if a function f(z) is analytic and single-valued inside and on a simple closed contour C, then the integral of f(z) over C is zero. Morera’s theorem, in contrast, sort of flips this idea on its head.
This theorem was brought to light by the Italian engineer and mathematician Giacinto Morera, who made significant contributions to the field of Linear Elasticity and the theory of functions in
complex variables
.
Understanding Morera’s Theorem
Morera’s theorem deals with the integration of complex functions in a simply connected domain D. Here’s how it goes –
Assume a function f(z) is continuous in a simply connected domain D. If for every closed contour C in the domain D, the integral of f(z) over C is zero, then f(z) is analytic in D.
In simpler terms, if a function f(z) is continuous in a simply connected domain D and the integral of f(z) over every closed contour C in the domain D is zero, then f(z) is analytic or holomorphic within D.
Proof of Morera’s Theorem
The proof of Morera’s theorem is quite fascinating. Let's suppose we have a continuous function f defined in a simply connected domain D and the integral of f(z) over C is zero, where C is a closed contour within D. The goal is to prove that f is analytic within D.
The proof includes a number of steps involving line integrals, path independence, and the definition of a continuous function. After a series of mathematical manipulations and transformations, we can establish that f(z) is indeed analytic in D.
Key Points about Morera’s Theorem
The function f has to be continuous in the domain D.
f(z) must exist for every z in D.
If C is any closed curve within domain D where the given function converges within C, then the integral of f(z) over C is zero.
If f is a continuous function within a simply connected domain D, such the integral of f on a closed contour C within D vanishes then the function is analytic.
What is proved by using Morera’s Theorem?
Morera’s theorem is used to prove whether a given function is analytic in given domain.
What are the conditions of Morera’s Theorem?
The conditions of Morera’s theorem are that the given function must be continuous within the domain and the closed curve integral on every closed contour within the domain vanishes
Morera’s theorem is converse of which theorem?
Morera’s theorem is converse of Cauchy-Goursat’s theorem.