TheMittagLefflerTheorem - crowlogic/arb4j GitHub Wiki

The Mittag-Leffler theorem is a central result in the field of complex analysis, which deals with the study of meromorphic functions. A meromorphic function is a function that is holomorphic (i.e., complex-differentiable) everywhere in its domain except for isolated poles, which are points where the function goes to infinity.

Named after the Swedish mathematician Gösta Mittag-Leffler, the theorem provides a way to decompose meromorphic functions into simpler components. The decomposition relies on the concept of a principal part, which is a local representation of a meromorphic function near one of its poles.

The Mittag-Leffler theorem can be stated as follows:

Let $f(z)$ be a meromorphic function defined on an open, connected domain $D$ in the complex plane, and let $P$ be the set of poles of $f(z)$ in $D$. Then, $f(z)$ can be expressed as the sum of a holomorphic function $g(z)$ and a finite or countably infinite sum of principal parts associated with the poles of $f(z)$.

Mathematically, the theorem can be written as:

$$f(z) = g(z) + \sum\left(\frac{a_n}{(z - z_n)^m}\right)$$

Here, $g(z)$ is a holomorphic function defined on $D$, $z_n$ are the poles of $f(z)$ in $D$, $a_n$ are complex coefficients, and $m$ is the order of the pole $z_n$. The sum runs over all the poles of $f(z)$ in the domain $D$.

The Mittag-Leffler theorem is significant because it enables us to analyze and manipulate meromorphic functions by breaking them down into simpler components. This decomposition is particularly useful in solving various problems in complex analysis, such as the evaluation of complex integrals and the study of the behavior of meromorphic functions near their singularities.