# Meromorphic functions

• greentea28a
In summary, a meromorphic function is a complex-valued function that is a combination of both holomorphic and singular functions. It is defined and analytic on almost all points in its domain, except for a finite number of isolated singularities where it is allowed to have poles. This makes it a more general class of functions compared to holomorphic functions. The singularities of a meromorphic function can be either poles or removable singularities. Meromorphic functions are used in complex analysis and number theory, and they have applications in areas such as differential equations, geometry, and physics. To determine if a function is meromorphic, you can check if it is analytic on almost all points in its domain and look for patterns in its behavior near its singularities

#### greentea28a

Hi,
Half of the Riemann Hypothesis is solved by meromorphic functions.

The other half, non-trivial zeros should lie on a critical line needs your creative work.

Can someone direct me to sites or papers with the previous in mind?

Best

This is poorly stated. The Riemann Hypothesis has nothing to do with meromorphic functions. Rather it is a statement about the zeros of the analytic continuation of the Euler sum.

## 1. What is a meromorphic function?

A meromorphic function is a complex-valued function that is defined and analytic on almost all points in its domain, except for a finite number of isolated singularities, where it is allowed to have poles. This means that a meromorphic function is a combination of both holomorphic (analytic) and singular (non-analytic) functions.

## 2. What is the difference between a meromorphic function and a holomorphic function?

A holomorphic function is a complex-valued function that is defined and analytic on all points in its domain, while a meromorphic function is only analytic on almost all points in its domain, with a finite number of isolated singularities. This means that a meromorphic function is a more general class of functions compared to holomorphic functions.

## 3. What are the singularities of a meromorphic function?

The singularities of a meromorphic function are points in its domain where the function is not analytic. These can be either poles, where the function approaches infinity, or removable singularities, where the function is undefined but can be made analytic by redefining the value at that point.

## 4. How are meromorphic functions used in mathematics?

Meromorphic functions are important in complex analysis and number theory. They are used to study the behavior of functions and their singularities, and they have applications in areas such as differential equations, geometry, and physics.

## 5. How can I determine if a function is meromorphic?

To determine if a function is meromorphic, you can check if it is analytic on almost all points in its domain, except for a finite number of isolated singularities where it is allowed to have poles. You can also look for patterns in the function's behavior near its singularities, such as if the function approaches infinity or has a removable singularity.