An analytic function is a mathematical function that is defined and continuous in a given region of the complex plane. It also has derivatives of all orders at each point within that region.
A pole in an analytic function is a point where the function is undefined or infinite. It is usually represented as a singularity in the complex plane and can cause the function to have discontinuities.
To find the location of a pole in an analytic function, you need to solve the equation of the function to determine the values that make the function undefined or infinite. These values will correspond to the location of the poles in the complex plane.
A simple pole is a point where the function has a single pole, while a multiple pole is a point where the function has more than one pole. The order of a pole refers to the number of poles at a given point, with a simple pole having an order of 1 and a multiple pole having an order greater than 1.
Poles can cause the function to have discontinuities, making it non-analytic at those points. They can also affect the convergence of a power series representation of the function, and the location of poles can provide valuable information about the behavior of the function in the complex plane.