Theta functions are essentially complex extensions of trigonometric functions, elliptic analogues of the exponential function. They are quasi-doubly periodic functions.
Theta functions are also "super-symmetric," meaning that if a specific type of mathematical function called a Moebius transformation is applied to the functions, they turn into themselves
This subject has numerous links to the theory of functions of a complex varibale, to the theory of partial differential equations, to number theory, and even to stastical mechanics.
Consider the function below, z is a complex variable, while t is a complex parameter where Re(t) > 0
Theta functions are also "super-symmetric," meaning that if a specific type of mathematical function called a Moebius transformation is applied to the functions, they turn into themselves
This subject has numerous links to the theory of functions of a complex varibale, to the theory of partial differential equations, to number theory, and even to stastical mechanics.
Consider the function below, z is a complex variable, while t is a complex parameter where Re(t) > 0
this no artifically concoted function. This is one of the basic functions of analysis, a theta function.
Four types of Theta Functions
here