What Is the Equivalence Point?

Let f (z) be the self-defense function of the fractional transformation group G. If two points in the plane can be used to make one point into another point, then these two points are called equivalent points about group G.

Let f (z) be the

A region D in the plane, if any two different points are not equivalent to each other, and any point in the plane can find its equivalent point in D, then D is called the basic region of the group G, also known as The basic region of the guard function f (z).

The self-defense function takes the same number of times in any value (including infinity) in the basic region. ^[1]

(automorphic function)

The self-defense function is a generalization of the concepts of circular function, hyperbolic function, and elliptic function.

Let X be a bounded connected open set in C, G be an automorphism (ie, biholomorphic bijective) group after X is given a tightly open topology, and is a discrete subgroup of G. If a meromorphic function f acts on If it is invariant, it is called self-defense function (about ).

If there exists a function (r, z) of × X to C, it is holomorphic with respect to zX, and is non-zero everywhere, so that for each : f (, z) = (r, z) f (z) (zX), then f is called the self-defense form (about ), is called the self-defense factor, it should satisfy the relationship ( , z) = (, Z) ( , z) (Note: f here is usually required to be completely pure, and the behavior of f at the cusp must have some appropriate conditions).