Branch point

Branch of a multivalued function

A branch of a multivalued analytic expression is a single-valued holomorphic choice on a domain obtained by excluding suitable branch cuts.

Let

W (z) = \int_{z_{0}}^{z} \frac{q ( t )}{P ( t )} d t,

where

P

a

, and let

A

be the limiting value of

W

a

on one sheet. Analytic continuation once around

a

changes the sign of the square root and hence of

W^{'}

. The continued primitive agrees at

a

with the original one, so it is

W ⟼ 2 A - W .

If continuation around two square-root branch points acts on a primitive as

R_{A} (W) = 2 A - W

and

R_{B} (W) = 2 B - W

, then

R_{B} \circ R_{A} (W) = W + 2 (B - A) .

Thus pairs of branch-point loops generate additive periods of an inverse function.

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A branch point is a concept primarily associated with complex analysis and algebraic geometry. Here are two contexts in which the term is commonly used: 1. **Complex Analysis**: In the context of complex functions, a branch point is a point where a multi-valued function (like the square root function or logarithm) is not single-valued. For example, consider the complex logarithm \( f(z) = \log(z) \).