Branch point

singular point of multi-valued character

An isolated singular point $ a $ of an analytic function $ f(z) $ of one complex variable $ z $ such that the analytic continuation of an arbitrary function element of $ f(z) $ along a closed path which encircles $ a $ yields new elements of $ f(z) $. More exactly, $ a $ is said to be a branch point if there exist: 1) an annulus $ V= \{ {z } : {0 < | z - a | < \rho } \} $ in which $ f(z) $ can be analytically extended along any path; 2) a point $ z _ {1} \in V $ and some function element of $ f(z) $ represented by a power series

$$ \Pi (z _ {1} ; r) = \ \sum _ {v = 0 } ^ \infty c _ {v} (z - z _ {1} ) ^ {v} $$

with centre $ z _ {1} $ and radius of convergence $ r > 0 $, the analytic continuation of which along the circle $ | z - a | = | z _ {1} - a | $, going around the path once in, say, the positive direction, yields a new element $ \Pi ^ { \prime } (z _ {1} ; r ^ \prime ) $ different from $ \Pi (z _ {1} ; r) $. If, after a minimum number $ k > 1 $ of such rounds the initial element $ \Pi (z _ {1} ; r) $ is again obtained, this is also true of all elements of the branch (cf. Branch of an analytic function) of $ f(z) $ defined in $ V $ by the element $ \Pi (z _ {1} ; r) $. In such a case $ a $ is a branch point of finite order $ k - 1 $ of this branch. In a punctured neighbourhood $ V $ of a branch point $ a $ of finite order this branch is represented by a generalized Laurent series, or Puiseux series:

$$ \tag{1 } f (z) = \ \sum _ {v = - \infty } ^ { {+ } \infty } b _ {v} (z - a) ^ {v/k} , \ z \in V. $$

If $ a = \infty $ is an improper branch point of a finite order, then the branch of $ f(z) $ is representable in some neighbourhood $ V ^ { \prime } = \{ {z } : {| z | > \rho } \} $ by an analogue of the series (1):

$$ \tag{2 } f (z) = \ \sum _ {v = - \infty } ^ { {+ } \infty } b _ {v} z ^ {-v/k} , \ z \in V ^ \prime . $$

The behaviour of the Riemann surface $ R $ of $ f(z) $ over a branch point of finite order $ a $ is characterized by the fact that $ k $ sheets of the branch of $ f(z) $ defined by the element $ \Pi (z _ {1} ; r) $ come together over $ a $. At the same time the behaviour of other branches of $ R $ over $ a $ may be altogether different.

If the series (1) or (2) contains only a finite number of non-zero coefficients $ b _ {v} $ with negative indices $ v $, $ a $ is an algebraic branch point or an algebraic singular point. Such a branch point of finite order is also characterized by the fact that as $ z \rightarrow a $ in whatever manner, the values of all elements of the branch defined by $ \Pi (z _ {1} ; r) $ in $ V $ or $ V ^ { \prime } $ tend to a definite finite or infinite limit.

Example: $ f (z) = z ^ {1/k} $, where $ k > 1 $ is a natural number, $ a = 0, \infty $.

If the series (1) or (2) contain an infinite number of non-zero coefficients $ b _ {v} $ with negative indices $ v $, the branch points of finite order $ a $ belong the class of transcendental branch points.

Example: $ f (z) = \mathop{\rm exp} (1/z) ^ {1/k} $, where $ k > 1 $ is a natural number, $ a = 0 $.

Finally, if it is impossible to return to the initial element after a finite number of turns, $ a $ is said to be a logarithmic branch point or a branch point of infinite order, and is also a transcendental branch point.

Example: $ f(z) = \mathop{\rm Ln} z, a = 0, \infty $.

Infinitely many sheets of the branch of $ f(z) $ defined by the element $ \Pi (z _ {1} ; r) $ come together over a logarithmic branch point.

In the case of an analytic function of several complex variables $ f(z) $, $ z = (z _ {1} \dots z _ {n} ) $, $ n \geq 2 $, a point $ a $ of the space $ \mathbf C ^ {n} $ or $ \mathbf C P ^ {n} $ is said to be a branch point of order $ m $, $ 1 \leq m \leq \infty $, if it is a branch point of order $ m $ of the, generally many-sheeted, domain of holomorphy of $ f(z) $. Unlike in the case $ n=1 $, branch points, just like other singular points of analytic functions (cf. Singular point), cannot be isolated if $ n \geq 2 $.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 2 , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)
[2]	B.A. Fuks, "Theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian)