Complex Analysis introduces multivalued functions, which assign multiple values to points in their domain. These functions challenge our understanding of well-defined functions and require special techniques for analysis. Branch points are crucial in studying multivalued functions.

Branch points are where a function isn't single-valued or analytic. They indicate multiple branches of the function and are classified by order. Understanding branch points is key to grasping the behavior of multivalued functions in the complex plane.

Multivalued Functions and Properties

Definition and Characteristics

• Multivalued functions assign more than one value to each point in their domain
  • Violate the uniqueness property of well-defined functions
  • Arise when inverting non-injective complex functions (complex logarithm)
  • Occur when considering functions with multiple branches (square root function, inverse trigonometric functions)
• The values of a multivalued function at a given point form a set called the function's value set at that point
  • Value sets containing more than one distinct value indicate the function is multivalued at that point

Comparison to Well-Defined Functions

• Multivalued functions do not satisfy the definition of a well-defined function
  • Well-defined functions must assign a unique value to each point in their domain
  • Multivalued functions violate this uniqueness property by assigning multiple values to points
• Understanding the differences between multivalued and well-defined functions is crucial for analyzing their properties
  • Well-defined functions are easier to study due to their uniqueness property
  • Multivalued functions require special techniques (branch cuts, Riemann surfaces) to analyze effectively

Branch Points in Complex Analysis

Definition and Significance

• Branch points are points in the complex plane where a multivalued function is not analytic or not single-valued
  • Function's value set at a branch point contains more than one distinct value
  • Function cannot be defined as a single-valued analytic function in any neighborhood of the branch point
• Branch points are classified as isolated singularities of a function
  • Differ from poles and essential singularities in their properties and behavior
• The presence of branch points indicates the function has multiple branches
  • Each branch represents a different "copy" of the function with distinct values

Classification and Examples

• Branch points are classified by their order and the number of distinct branches they generate
  • The order of a branch point is the smallest positive integer $n$ such that the function's value set remains unchanged after traversing a closed path around the point $n$ times
  • The number of distinct branches at a branch point is equal to the order of the branch point
• Common examples of functions with branch points include:
  • Logarithmic functions (branch point at the origin)
  • Root functions (branch points at zeros of the argument inside the radical)
  • Inverse trigonometric functions (branch points at critical values)

Behavior Around Branch Points

Monodromy and Value Set Changes

• Analyzing the behavior of a multivalued function near a branch point involves considering the function's value set as one traverses a closed path around the point
  • If the function's value set changes after completing a closed path, it indicates the presence of multiple branches
  • The change in the function's value set upon completing a closed path is called the monodromy of the function at the branch point
• The monodromy can be used to determine the number of distinct branches of the function at the branch point
  • Each distinct value in the function's value set after completing a closed path corresponds to a different branch

Visualization with Riemann Surfaces

• Riemann surfaces are used to visualize the behavior of multivalued functions around branch points
  • Each branch of the function is represented as a separate sheet on the Riemann surface
  • The sheets are connected at the branch points, allowing for a continuous representation of the function
• Traversing a closed path around a branch point on the Riemann surface involves moving between the sheets
  • The number of sheets and their connectivity reflect the number of branches and the monodromy of the function

Branch Point Determination

Algebraic Functions

• For functions defined by algebraic expressions, branch points often occur at zeros of the argument inside a radical or logarithm
  • Example: The function $f(z) = \sqrt{z}$ has a branch point at $z = 0$ because the argument inside the square root vanishes
• Rational functions can only have branch points at poles and zeros of the denominator
  • Example: The function $f(z) = \sqrt{\frac{1}{z}}$ has a branch point at $z = 0$ because it is a zero of the denominator

Transcendental Functions

• Logarithmic functions have a branch point at the origin and, in some cases, at infinity
  • Example: The complex logarithm $\log(z)$ has a branch point at $z = 0$
• Root functions have branch points at zeros of the argument inside the radical
  • Example: The cube root function $\sqrt[3]{z}$ has branch points at $z = 0$ and $z = \infty$
• Inverse trigonometric functions have branch points at their critical values
  • Example: The function $\arcsin(z)$ has branch points at $z = \pm 1$