Analytic continuation is a powerful technique in complex analysis that extends functions beyond their original domains. It's like giving a function superpowers, allowing it to explore new territories while staying true to its analytic nature.

This topic connects to Riemann surfaces, which are the playgrounds where multivalued functions can roam freely. By understanding analytic continuation, we unlock the secrets of these mysterious surfaces and gain deeper insights into complex functions.

Analytic Continuation

Definition and Properties

• Analytic continuation is a technique used to extend the domain of a complex function beyond its original domain of definition while preserving the function's analyticity
• The extended function, called the analytic continuation, is unique if it exists and agrees with the original function on the original domain
• Analytic continuation is based on the idea that two analytic functions that agree on an open subset of their domains must agree on the intersection of their domains
• The process of analytic continuation involves finding a chain of overlapping disks or regions, each containing a power series representation of the function, to extend the domain (e.g., extending the complex logarithm function to a Riemann surface covering the entire complex plane)

Geometric Interpretation and Applications

• Analytic continuation can be used to define functions on larger domains, such as the Riemann surface, which is a geometric representation of the domain of a multivalued function
• Riemann surfaces provide a way to visualize and study the properties of multivalued functions, such as branch points and branch cuts (e.g., the Riemann surface of the complex square root function)
• Analytic continuation is a powerful tool in complex analysis, with applications in various fields such as quantum mechanics, statistical mechanics, and number theory
• Many special functions, such as the gamma function, zeta function, and hypergeometric functions, can be defined on larger domains using analytic continuation, extending their applicability and revealing their deeper properties

Extending Function Domains

Process of Analytic Continuation

• To perform analytic continuation, start with a complex function defined on an open disk or region and find its power series representation
• Use the power series to define the function on a larger disk or region that overlaps with the original domain, ensuring that the series converges on the new domain
• Repeat the process of finding power series representations and extending the domain until the desired domain is covered or no further continuation is possible
• The choice of the path along which analytic continuation is performed can affect the final result, especially in the presence of singularities or branch points (path dependence)

Examples and Applications

• The complex exponential function, $e^z$, can be analytically continued to the entire complex plane using its power series representation, $\sum_{n=0}^{\infty} \frac{z^n}{n!}$
• The complex logarithm function, initially defined on the slit plane $\mathbb{C} \setminus (-\infty, 0]$, can be analytically continued to a Riemann surface that covers the entire complex plane
• Analytic continuation can be used to extend the domain of special functions, such as the gamma function, zeta function, and hypergeometric functions, beyond their original domains of definition
• In physics, analytic continuation is used to study the behavior of correlation functions and partition functions in different regimes, such as the continuation from Euclidean to Minkowski spacetime

Uniqueness and Existence

Uniqueness Theorem

• The uniqueness theorem for analytic continuation states that if two analytic functions agree on an open subset of their domains, then they must agree on the intersection of their domains
• As a consequence of the uniqueness theorem, if an analytic continuation of a function exists, it is unique
• The uniqueness theorem highlights the rigidity of analytic functions and the strong constraints imposed by analyticity

Existence and Obstacles

• The existence of an analytic continuation is not always guaranteed and depends on the properties of the original function and the desired extended domain
• Obstacles to the existence of analytic continuation include singularities (poles, essential singularities), branch points, and natural boundaries of the function
• The presence of singularities or branch points can lead to the need for Riemann surfaces to properly define the analytic continuation (e.g., the Riemann surface of the complex logarithm)
• Natural boundaries are curves or regions beyond which a function cannot be analytically continued, often arising from the accumulation of singularities (e.g., the unit circle as a natural boundary for the Taylor series of $\frac{1}{1-z}$)

Monodromy Group of Multivalued Functions

Multivalued Functions and Monodromy

• A multivalued function is a function that assigns multiple values to each point in its domain, arising from the presence of singularities or branch points
• The monodromy group of a multivalued function describes how the function's values permute when analytically continued along closed paths around singularities or branch points
• The monodromy group provides insight into the structure and properties of the Riemann surface associated with the multivalued function
• Understanding the monodromy group is crucial for studying the global behavior of multivalued functions and their analytic continuations

Determining the Monodromy Group

• To determine the monodromy group, identify the singularities and branch points of the multivalued function
• Perform analytic continuation along closed paths that encircle the singularities or branch points, keeping track of how the function's values change along the path
• The permutations of the function's values obtained from analytic continuation along all possible closed paths form the monodromy group
• The generators of the monodromy group correspond to the analytic continuations along fundamental loops around the singularities or branch points
• The monodromy group is a subgroup of the permutation group acting on the function's values and reflects the connectivity and structure of the Riemann surface