Glossary

11.2 Analytic continuation

5 min read•august 20, 2024

is a powerful technique in complex analysis that extends the domain of analytic functions. It allows us to study functions beyond their initial definitions, preserving their properties and revealing hidden connections.

This method is crucial for understanding multivalued functions, exploring Riemann surfaces, and applying the . It also plays a key role in defining important functions like the and , which have wide-ranging applications in mathematics.

Definition of analytic continuation

  • Analytic continuation extends the domain of a complex f(z)f(z) beyond its original domain of definition while preserving its analyticity
  • Allows for the study of complex functions defined by or other representations in a larger domain
  • The extended function agrees with the original function on the initial domain and satisfies the in the extended region

Uniqueness of analytic functions

  • If two analytic functions f(z)f(z) and g(z)g(z) agree on an open set, they are identical everywhere in their common domain of definition
  • Consequence of the for analytic functions
  • Ensures that an analytic continuation of a function, if it exists, is unique

Analytic continuation along curves

Continuation along simple curves

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  • Analytic continuation can be performed along simple curves, such as line segments or circular arcs
  • The function is extended by evaluating the power series representation or integral representation along the curve
  • The continuation is valid as long as the curve does not pass through any singularities of the function

Continuation along closed curves

  • Analytic continuation along closed curves can lead to different values of the function at the starting point ()
  • The value of the function after a complete traversal of a closed curve may differ from the initial value by a constant factor or a branch cut
  • Monodromy occurs when the function is multivalued and has or

Monodromy theorem

Monodromy vs analytic continuation

  • Monodromy refers to the phenomenon where the value of a function changes after analytic continuation along a closed curve
  • Analytic continuation extends the function to a larger domain while preserving analyticity
  • Monodromy is a consequence of the multivalued nature of some complex functions and the presence of branch points or branch cuts

Riemann surfaces

Riemann surfaces for multivalued functions

  • Riemann surfaces provide a geometric representation of multivalued complex functions
  • They are constructed by gluing together multiple copies of the complex plane, each representing a branch of the function
  • Riemann surfaces allow for a single-valued and continuous representation of the function
  • Examples include the for the logarithm function and the square root function

Schwarz reflection principle

Reflection across real axis

  • The Schwarz reflection principle states that if a complex function f(z)f(z) is analytic in the upper half-plane and real-valued on the real axis, then it can be extended to the lower half-plane by reflection
  • The extended function f~(z)\tilde{f}(z) is defined as f~(z)=f(zˉ)‾\tilde{f}(z) = \overline{f(\bar{z})} for Im(z)<0\text{Im}(z) < 0
  • The resulting function is analytic in the entire complex plane, except possibly on the real axis

Reflection across circles and lines

  • The Schwarz reflection principle can be generalized to reflection across circles and lines
  • If a function is analytic inside a circle and real-valued on the circle, it can be extended to the exterior of the circle by reflection
  • Similarly, if a function is analytic in a half-plane bounded by a line and real-valued on the line, it can be extended to the other half-plane by reflection

Applications of analytic continuation

Zeta function

  • The Riemann zeta function ζ(s)\zeta(s) is initially defined for Re(s)>1\text{Re}(s) > 1 by the series ζ(s)=∑n=1∞1ns\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
  • Analytic continuation extends the zeta function to the entire complex plane, except for a simple pole at s=1s = 1
  • The extended zeta function satisfies the functional equation ζ(s)=2sπs−1sin⁡(πs2)Γ(1−s)ζ(1−s)\zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s)

Gamma function

  • The gamma function Γ(z)\Gamma(z) is defined as Γ(z)=∫0∞tz−1e−tdt\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} dt for Re(z)>0\text{Re}(z) > 0
  • Analytic continuation extends the gamma function to the entire complex plane, except for non-positive integers
  • The gamma function satisfies the functional equation Γ(z+1)=zΓ(z)\Gamma(z+1) = z\Gamma(z)

Dirichlet L-functions

  • L(s,χ)L(s, \chi) are generalizations of the Riemann zeta function associated with Dirichlet characters χ\chi
  • They are initially defined for Re(s)>1\text{Re}(s) > 1 by the series L(s,χ)=∑n=1∞χ(n)nsL(s, \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s}
  • Analytic continuation extends Dirichlet L-functions to the entire complex plane, with possible poles at s=1s = 1

Limitations of analytic continuation

Natural boundaries

  • Some functions have beyond which analytic continuation is impossible
  • Natural boundaries are curves or regions in the complex plane that limit the domain of analyticity
  • Examples include the unit circle for the series ∑n=1∞zn!\sum_{n=1}^{\infty} z^{n!} and the imaginary axis for the series ∑n=1∞znnn\sum_{n=1}^{\infty} \frac{z^n}{n^n}

Singularities and branch cuts

  • Singularities and branch cuts can obstruct analytic continuation
  • Functions with essential singularities or branch points cannot be analytically continued across these points
  • Branch cuts are curves in the complex plane across which the function is discontinuous or multivalued

Techniques for finding analytic continuations

Power series expansions

  • Power series expansions can be used to find analytic continuations of functions
  • The function is represented as a power series ∑n=0∞an(z−z0)n\sum_{n=0}^{\infty} a_n (z-z_0)^n around a point z0z_0
  • The series is then extended to a larger domain by considering its convergence properties

Integral representations

  • Integral representations, such as the Cauchy integral formula, can be used to find analytic continuations
  • The function is expressed as a contour integral f(z)=12πi∮Cf(ζ)ζ−zdζf(z) = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{\zeta - z} d\zeta
  • The contour of integration is deformed to extend the domain of the function

Functional equations

  • Functional equations satisfied by the function can be used to find analytic continuations
  • Examples include the functional equations for the gamma function and the Riemann zeta function
  • The functional equations relate the values of the function at different points, allowing for extension to a larger domain

Key Terms to Review (21)

Analytic continuation: Analytic continuation is a technique in complex analysis that extends the domain of a given analytic function beyond its original region of convergence. This process allows mathematicians to define a function on a larger domain while preserving its analytic properties, effectively creating a new representation of the same function. By using this method, various important functions, like the exponential and logarithmic functions, can be explored in more depth across different contexts, revealing hidden structures and relationships.
Analytic function: An analytic function is a complex function that is locally represented by a convergent power series, meaning it is differentiable in some neighborhood of each point in its domain. This property connects deeply with concepts such as differentiability, Cauchy-Riemann equations, and integral theorems, revealing the intricate structure of functions within the complex number system and their behavior in the complex plane.
Bernhard Riemann: Bernhard Riemann was a 19th-century German mathematician whose work laid the foundations for many areas of modern mathematics, particularly in complex analysis and number theory. His concepts, including Riemann surfaces and the Riemann zeta function, are fundamental in understanding various aspects of both pure and applied mathematics.
Branch cuts: Branch cuts are lines or curves in the complex plane that are used to define a single-valued branch of a multi-valued complex function, such as the logarithm or roots of complex numbers. They help to create a continuous path for analytic continuation by restricting the domain and avoiding discontinuities that arise from the multi-valued nature of these functions. By introducing branch cuts, we can effectively manage how these functions behave in different regions of the complex plane.
Branch Points: Branch points are specific points in the complex plane where a multi-valued function, such as a complex logarithm or root, switches from one branch to another. These points are crucial in defining the branch cuts necessary to create single-valued analytic functions, allowing us to handle the discontinuities that arise when dealing with such functions in complex analysis.
Cauchy-Riemann Equations: The Cauchy-Riemann equations are a set of two partial differential equations that are essential for determining whether a complex function is analytic (differentiable in the complex sense). They establish a relationship between the real and imaginary parts of a complex function, showing that if a function satisfies these equations, it has a derivative at that point, which leads to important results in complex analysis.
Continuity: Continuity in complex analysis refers to the property of a function that ensures it behaves predictably as its input approaches a certain point. It signifies that small changes in the input of the function lead to small changes in the output, which is essential for establishing concepts like limits, differentiability, and integrability in the complex plane.
Dirichlet L-functions: Dirichlet L-functions are complex functions that generalize the Riemann zeta function and are associated with Dirichlet characters. They are significant in number theory, especially in studying properties of primes in arithmetic progressions and have deep connections to analytic continuation, allowing them to be extended beyond their initial domain of convergence.
Gamma Function: The gamma function is a special function that extends the factorial function to complex and real numbers, defined for all complex numbers except the non-positive integers. It is denoted as $$\\Gamma(z)$$ and has important properties, such as $$\\Gamma(n) = (n-1)!$$ for any positive integer n. This function plays a significant role in many areas of mathematics, especially in the context of entire functions and analytic continuation.
Henri Poincaré: Henri Poincaré was a French mathematician, theoretical physicist, and philosopher known for his foundational contributions to various areas of mathematics and science, especially in complex analysis. His work laid the groundwork for many modern theories, particularly in fields like dynamical systems and topology, and influenced the development of several important concepts such as analytic continuation and conformal mappings.
Holomorphic function: A holomorphic function is a complex function that is differentiable at every point in its domain, which means it is not only continuous but also has a derivative that exists throughout. This property allows holomorphic functions to exhibit remarkable characteristics, such as being expressible as power series and having derivatives of all orders, making them a central focus in complex analysis.
Identity theorem: The identity theorem states that if two analytic functions agree on a set of points that has a limit point within their domain, then the two functions are identical on the entire connected component of that domain. This powerful result highlights the uniqueness of analytic functions, indicating that they cannot cross each other without being the same function throughout their common domain.
Monodromy: Monodromy refers to the concept in complex analysis that describes how a multi-valued function behaves when you travel around a loop in its domain. It highlights the relationship between different branches of a multi-valued function, showing how they are connected as you move along paths in the complex plane. Monodromy is crucial for understanding analytic continuation and the behavior of functions defined on Riemann surfaces.
Monodromy Theorem: The Monodromy Theorem is a fundamental result in complex analysis that describes how multi-valued functions can be understood through their analytic continuations along different paths in the complex plane. It establishes a connection between the behavior of these functions and the concept of monodromy, which refers to how the values of a function can change when you traverse loops in its domain. Essentially, it provides insight into how different branches of a function relate to one another and assures that analytic continuation respects these relationships.
Natural Boundaries: Natural boundaries are geographical features that serve as barriers to define the limits of territories or regions, often formed by rivers, mountains, or lakes. These features are important in the context of analytic continuation, as they can limit the domain of a complex function and prevent it from being analytically continued across these boundaries.
Power Series: A power series is an infinite series of the form $$ ext{P}(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + ...$$ where each term consists of coefficients multiplied by increasing powers of the variable. Power series are important in complex analysis as they allow functions to be expressed in terms of their derivatives at a point, connecting them to concepts such as convergence, analytic functions, and function approximation.
Riemann Surface: A Riemann surface is a one-dimensional complex manifold that allows for the extension of complex functions beyond their traditional boundaries. They enable the visualization of multi-valued functions, like the square root or logarithm, in a way that is both structured and manageable. By providing a means to resolve branch points and identify different sheets of a function, Riemann surfaces play a crucial role in understanding conformal mappings, analytic properties, and more complex relationships between functions.
Riemann zeta function: The Riemann zeta function is a complex function defined for complex numbers, originally introduced to understand the distribution of prime numbers. It connects various areas of mathematics, particularly number theory and complex analysis, through its properties as an entire function, its analytic continuation, and its relationship with the zeros and the Riemann hypothesis.
Schwarz reflection principle: The Schwarz reflection principle is a powerful concept in complex analysis that allows one to extend a function's domain by reflecting it across a line of symmetry. This principle shows how if a function is analytic in a region and continuous up to its boundary, and if it satisfies certain conditions on that boundary, you can reflect the values of the function to generate new values in the complex plane, thus aiding in analytic continuation. It directly ties into the behavior of harmonic functions and showcases the intrinsic connections between symmetry and analytic properties.
Singularity: In complex analysis, a singularity refers to a point in the complex plane where a function ceases to be analytic, meaning it cannot be expressed as a power series around that point. Singularities can be classified as removable, poles, or essential, and understanding their nature is crucial for evaluating integrals, applying Cauchy's integral formula, and exploring properties of functions like analytic continuation and Hadamard factorization.
Uniform Convergence: Uniform convergence is a type of convergence for sequences of functions where the speed of convergence is uniform across a set of points. This means that for every point in the domain, the functions in the sequence get uniformly close to the limit function as the sequence progresses, allowing for certain nice properties such as the interchange of limits and integration or differentiation to hold.
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