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Holomorphic Function (Analytic): Definition, Examples

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Contents (Click to skip to that section):

  1. What is a Holomorphic Function?
  2. Holomorphic Functional Calculus

What is a Holomorphic Function?

Holomorphic functions (also called analytic functions) usually refer to functions that are infinitely differentiable; They are a big part of complex analysis (the study of functions of complex numbers).

Important Note About Ambiguity and Holomorphic Functions

if you’re working with a holomorphic function, make sure you know the author’s intent, and which definitions they are working with. “Holomorphic” is one of those terms that has many grey areas.

  • Some authors define a holomorphic function as one that is differentiable, and an analytic function if they have a power series expansion for each point of their domain. Other authors use both terms interchangeably, perhaps because a few theorems exist that prove all analytic functions are holomorphic and all holomorphic functions are analytic.
  • Which term you use may also depend on your field: According to Eric Weisstein, “Holomorphic function” (or “holomorphic map”) is usually preferred by mathematicians; “Analytic function” seems to be the term of choice in physics, engineering and in a few older texts (e.g. Morse and Feshbach 1953, pp. 356-374; Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).
  • To make things a little more complicated, other conditions are added to the definitions by some authors. For example, Ji Shanyu notes that “…in some books, the, C1- smoothness condition is added to the definition of holomorphic function.”

Definition of Holomorphic Function

As noted above, several definitions exist. The following comes from W. Rudin, Real and complex analysis (chapter 10) and H. Priestley, Introduction to complex analysis, as summarized by T. Perutz. It’s one of the most succinct definitions you’ll find:


Let G be an open set in . A function f : G → C is called holomorphic if, at every
point z ∈ G, the complex derivative
holomorphic function definition

exists as a complex number.

Where:

  • = complex realm

Essentially, the definition relies on just the existence of complex derivatives at every point. This isn’t much different from a regular differentiable function (the difference being that the derivatives are “regular” instead of complex). “Complex differentiable” is almost the same as differentiable, but with some constraints. Specifically, it needs to satisfy Cauchy-Riemann equations.

In comparison, Wikipedia has this definition:

…a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighbourhood of the point.

If you know what a neighborhood is, and what it means to be complex differentiable, then that definition is fairly straightforward, albeit a little wordy.

Holomorphic Function Example: Dedekind Eta Function

dedekind eta function graph

The Dedekind η function in the upper half-plane.


The Dedekind eta function (also called the η function) is defined in the upper half-plane of complex numbers, with a positive imaginary part.

The function was first described by Dedekind in 1877, and can be mathematically defined in a few different ways. For example, where z is a complex-number it can be represented by infinite products, as follows (Rademacher & Grosswald, 1972):
dedekind eta function

Another popular way to quantify the η function is as (2π)-½) times the 24th root of Ramanujan’s Delta Function (Unterberger, 2008).

The function is holomorphic and non-vanishing on the upper-half complex plane (ℍ). To check, all that’s needed is to show Σqn has both absolute convergence and uniform convergence on compact subsets of ℍ (Masdeau, 2014). The function can’t be continued analytically past ℍ.

A Bizarre Holomorphic Function and Its Uses

The Dedekind eta function is a primary example of a modulo form, which essentially means it falls into the category of “bizarre” functions. The η function appears in esoteric fields like bosonic string theory and supergravity theory.

One of the major uses of the Dedekind eta function is in elliptic function theory and modulo forms, where one important use is to describe the pattern of an elliptic function’s periods (Rademacher & Grosswald, 1972). In addition, many functions can be expressed as Dedekind eta functions, including class invariants, theta-functions and the Rogers-Ramanujan continued fraction (Bustoz et al., 2001).

The generalized Dedekind eta function can be used to construct modular functions of different weights (Vestal, 2001).

Univalent Function

In general, a univalent function (sometimes called analytic univalent) is an analytic function that maps one input to one output (i.e. it is injective) on the complex plane. In other words, it’s a complex function that has no overlap.

More Narrow Definitions

Univalent Functions can also be defined more narrowly, as a function that maps different points to different points in the unit disk 𝔻 (Ravichandran, 2012).

In notation (Thomas, 1967), a function f(z) in a domain 𝔻 is univalent if w = f(z) takes on different values of w for different z ∈ 𝔻. In other words, f(z1) = f(z2) implies z1 = z2 for z1, z2 ∈ 𝔻. This tells you that a function f(z) defines a one to one correspondence between points of 𝔻 and a domain in the w-plane.

One example:
univalent function

Where:

φa is univalent when |a| < 1.

*A complex conjugate has a real part and an imaginary part with the same magnitude but different signs. For example, the complex conjugate of a + bi is abi (where a and b are two real numbers).

Some authors define univalent functions as being injective and meromorphic in the unit disk (e.g. Pommerenke, 1985). A meromorphic function is the ratio of two analytic functions, which are analytic with the exception of “poles“— isolated singularities.

Neighborhoods of univalent functions

An interesting feature of univalent functions is that their entire neighborhoods are also univalent functions (Pascu & Pascu, 2009). In other words, a small deviation in the function will not result in a non-univalent function.

Subclasses

Univalent functions can be categorized into various subclasses. For example:

  1. Regular starlike: When mapping from the unit disk Δ to the complex plane ℂ, a function is starlike if and only if
    starlike function
    (Nevanlinna, 1921).
  2. Close-to-convex: the definition is the same as regular starlike, except the function is convex if and only if:,
    convex

What is a Universal Function?

Loosely speaking, a universal function is a function that imitates any other function. The definition differs depending on what field you’re working in; sometimes, the term is used by some authors to mean a “very useful function.”

Complicating the definition is that, in addition to the multiple field-specific definitions (some are listed below), not everyone agrees on the same definition for holomorphic functions (which are an integral part of the mathematical definition below). The takeaway: it can be difficult to be specific about an exact definition unless you delve into a particular author’s writings.

Universal Functions in Neural Networks

In artificial neural networks (NNs), a universal function (also called a universal function approximator) is a two-layer neural network that can approximate any other function with a very small error.

The definition sounds simple, but NNs are notoriously difficult to implement, work with, and comprehend. Shubham Panchal on Medium.com notes

“They contain crazy math and require expertise to be fine-tuned.”

Delving into the “crazy math” of deep learning is beyond the scope of this site. However, for an introduction to neural networks, I recommend you start here: A Basic Introduction to Neural Networks.

Mathematical Definition of a Universal Function

In mathematics, the definition specifically relates to the approximation of holomorphic functions.

Various definitions exist, which may be due to the differing definitions of holomorphic functions.

This first, fairly straightforward definition is from Larson et al.:
A function of two variables F(x, y) is a universal function if two functions h(x) and k(y) exist for every function G(x, y) (and for all x, y) as follows:

G(x, y) = F(h(x), k(y))

Chee (1978) defines it more completely in terms of regions (Ω) in complex variable spaces (ℂ):

A function g ∈ 𝓕 is a universal function of 𝓕, relative to {φ} if for any f ∈ 𝓕, there exists a sequence {φk}1 from {φ} such that:

universal function

Where:

  • Φ = a family of holomorphic automorphisms of Ω
  • 𝓕 = a family of holomorphic functions in Ω

Holomorphic Functional Calculus

Holomorphic functional calculus (also called analytic functional calculus) was developed by Arens-Calderon, Shilov, and Waelbroeck. It connects the theory of complex variables with the theory of commutative Banach algebras [1]. A Banach algebra is an associative algebra A that is also a Banach space—a normed linear space complete in the metric induced by the norm [2]. Commutative Banach algebra is a Banach algebra A with identity over the ℂ where xy = yx for all x,yA [3].

The purpose of a functional calculus is to give a meaning to the complex function 𝑓(𝑇), where the operator 𝑇 is defined on a Hilbert space, a vector space with an inner product, an operation which allows for the definition of angles and lengths [4].

Formal Definition of Holomorphic Functional Calculus

Holomorphic functional calculus gives a systematic way of applying analytic functions to elements of Banach algebras. Formally, it can be defined as [5, 6]:

Let A be a unital Banach algebra and let a A. For every f O(a), element f(a) ∈ A is defined as follows: Let U ⊃ σ(a) be an open set where f is a defined holomorphic function, and let V be an open bounded set such that σ(a) ⊂ VVU and such that ∂ V is a finite union of C1 curves; define

f(a) = ∫∂V f(z)(z – a)-1dz.

Where:

Holomorphic Functional Calculus: References

[1] Schanuel, S. & Zame, W. (1982). Normality of the Functional Calculus. Retrieved June 25, 2021 from: https://academic.oup.com/blms/article-abstract/14/3/218/292532?redirectedFrom=PDF
[2] Hunter, J. & Nachtergaele, N. Applied Analysis. Chapter 5: Banach Spaces. Retrieved June 25, 2021 from: https://www.math.ucdavis.edu/~hunter/book/ch5.pdf
[3] Commutative Banach Algebra. Encyclopedia of Mathematics. Retrieved June 25, 2021 from: https://encyclopediaofmath.org/wiki/Commutative_Banach_algebra
[4] Ariza, H. et al. (2019). Holomorphic functional calculus for sectorial operators. IX Escuela-Taller de Análisis Funcional.
[5] Shalit, O. Advanced Analysis, Notes 18: The Holmorphic Functional Calculus I (motivation, definition, line integrals of holomorphic Banach-space valued functions). Retrieved June 25, 2021 from: https://noncommutativeanalysis.wordpress.com/2014/01/02/advanced-analysis-notes-18-the-holomorphic-functional-calculus-i/
[6] Shalit, O. Advanced Analysis, Notes 19: The Holomorphic Functional Calculus II (Definition and Basic Properties). Retrieved June 25, 2021 from: https://noncommutativeanalysis.wordpress.com/2014/01/05/advanced-analysis-notes-19-the-holomorphic-functional-calculus-ii-definition-and-basic-properties/

Other References

Bustoz, J. et al. (Eds.) Special Functions 2000: Current Perspective and Future Directions. NATO Science Series II: Mathematics, Physics and Chemistry (Book 30) 2001.
Chee, P. (1978). Universal Functions in Several Complex Variables. J. Austral. Math. Soc. (Series A) 28, 189-196.Retrieved December 9, 2019 from: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/381E870BD1348C100ADEB7530854FF6B/S1446788700015676a.pdf/div-class-title-universal-functions-in-several-complex-variables-div.pdf
Hundley, D. Neural Nets. Retrieved December 9, 2019 from: http://people.whitman.edu/~hundledr/courses/M339/FFNeural.pdf
Larson, P. et al. Universal Functions. Retrieved December 9, 2019 from: https://www.math.wisc.edu/~miller/res/univ.pdf
Masdeau, M. Modular Forms. 2010. Retrieved May 21, 2020 from: http://mat.uab.cat/~masdeu/files/main.pdf
Perutz, T. A rapid review of complex function theory. Retrieved October 11, 2019 from: https://web.ma.utexas.edu/users/perutz/CxAn.pdf
Shanyu, Ji. Chapter 3: Holomorphic Functions. Retrieved October 11, 2013 from: https://www.math.uh.edu/~shanyuji/Complex/complex-1/cx-17-online.pdf
Conway, J. Functions of One Complex Variable I.
Knopp, K. “Analytic Continuation and Complete Definition of Analytic Functions.” Ch. 8 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 83-111, 1996.
Morse, P. M. and Feshbach, H. “Analytic Functions.” §4.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 356-374, 1953.
Nevanlinna, R. (1921). Über die konforme Abbildund Sterngebieten
Oversikt av Finska–Vetenskaps Societen Forhandlingar, 63(A) (6), pp. 48-403
Pascu,M. & Pascu, N. (2009). Neighborhoods of univalent functions. Retrieved December 19 from: https://arxiv.org/abs/0910.5456
Pemmerenke, Ch. (1985). On the Integral Means of the Derivative of a Univalent Function. Journal of the London Mathematical Society, Volume s2-32, Issue 2, October 1985, Pages 254–258, https://doi.org/10.1112/jlms/s2-32.2.254
Rademacher, H. & Grosswald, E. Dedekind Sums (The Carus Mathematical Monographs, No. 16). 1972. Mathematical Association of America.
Ravichandran, V. (2012). Geometric Properties of Partial Sums of Univalent Functions. Retrieved December 19, 2019 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.746.6182&rep=rep1&type=pdf
Sokół, J. (2011). A certain class of starlike functions
Study, E. (1913). Konforme Abbildung Einfachzusammenhangender Bereiche
B. C. Teubner, Leipzig und Berlin
Thomas, D. (1967). Starlike and close-to-convex functions. Retrieved December 19, 2019 from: https://spiral.imperial.ac.uk/bitstream/10044/1/17592/2/Thomas-DK-1967-PhD-Thesis.pdf
Unterberger, A. Quantization and Arithmetic. 2008. Springer Science and Business Media.
Vestal, D. Construction of weight two eigenforms via the generalized dedekind eta function. Rocky Mountain Journal of Mathematics. Vol 31. Number 2. 2001.
Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
Weisstein, Eric W. “Holomorphic Function.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HolomorphicFunction.html


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