Types of Functions >
- Meromorphic Function Definition
- Examples of Meromorphic Functions
- Meromorphic Asymptotes
- Meromorphic Behavior
- Derivatives of Meromorphic Functions
- Elliptic Functions
A meromorphic function is the ratio of two analytic functions which are analytic except for isolated singularities, called “poles.” The word meromorphic comes from the Greek words meros (“part”) and morphe (“form” or “appearance”).
A “pole” is where the function is undefined, or approaches infinity. Like their real valued function counterparts, analytic functions will have poles when the denominator in the function equals zero (as long as the numerator is not also zero). When graphed, the poles create vertical asymptotes.
The gamma function is meromorphic in the whole complex plane.
|1 / z||everywhere except z = 0|
|1 / (1 + z2)||Everywhere except z = ±1|
|1 + z + z2||Everywhere|
The elliptic lambda is the fundamental modular function, taking values 0, 1, and infinity on the cusps.
The term “modular” comes from the moduli space of complex curves (a.k.a. Riemann surfaces) of genus 1 (Zagier, 1991).
An example of a modular function is the j-invariant. In fact, every modular function is a rational function of the j-invariant (Snowden, 2020).
Formal Definition of a Modular Function
A modular function can be defined in several ways. One way is to view these as complex-valued functions on ℍ which are (Zagler, 1991):
- Invariant under the action τ ↦ (aτ + b)/(cτ + d) of Γ1 on ℍ,
- Holomorphic on ℍ,
- Satisfy suitable growth conditions at infinity.
A modular function can also be defined as one with the following properties:
- f(μ(z)) = f(z) for all μ in the modular group Γ.
- f has a Fourier expansion of the form
Apostol, T. (1997). Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag.
Borwein, J. & Borwein, P. (1987). “Elliptic Modular Functions.” §4.3 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112-116.
Milanovich, D. (2020). Modular Functions and Picard’s Little Theorem. Retrieved November 12, 2020 from: https://sites.math.washington.edu/~morrow/336_20/papers19/Daniel.pdf
Prasad, D. (2000). Introduction to modular forms. Retrieved November 12, 2020 from: http://www.math.tifr.res.in/~dprasad/mf2.pdf
Rankin, R. (1997). Modular Forms and Functions. Cambridge University Press.
Schoeneberg, B. (1974). Elliptic Modular Functions: An Introduction. Berlin: New York: Springer-Verlag.
Snowden, A. (2020). Lecture 13: Modular forms. Retrieved November 12, 2020 from: http://www-personal.umich.edu/~asnowden/teaching/2013/679/L13.html
Zagler, D. (1991). Modular Forms of One Variable. Retrieved November 12, 2020 from: https://people.mpim-bonn.mpg.de/zagier/files/tex/UtrechtLectures/UtBook.pdf
As stated above, what makes a meromorphic function unique is that it contains singularities that tend to infinity. If you’re having trouble visualizing this, you can think of a “pole” in terms of asymptotes. A pole (zero) will create a vertical asymptote on a graph; The asymptotes of a meromorphic function are defined as the absolute value of z goes to infinity along a specified path, the function value tends to that asymptote. In notation, that’s:
As |z| →, f(z) → a.
While the term “meromorphic function” applies only to complex-valued functions, rational functions can show “meromorphic” behavior (i.e. there’s a place where a pole/zero creates an asymptote). A rational function is the division of two polynomials; Every rational function will have a pole and a vertical asymptote when there is division by zero.
A more precise definition
- The limit at each singularity is infinity,
- The singularity is surrounded by a neighborhood where the function is analytic, with the exception of the singularity itself.
This can be defined in notation as follows (Elajolet and Sedgewick, 2009):
“A function h(z), defined in Ω is meromorphic at z0 in Ω if and only if in a neighborhood of z0 ≠ z0 it can be represented as f(z)/g(z), where f(z) and g(z) are analytic at z0.”
Or, stated more simply:
It follows then, that if the function is basically a set of neighborhoods with sprinkled asymptotes, the derivative can be found for one neighborhood at a time, assuming the values are finite (with the exception of at most one singularity).
Elliptic functions are meromorphic functions and doubly periodic.
- A meromorphic function is the ratio of two analytic functions which are analytic except for isolated singularities, called “poles.”
- A doubly periodic function has two periods (ω1 and ω2), such that
f(z + ω1) = f (z + ω1) = f(z)
The fundamental pair of periods (demoted by omega, ω) span a parallelogram in the complex plane.
The ratio of the two periods, A/B cannot be purely real. Otherwise, the function is singly periodic (if the ratio is rational) and constant (if the ratio is irrational).
Fundamental Parallelograms of Elliptic Functions
The building blocks of elliptic functions are period parallelograms. With a real-valued periodic function, a period, or interval, repeats. With an elliptic function, a parallelogram repeats.
The fundamental period parallelogram of an elliptic function has vertices 0, 2ω1, 2ω1 + 2ω2, and 2ω2, where ω1 and ω2 are the function’s smallest periods. If you translate the fundamental parallelogram by integer multiples of periods ω1 and ω2, you get a period parallelogram.
Properties of Elliptic Functions
Elliptic functions obey the following properties. A “cell” is a period parallelogram where the function is not multi-valued:
- There are a finite number of poles in a cell; A cell with no poles is a constant.
- There are a finite number of roots in a cell.
- The sum of residues in any cell equals zero.
- The number of zeros (the order) equals the number of poles in f(z).
Types of Elliptic Functions
- Sine amplitude elliptic function — sn(x, k),
- Cosine amplitude elliptic function — cn(x, k),
- Delta amplitude elliptic function — dn(x, k).
Abelian functions (also called hyperelliptic functions) are a vast generalization of elliptic functions to more than one complex variable. While elliptic functions are associated with elliptic surfaces (called Riemann surfaces of genus 1), Abelian functions are associated to Riemann surfaces of higher genus (Doconinck et al., 2003). Both elliptic functions and Abelian functions can be written as a ratio of homogeneous polynomials of an auxiliary function, the Riemann theta function.
Historical Notes on Abelian Functions
Abelian functions have four periods (i.e. the values repeat as you move in one of four directions) and two variables. One way to define them is (Baker 1907, p. 21):
Note: Baker’s historical texts are freely available on the University of Michigan Historical Math Collection website.
While quite popular in the 19th century, the term “Abelian function” fell out of favor for a long time. The functions were rarely used outside of theoretical physics and applied mathematics, where many solutions of differential equations are written in terms of Abelian functions. However, in 1997, Buchstaber et al. published a review in that “recapitulated and developed” classical Abelian function theory in terms of multi-dimensional sigma-functions. The review is freely available as a pdf here.
Abelian Integrals and Abelian Functions
Abelian functions are obtained by inverting an arbitrary algebraic integral, or a combination of those types of integrals (Papadopoulos, 2017). Algebraic integrals have the form:
∫ R(x, y)dx
Abelian integrals are any integral of an algebraic function which can’t be reduced to elliptic form. These integrals give rise to Abelian functions, defined as symmetric hyperelliptic functions, composed of multi-variable inverses of Abelian integral.
Baker, H. F. An Introduction to the Theory of Multiply Periodic Functions. London: Cambridge University Press, 1907. Available here.
Cima, J. & Schober, G. On Spaces of Meromorphic Functions. Rocky Mountain Journal of Mathematics. Volume 9, Number 3. Summer 1979. Retrieved December 8, 2019 from: https://projecteuclid.org/download/pdf_1/euclid.rmjm/1250129198
Deconinck, B. et al. (2003). Computing Riemann Theta Functions. Mathematics of Computation. Volume 73, Number 247.
Derbyshire, J. (2003). Prime Obsession. Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. National Academic Press.
Elajolet, P. & Sedgwick, R. (2009). Analytic Combinatorics. Cambridge University Press.
Hazelwinkel, M. (2012). Encyclopaedia of Mathematics, Volume 1. Springer, Netherlands.
Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus fratrum Borntraeger, 1829.
Ludmark, H. Complex Analysis. Retrieved December 20, 2019 from: http://users.mai.liu.se/hanlu09/complex/elliptic/
Papadopoulos, A. (2017). Looking Backward from Euler to Riemann. Retrieved September 24, 2020 from: http://arxiv-export-lb.library.cornell.edu/pdf/1710.03982
Smith, K. (2013). Elementary Functions.
Wells, R. (2015). The Origins of Complex Geometry in the 19th Century. Retrieved September 24, 2020 from: https://arxiv.org/pdf/1504.04405.pdf
Learn more about Meromorphic Function: Nine Introductions in Complex Analysis. In North-Holland Mathematics Studies, 2008.
Elliptic Function. Retrieved December 20, 2019 from: https://archive.lib.msu.edu/crcmath/math/math/e/e093.htm
Tkachev, V. Introductory course. Retrieved December 20, 2019 from: http://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf