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Encyclopedia > Elliptic function

In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only). Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Partial plot of a function f. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... For other uses, see Curve (disambiguation). ... The ellipse and some of its mathematical properties. ...


Formally, an elliptic function is a meromorphic function f defined on C for which there exist two non-zero complex numbers a and b such that In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...

f(z + a) = f(z + b) = f(z)   for all z in C

and such that a/b is not real. From this it follows that In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

f(z + ma + nb) = f(z)   for all z in C and all integers m and n.

In developments of the theory of elliptic functions, modern authors mostly follow Karl Weierstrass: the notations of Weierstrass's elliptic functions based on his wp-function are convenient, and any elliptic function can be expressed in terms of these. Weierstrass became interested in these functions as a student of Christoph Gudermann, a student of Carl Friedrich Gauss. The elliptic functions introduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are more complex; but important both for the history and for general theory. The primary difference between these two theories is that the Weierstrass functions have high-order poles located at the corners of the periodic lattice, whereas the Jacobi functions have simple poles. The development of the Weierstrass theory is easier to present and understand, having fewer complications. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Karl Weierstraß Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Biography Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... Christoph Gudermann (March 25, 1798 - September 25, 1852) was born in Vienenburg, Germany. ... (help· info) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ... Carl Gustav Jakob Jacobi for the mathematician (1804-1851). ... In mathematics, theta functions are special functions of several complex variables. ... See lattice for other meanings of this term, both within and without mathematics. ...


More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, examples of which include the j-invariant, the Eisenstein series and the Dedekind eta function. In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ... A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ... Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ... In mathematics, Eisenstein series are particular modular forms with infinite series expansions that may be written down directly. ... The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...


Definition and properties

Any complex number ω such that f(z + ω) = f(z) for all z in C is called a period of f. If the two periods a and b are such that any other period ω can be written as ω = ma + nb with integers m and n, then a and b are called fundamental periods. Every elliptic function has a pair of fundamental periods, but this pair is not unique, as described below. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ...


If a and b are fundamental periods describing a lattice, then exactly the same lattice can be obtained by the fundamental periods a' and b' where a' = p a + q b and b' = r a + s b where p, q, r and s being integers satisfying p s - q r = 1. That is, the matrix begin{pmatrix} p & q  r & s end{pmatrix} has determinant one, and thus belongs to the modular group. In other words, if a and b are fundamental periods of an elliptic function, then so are a' and b' . In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...


If a and b are fundamental periods, then any parallelogram with vertices z, z + a, z + b, z + a + b is called a fundamental parallelogram. Shifting such a parallelogram by integral multiples of a and b yields a copy of the parallelogram, and the function f behaves identically on all these copies, because of the periodicity. A parallelogram. ...


The number of poles in any fundamental parallelogram is finite (and the same for all fundamental parallelograms). Unless the elliptic function is constant, any fundamental parallelogram has at least one pole, a consequence of Liouville's theorem. Liouvilles theorem in complex analysis states that every bounded (i. ...


The sum of the orders of the poles in any fundamental parallelogram is called the order of the elliptic function. The sum of the residues of the poles in any fundamental parallelogram is equal to zero, so in particular no elliptic function can have order one. In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. ...


The number of zeros (counted with multiplicity) in any fundamental parallelogram is equal to the order of the elliptic function.


The derivative of an elliptic function is again an elliptic function, with the same periods. The set of all elliptic functions with the same fundamental periods form a field. In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...


The Weierstrass elliptic function wp is the prototypical elliptic function, and in fact, the field of elliptic functions with respect to a given lattice is generated by wp and its derivative wp'. In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ...


References

  • Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 16.)
  • Naum Illyich Akhiezer, Elements of the Theory of Elliptic Functions, (1970) Moscow, translated into English as AMS Translations of Mathematical Monographs Volume 79 (1990) AMS, Rhode Island ISBN 0-8218-4532-2
  • Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976. ISBN 0-387-97127-0 (See Chapter 1.)
  • Albert Eagle, The elliptic functions as they should be. Galloway and Porter, Cambridge, England 1958.
  • E. T. Whittaker and G. N. Watson. A course of modern analysis, Cambridge University Press, 1952

  Results from FactBites:
 
Elliptic function - Wikipedia, the free encyclopedia (718 words)
Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives.
More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, examples of which include the j-invariant, the Eisenstein series and the Dedekind eta function.
The derivative of an elliptic function is again an elliptic function, with the same periods.
Elliptic integral - Wikipedia, the free encyclopedia (655 words)
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler.
Elliptic integrals are often expressed as functions of a variety of different arguments.
Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular: we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.
  More results at FactBites »

 
 

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