Everything about Elliptic Function totally explained
In
complex analysis, an
elliptic function is a
function defined on the
complex plane which is
periodic in two directions (a
doubly-periodic function). 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.
Formally, an elliptic function is a
meromorphic function f defined on
C for which there exist two non-zero complex numbers
a and
b with
a/
b not
real, such that
» f(
z +
a) =
f(
z +
b) =
f(
z) for all
z in
C
wherever
f(z) is defined. From this it follows that
» 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
-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 complicated but important both for the history and for general theory. The primary difference between these two theories is that the Weierstrass functions have second-order and higher-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.
More generally, the study of elliptic functions is closely related to the study of
modular functions and
modular forms, a relationship proven by the
modularity theorem. Examples of this relationship include the
j-invariant, the
Eisenstein series and the
Dedekind eta function.
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 isn't unique, as described below.
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
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' .
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.
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.
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.
The number of zeros (counted with multiplicity) in any fundamental parallelogram is equal to the order of the elliptic function.
The set of all elliptic functions with the same fundamental periods form a
field.
The
derivative of an elliptic function is again an elliptic function, with the same periods.
The
Weierstrass elliptic function is the prototypical elliptic function, and in fact, the field of elliptic functions with respect to a given lattice is generated by
and its derivative
.
Further Information
Get more info on 'Elliptic Function'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://elliptic_function.totallyexplained.com">Elliptic function Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
