Kamov Ka-50

In Analyze complexes, a elliptic function is, coarsely speaking, a function definite on the Plan complex which is doubly periodic (periodical in two directions). It can be seen as similar to a goniometrical Function (which has only one period).

Description

Formally, an elliptic function is a Fonction méromorphe F definite on

\ mathbb {C}

for which there exist two nonnull complex numbers has and B such as:

${has \ over B} \ not \ in \ mathbb {R}$
$f (z+a)=f (z+b)=f (Z) \ \ forall Z \ in \ mathbb {C}$

Of this, it follows that: $f (z+ma+nb) =f (Z) \ \ forall Z \ in \ mathbb {C}, \ forall m \ in \ mathbb {Z}, \ forall N \ in \ mathbb {Z}$

The most important class of elliptic functions is that of the elliptic functions of Weierstrass; any elliptic function can be expressed using those.

The elliptic functions are the reciprocal applications of the elliptic integral functions , and it is in this way that historically they were introduced.

All Complex number ω such as $f (z+ \ Omega) =f (Z) \ \ forall Z \ in \ mathbb {C}$ is called “period” of F . If the two periods has and B is such as any other period ω can be written in the form ω = my + Nb with m and N whole, then has and B is called “fundamental periods”. Any elliptic function has a pair of fundamental periods, but it is not single.

If has and B is fundamental periods, then all Parallélogramme of tops of affix Z , Z + has , Z + B , Z + has + B is called a “fundamental parallelogram”. To relocate such a parallelogram of a multiple entirety of has and B gives a parallelogram of the same type, and the function F behaves identically on this relocated parallelogram, because of the periodicity.

The number of poles in any fundamental parallelogram is finished (and the same one for any fundamental parallelogram). Unless the elliptic function is not constant, any parallelogram contains at least a pole, consequence of the theorem of Liouville.

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 a fundamental parallelogram is equal to zero, therefore in particular no elliptic function can have an order equal to one.

The Dérivée from an elliptic function is still an elliptic function, of the same period. The whole of all the elliptic functions of same fundamental periods forms a body. More precisely, a couple of periods being given, any elliptic function admitting this couple of periods can be defined on some Surface of Riemann: the complex Torus obtained by sticking together of the couples on sides opposite of the fundamental parallelogram. The elliptic functions are then the functions méromorphes on this torus. In addition, the function of Weierstrass associated with this couple with periods, and its derivative parameterize a certain complex curve: a elliptic Curve.

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