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Elliptic Functions


Post Sun Mar 16, 2014 9:36 am
Shobhit Site Admin
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Introduction to Elliptic Functions

1. Doubly Periodic Functions

Let $\omega_1,\omega_2$ be any two numbers(real or complex) whose ratio is not real. A function which satisfies the equations $$f(z+2\omega_1)=f(z+2\omega_2)=f(z)$$ for all values of $z$ for which $f(z)$ exists is called a doubly periodic function with periods $2\omega_1,2\omega_2$. A doubly periodic function which is analytic(except at poles) and which has no singularities other than poles in the finite part of the plane is called an elliptic function.

2. Period Parallelograms

Suppose that in the plane of $z$ we mark the points $0,\omega_1,2\omega_2,2\omega_1+2\omega_2$, we obtain a parallelogram. If there is no point $\omega$ inside or on the boundary of the parallelogram(except the vertices) such that $$f(z+\omega)=f(z)$$ for all values of $z$ then this parallelogram is called the fundamental period parallelogram for an elliptic function with periods $\omega_1,2\omega_2$. The $z$ plane may be covered with a network of parallelograms equal to the fundamental period-paralellogram and similarly situated, each of the points $2m\omega_1+2n\omega_2$ being a vertex of four parallelograms.

3. Liouville's Theorem An elliptic function with no poles is merely a constant.

Post Sun Mar 16, 2014 9:46 am
Shobhit Site Admin
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4. Construction of Elliptic Functions

The Weierstrass P Function with periods $2\omega_1$ and $2\omega_2$ is defined as $$\wp(z)=\frac{1}{z^2}+\sum_{n^2+m^2\neq 0} \left(\frac{1}{(z-2m\omega_1-2n\omega_2)^2}-\frac{1}{(2m\omega_1+2n\omega_2)^2} \right)$$ The summation extends over all integer values of $m$ and $n$ except simultaneous zero values. It can be shown that $\wp(z+2\omega_1)=\wp(z)$ and $\wp(z+2\omega_2)=\wp(z)$ and has no singularities but poles. (See next section for details)

Post Tue Apr 22, 2014 8:37 am
Shobhit Site Admin
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5. Periodicity Properties of the Weierstrass P Function

Since the series for $\wp(z)$ is a uniformly convergent series of analytic functions, term by term differentiation is legitimate, and so $$\wp'(z)=\frac{d}{dz}\wp(z)=-\frac{2}{z^3}-2\sum_{m^2+n^2\neq 0} \frac{1}{(z-2m\omega_1-2n\omega_2)^3}$$
The set of points $-2m\omega_1-2n\omega_2$ is the same as $2m\omega_1+2n\omega_2$ and so the terms of $\wp'(-z)$ are the same as $-\wp'(z)$ in different order. Thus the function $\wp'(z)$ is an odd function of $z$ i.e. $$\wp'(-z)=-\wp'(z)$$.Similarly the terms of $\wp(z)$ are the same as $\wp(-z)$ in the different order. Hence $$\wp(-z)=\wp(z)$$ Further $$\wp'(z+2\omega_1)=-2\sum_{\begin{matrix} m,n=-\infty \end{matrix}}^\infty \frac{1}{(z-2m\omega_1-2n\omega_2+2\omega_1)^3}$$But the set of points $2m\omega_1+2n\omega_2-2\omega_1$ is same as $2m\omega_1+2n\omega_2$, so the series for $\wp'(z+2\omega_1)$ is a derangement of the series for $\wp'(z)$. Thus $$\wp'(z+2\omega_1)=\wp'(z)$$ If we integrate the above equation, we get $$\wp(z+2\omega_1)=\wp(z)+A$$ where $A$ is a constant. Putting $z=-\omega_1$ and using the fact that $\wp(z)$ is an even function we get $A=0$, so that $$\wp(z+2\omega_1)=\wp(z)$$ In a like manner $\wp(z+2\omega_2)=\wp(z)$. Since $\wp(z)$ has no singularities but poles, it follows that $\wp(z)$ is an elliptic function.

Post Tue Apr 22, 2014 9:20 am
Shobhit Site Admin
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6. Differential Equation satisfied by \(\displaystyle \wp(z)\)

The function $\wp(z)-z^{-2}$ is analytic in a region where $0$ is an internal point, and it is an even function of $z$. Consequently by Taylor's theorem we have an expansion of the form $$\wp(z)-z^{-2}=\frac{1}{20}g_2 z^2+\frac{1}{28}g_3 z^4 +O(z^6)$$
valid for sufficiently small values of $|z|$. It is easy to see that $$
\begin{align*}
g_2 &= 60\sum_{m^2+n^2\neq 0}\frac{1}{(2m\omega_1+2n\omega_2)^4} \\
g_3 &= 140\sum_{m^2+n^2\neq 0}\frac{1}{(2m\omega_1+2n\omega_2)^6}
\end{align*}
$$ Thus $$\wp(z)=z^{-2}+\frac{1}{20}g_2 z^2+\frac{1}{28}g_3 z^4 +O(z^6)$$ Differentiating this result, we have $$\wp'(z)=-2z^{-3}+\frac{1}{10}g_2 z+\frac{1}{7}g_3 z^3+O(z^5)$$ Cubing and squaring these respectively we get $$\begin{align*}
\wp(z)^3 &= z^{-6}+\frac{3}{20}g_2 z^{-2}+\frac{3}{28}g_3+O(z^2) \\
\wp'(z)^2 &= 4z^{-6}-\frac{2}{5}g_2 z^{-2}-\frac{4}{7}g_3+O(z^2)
\end{align*}$$ Hence $$ \wp'(z)^2-4\wp(z)^3 +g_2 \wp(z)+g_3= O(z^2)$$
That is to say, the function $\wp'(z)^2-4\wp(z)^3 +g_2 \wp(z)+g_3$ which is obviously an elliptic function is analytic at the origin and at all congruent points. But such points are the only possible singularities of the function, so it is an elliptic function with no singularities; it is therefore a constant (Liouville's Theorem). On making $z\to 0$ we see that this constant is $0$.

Thus finally the function $\wp(z)$ satisfies the differential equation $$\wp'(z)^2=4\wp(z)^3 -g_2 \wp(z)-g_3$$


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