Franklin's Notes

Doubly periodic function

A function $f:\mathbb C\to\mathbb C$ is said to be periodic with period $\omega$ if $f(z)=f(z+\omega)$ for all $z\in\mathbb C$. Intuitively, the values of $f$ "repeat" in intervals of $\omega$. If $f$ has period $\omega$, we can divide the complex plane into countably infinitely many "strips" so that the values taken by $f$ are the same in each strip:

If $\omega$ is a period, then so are $2\omega,3\omega,$ and more generally $n\omega$ for all $n\in\mathbb N$. A fundamental period of $f$ is a period which is not an integer multiple of another period with smaller magnitude. A doubly periodic function is a function which is periodic with two distinct fundamental periods that are not opposites (for if $\omega$ is a fundamental period, then so is $-\omega$). These periods are often denoted $\omega_1,\omega_2$. If these two periods are not real multiples of each other, then the complex plane can be divided into parallelograms, called fundamental parallelograms, in which the values of $f$ repeat:

In the image above, the function $f$ takes the same value at all of the green points. The purple and blue vectors represent the fundamental periods $\omega_1$ and $\omega_2$.

As it happens, the holomorphic doubly periodic functions $f$ have some very nice properties that make them easy to classify. For instance:



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