In mathematics, a doubly periodic function is a function defined at all points on the complex plane and having two "periods", which are complex numbers u and v that are linearly independent as vectors over the field of real numbers. That u and v are periods of a function ƒ means that
for all values of the complex number z.
The doubly periodic function is thus a twodimensional extension of the simpler singly periodic function, which repeats itself in a single dimension. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine. In the complex plane the exponential function e^{z} is a singly periodic function, with period 2πi.
As an arbitrary mapping from pairs of reals (or complex numbers) to reals, a doubly periodic function can be constructed with little effort. For example, assume that the periods are 1 and i, so that the repeating lattice is the set of unit squares with vertices at the Gaussian integers. Values in the prototype square (i.e. x + iy where 0 ≤ x < 1 and 0 ≤ y < 1) can be assigned rather arbitrarily and then 'copied' to adjacent squares. This function will then be necessarily doubly periodic.
If the vectors 1 and i in this example are replaced by linearly independent vectors u and v the prototype square becomes a prototype parallelogram, which still tiles the plane. And the "origin" of the lattice of parallelograms does not have to be the point 0; the lattice can start from any point. In other words, we can think of the plane and its associated functional values as remaining fixed, and mentally translate the lattice to gain insight into the function's characteristics.
However, what is usually meant is a "smooth" complex function, a mapping satisfying the Cauchy–Riemann equations and providing an analytic function (away from some reasonable number of singularities), in other words, a meromorphic function. Quite a bit of information about such a function can be obtained by applying some basic theorems from complex analysis.
See elliptic function for a more complete account of doubly periodic functions that are meromorphic on the complex plane, and fundamental pair of periods for an account of the lattices involved. Also see Jacobi's elliptic functions and Weierstrass's elliptic functions.
