When does a function have an addition theorem?
The addition theorem for cosine says that
and the addition theorem for hyperbolic cosine is analogous, though with a sign change.
An addition theorem is a theorem that relates a function's value at x + y to its values at x and at y. The squaring function satisfies a very simple addition theorem
and the Jacobi function sn satisfies a more complicated addition theorem.
Which functions have addition theorems? Before we can answer this question we need to be more precise about what an addition theorem is. We've said that an addition theorem for relates (x + y) to (x) and (y). But what exactly do we mean by relate"? What counts as a relation?
Also, the examples above don't exactly satisfy this definition. The addition law for cosines, for example, relates cos(x + y) to the values of cos(x) and cos(y) but also to sin(x) and sin(y). Somehow that feels OK because sine and cosine are related. But here again we're talking about things being related without saying exactly what we mean.
Weierstrass (1815-1897) made the idea of an addition theorem precise and classified functions having addition theorems. A function satisfies an algebraic addition theorem if there is a polynomial F in three variables such that
For example, if (x) = x^2 then
and so we could take F to be
Similarly, if (x) = cos x then
and so we could take F to be
Now for Weierstrass' theorem. A merimorphic function (z) has an algebraic addition theorem if and only if it is an elliptic function of z, a rational function of z, or a rational function of exp(z).
A merimorphic function is one that is analytic everywhere except at isolated singularities. To put it another way, we assume has a convergent power series everywhere in the complex plane except at isolated points.
The examples above illustrate all three cases of Weierstrass' theorem. The function sn(z) is elliptic, the function z^2 is rational, and the functions cos(z) and cosh(z) are rational functions of exp(iz).
Other kinds of addition theoremsAlgebraic addition theorems are not the only kind of addition theorems. For example, Bessel functions satisfy a different kind of addition theorem:
This theorem relates the value of a Bessel function at x + y to the values of other Bessel functions at x and at y, but it is not an algebraic addition theorem because the right hand side is an infinite sum and because the Bessel functions are not algebraically related to each other.
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