• Complex Analysis: #25 Some Infinite Products

Genus zero:
Given a sequence of non-zero points a_n with ∑ 1/|a_n| converging, we can simply say that

gives us an example of a genus zero function with zeros just where we want them. Of course if we also want a zero at 0, then we can multiply this product with the factor Cz^l say, where l is the order of this zero, and C is a non-zero constant which we can choose as we like.


The sine function:
Being more concrete, let us look at the function sin πz. According to the definition of the sine function, we have

But this can only be zero if e^iπz  = e^−iπz. Writing z = x + iy, this means that e^−ye^iπx = e^ye^−iπx. In particular, y = 0 so that the zeros are just the familiar zeros which we know from real analysis, namely the integers, ℤ. As we know, the series ∑ 1/n² converges, thus the complex sine function must have genus one. Writing

the problem is then to determine the function g. For this, we take the logarithmic derivatives of both sides. We obtain

so that the sum is uniformly convergent in compact subsets of ℂ which do not contain points of ℤ, thus showing that the derivative of the sum is the sum of the derivatives.)

Therefore

where C is a constant. On the other hand, we have both cot(w) = −cot(−w) and χ(w) = −χ(−w). That is, they are both anti-symmetric. But this can only be true if the constant C = 0. But this implies that our function g satisfies the equation g '(z) = 0; that is, g is a constant. Since