Thursday, May 22, 2014

Complex Analysis: #25 Some Infinite Products

  • Complex Analysis: #25 Some Infinite Products

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

Complex Analysis: #25 Some Infinite Products equation pic 1

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 Czl 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

Complex Analysis: #25 Some Infinite Products equation pic 2

But this can only be zero if eiπz  = e−iπz. Writing z = x + iy, this means that e−yeiπx = eye−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

Complex Analysis: #25 Some Infinite Products equation pic 3

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

Complex Analysis: #25 Some Infinite Products equation pic 4

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

Complex Analysis: #25 Some Infinite Products equation pic 5

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

Complex Analysis: #25 Some Infinite Products equation pic 6

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