Entire function

[Liu]

I have just finished the first read of Chapter 5 of Stein’s book:
complex analysis. Roughly put, entire functions are decided by there
zeros. If the number of zeros are finite, then the function can be
constructed easily.

But when it comes to the situation when the number of zeros are
infinite, it is much more complicated. However, we are lucky enough to
have the following theorem:

Theorem 4.1(weierstrass infinite products) Given any sequence \{a_n\}
of complex numbers with |a_n|\to \infty as n\to \infty, there exists
an entire function f that vanishes at all z=a_n and nowhere else. Any
other such entire function is of the form f(z)e^{g(z)}, where g is
entire.

To prove this theorem, we should observe an important identity first:

Basically what we will do to construct our entire function is to make
our \{a_n\} act just like these integers. A problem we should pay
attention to is that the speed with which \{|a_n|\} grows. Till now we
can see, some very detailed analysis must be done in order to get this
problem solved, and that is what the whole story goes. So, there are
lots of paragraphs dealing with the growing speed. The first section,
namely Jensen’s formula is a devise to do this. With the help of
Theorem 2.1, Hadamard’s factorization theorem goes even further,
strengthening the weierstrass infinite products.

see also: http://liuxiaochuan.wordpress.com/2009/11/09/entire-functions/

[Akhil Mathew]

I don't think what you said about entire functions with finitely many
zeros on your blog is true- e^z is an example with no zeros that is
not a polynomial.
Maybe you meant to say "a polynomial times a function e^h(z) where h
is entire"?

[Liu Xiaochuan]

Yeah, that's right. I didn't make it very strictly. Thanks for your
comment. I will modify it a little.

--
Xiaochuan Liu
lxc...@gmail.com
Nankai University