Examples of real analytic functions with finite number of zeros that are not polynomials Can you please give me examples of real analytic functions with finite number of zeros that are not polynomials? I know that there are $\mathrm{e}^x$ and $\operatorname{sh}x$, but I can't think of anything else.
It would be even better if there was a method of constructing a large class of such functions.
 A: *

*$e^x + $ any polynomial. 

*Any positive power series that whose coefficients decay even faster than the Taylor series of $e^x$.


You can concoct numerous other examples by combining other analytic functions and perturbing their coefficients. Perhaps you mean to ask something a little different...
A: Edit: I answered a companion question (for entire functions). I suppose the underlying question (as sort of explained in the comments) is to characterize the real analytic $f$ without zeroes. 

Weierstrass factorization theorem says the following: Suppose that $f$ is entire. Then $f$ has only countably many zeros (possibly only finitely many), counting multiplicities; if $0$ is a zero of multiplicity $k$, and $z_1,z_2,\dots$ enumerates the distinct non-zero zeros of $f$ (of multiplicities $m_1,m_2,\dots$, respectively), then there is an entire function $g(z)$ and a sequence $p_1,p_2,\dots$ of integers such that 
 $$ f(z)=z^k e^{g(z)}\prod_{n\ge 1}E_{p_n}(z/z_n), $$
where the $E_k(z)$ are the elementary factors, given by
 $$ E_0(z)=1-z $$ and, for $n>0$, $$E_n(z)=(1-z)exp(z+\frac{z^2}2+\dots+\frac{z^n}n).$$
(There is also a more general version of this result for functions that are not entire.)
Note that if $f$ has only finitely many zeroes, then it follows that $f$ has the form $p(z)e^{h(z)}$ for some entire $h$ and polynomial $p$. 
Also, this indicates the enormous variety of entire functions with given zeroes at prescribed locations, and gives a hint of how to build any. (There are general estimates on the sizes of the elementary factors and general results on convergence of infinite products that can be combined to check in specific cases that a product as above indeed defines an entire function.)
A: If I remember correctly, an entire function i.e. function analytic everywhere, has exactly finitely many zeros if and only if it is of the form
$$ p(z)e^{g(z)}$$ where $p(z)$ is a polynomial and  $g(z)$ is entire.
In fact, $p(z)=A(z-z_1)^a(z-z_2)^b\cdots (z-z_n)^c$ for some constant $A$, if $z_1,z_2,\ldots, z_n$ are zeros of the function with orders $a,b,\ldots, c$ respectively.
