The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real.
in the Laguerre–Pólya class are: A function is of Laguerre–Pólya class if and only if three conditions are met: with b and c real and c non-positive.
(The non-negative integer m will be positive if E(0)=0.
Note that if the number of zeros is infinite one may have to define how to take the infinite product.)
Here is one series of polynomials having all real roots: And here is another: This shows the buildup of the Hadamard product for cosine.
If we replace z2 with z, we have another function in the class: Another example is the reciprocal gamma function 1/Γ(z).