The Laguerre–Pólya class is the class of

entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any ...

s consisting of those functions which are locally the limit of a series of polynomials whose roots are all real. "Approximation by entire functions belonging to the Laguerre–Pólya class"
by D. Dryanov and Q. I. Rahman, ''Methods and Applications of Analysis'' 6 (1) 1999, pp. 21–38. Any function of Laguerre–Pólya class is also of Pólya class. The product of two functions in the class is also in the class, so the class constitutes a

monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...

under the operation of function multiplication. Some properties of a function

E(z)

in the Laguerre–Pólya class are: *All

root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...

s are real. *

, E(x+iy), =, E(x-iy),

for ''x'' and ''y'' real. *

, E(x+iy),

is a non-decreasing function of ''y'' for positive ''y''. A function is of Laguerre–Pólya class if and only if three conditions are met: *The roots are all real. *The nonzero zeros ''z_n'' satisfy :

\sum_n\frac

converges, with zeros counted according to their multiplicity) * The function can be expressed in the form of a Hadamard product :

z^m e^\prod_n \left(1-z/z_n\right)\exp(z/z_n)

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 In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound ...

Examples

Some examples are

\sin(z), \cos(z), \exp(z), \exp(-z), \text\exp(-z^2).

On the other hand,

\sinh(z), \cosh(z), \text \exp(z^2)

are ''not'' in the Laguerre–Pólya class. For example, :

\exp(-z^2)=\lim_(1-z^2/n)^n.

Cosine can be done in more than one way. Here is one series of polynomials having all real roots: :

\cos z=\lim_((1+iz/n)^n+(1-iz/n)^n)/2

And here is another: :

\cos z=\lim_\prod_^n \left(1-\frac\right)

This shows the buildup of the Hadamard product for cosine. If we replace ''z''² with ''z'', we have another function in the class: :

\cos \sqrt z=\lim_\prod_^n \left(1-\frac z\right)

Another example is the reciprocal gamma function 1/Γ(z). It is the limit of polynomials as follows: :

1/\Gamma(z)=\lim_\frac 1(1-(\ln n)z/n)^n\prod_^n(z+m).

References

{{DEFAULTSORT:Laguerre-Polya class Analytic functions