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 fin ...

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 branch of mathematics, 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 0. Monoids ...

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 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 below the sur ...

s are real. *

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

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

, E(x+iy),

is a

non-decreasing function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...

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.)

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 In mathematics, the reciprocal gamma function is the function :f(z) = \frac, where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As a ...

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