Editing Entire function (section)

==Other examples==
According to [[J. E. Littlewood]], the [[Weierstrass sigma function]] is a 'typical' entire function. This statement can be made precise in the theory of random entire functions: the asymptotic behavior of almost all entire functions is similar to that of the sigma function. Other examples include the [[Fresnel integral]]s, the [[Jacobi theta function]], and the [[reciprocal Gamma function]].  The exponential function and the error function are special cases of the [[Mittag-Leffler function]]. According to the fundamental [[Paley–Wiener theorem|theorem of Paley and Wiener]], [[Fourier transform]]s of functions (or distributions) with bounded support are entire functions of order <math>1</math> and finite type.

Other examples are solutions of linear differential equations with polynomial coefficients. If the coefficient at the highest derivative is constant, then all solutions of such equations are entire functions. For example, the exponential function, sine, cosine, [[Airy function]]s and [[Parabolic cylinder function]]s arise in this way. The class of entire functions is closed with respect to compositions. This makes it possible to study [[holomorphic dynamics|dynamics of entire functions]].

An entire function of the square root of a complex number is entire if the original function is [[even function|even]], for example <math>\cos(\sqrt{z})</math>.

If a sequence of polynomials all of whose roots are real converges in a neighborhood of the origin to a limit which is not identically equal to zero, then this limit is an entire function. Such entire functions form the [[Laguerre–Pólya class]], which can also be characterized in terms of the Hadamard product, namely, <math>f</math> belongs to this class [[if and only if]] in the Hadamard representation all <math>z_n</math> are real, <math>\rho\leq 1</math>, and 
<math>P(z)=a+bz+cz^2</math>, where <math>b</math> and <math>c</math> are real, and <math>c\leq 0</math>. For example, the sequence of polynomials

<math display="block">\left (1-\frac{(z-d)^2}{n} \right )^n</math>

converges, as <math>n</math> increases, to <math>\exp(-(z-d)^2)</math>. The polynomials

<math display="block"> \frac{1}{2}\left ( \left (1+\frac{iz}{n} \right )^n+ \left (1-\frac{iz}{n} \right )^n \right )</math>

have all real roots, and converge to <math>\cos(z)</math>. The polynomials

<math display="block"> \prod_{m=1}^n \left(1-\frac{z^2}{\left ( \left (m-\frac{1}{2} \right )\pi \right )^2}\right)</math>

also converge to <math>\cos(z)</math>, showing the buildup of the Hadamard product for cosine.