Special function

A special function is a named family of functions with a developed theory of identities, transformations, asymptotic behavior and numerical evaluation, often arising repeatedly as solutions of differential equations or integral representations. The boundary with elementary functions is conventional rather than a strict mathematical classification.

For example, the regular solution of $x^2{y}^{\prime \prime }+x{y}^{\prime }+x^{2}y=0$ normalized by $y(0)=1$ is the order-zero Bessel function $J_0$. Such a function packages a recurring radial-wave calculation into a reusable object. The Gamma function similarly extends factorial values through $\Gamma (n+1)=n!$ for nonnegative integers, allowing identities involving integer parameters to be studied continuously.