= 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 $x^2y''+xy'+x^2y=0$ 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.
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