Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.
In algorithmic languages, rational approximations are typically used, although they may behave badly in the case of complex argument(s).
Applications to the physical sciences and engineering determined the relative importance of functions.
For this purpose, the main techniques are: More theoretical questions include: asymptotic analysis; analytic continuation and monodromy in the complex plane; and symmetry principles and other structural equations.
The classic Whittaker and Watson (1902) textbook[4] sought to unify the theory using complex analysis; the G. N. Watson tome A Treatise on the Theory of Bessel Functions pushed the techniques as far as possible for one important type, including asymptotic results.
The later Bateman Manuscript Project, under the editorship of Arthur Erdélyi, attempted to be encyclopedic, and came around the time when electronic computation was coming to the fore and tabulation ceased to be the main issue.
Hypergeometric series, observed by Felix Klein to be important in astronomy and mathematical physics,[5] became an intricate theory, requiring later conceptual arrangement.
Further, work on algebraic combinatorics also revived interest in older parts of the theory.
Conjectures of Ian G. Macdonald helped open up large and active new fields with a special function flavour.
In number theory, certain special functions have traditionally been studied, such as particular Dirichlet series and modular forms.