orthogonal functions with respect to weighted summation
1—10 of 1004 matching pages
… ►(For other notation see Notation for the Special Functions.) ►
►The main functions treated in this chapter are the Airy functions
and , and the Scorer functions
and (also known as inhomogeneous Airy functions).
►Other notations that have been used are as follows: and for and (Jeffreys (1928), later changed to
and ); , (Fock (1945)); (Szegő (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii).
derivatives with respect to argument.
§23.15(i) General Modular Functions… ►
Elliptic Modular Function… ►
Dedekind’s Eta Function (or Dedekind Modular Function)… ►
§5.15 Polygamma Functions►The functions , , are called the polygamma functions. In particular, is the trigamma function; , , are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For see §24.2(i). …
§20.2(i) Fourier Series… ►Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to . ►
§20.2(ii) Periodicity and Quasi-Periodicity… ►The theta functions are quasi-periodic on the lattice: … ►
§15.2(i) Gauss Series►The hypergeometric function is defined by the Gauss series … … ►
§15.2(ii) Analytic Properties… ►Because of the analytic properties with respect to , , and , it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. …
… ► … ►The interval is mapped one-to-one to the interval , with the points and corresponding to and , respectively. … ►The interval is mapped one-to-one to the interval , with the points , , and corresponding to , , and , respectively. … ►
§14.20(x) Zeros and Integrals… ►For integrals with respect to involving , see Prudnikov et al. (1990, pp. 218–228).
§14.19 Toroidal (or Ring) Functions►
§14.19(i) Introduction… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates , which are related to Cartesian coordinates by … ►
§14.19(iv) Sums… ►
§14.19(v) Whipple’s Formula for Toroidal Functions…
§5.2(i) Gamma and Psi Functions►
Euler’s Integral… ►It is a meromorphic function with no zeros, and with simple poles of residue at . … ►
5.2.2 .… ►