relation%20to%20q-hypergeometric%20functions
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11: 14.19 Toroidal (or Ring) Functions
§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
…12: 17.6 Function
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Andrews–Askey Sum
… ►Rogers–Fine Identity
… ►Heine’s Contiguous Relations
… ►§17.6(v) Integral Representations
… ►§17.6(vi) Continued Fractions
…13: 8 Incomplete Gamma and Related
Functions
Chapter 8 Incomplete Gamma and Related Functions
…14: 9.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►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). | |
… | |
primes | derivatives with respect to argument. |
15: 9.12 Scorer Functions
§9.12 Scorer Functions
… ► is a numerically satisfactory companion to the complementary functions and on the interval . is a numerically satisfactory companion to and on the interval . … ► … ►As , and with denoting an arbitrary small positive constant, …16: 20.2 Definitions and Periodic Properties
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§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: … ►§20.2(iv) -Zeros
…17: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
… ►The branch obtained by introducing a cut from to on the real -axis, that is, the branch in the sector , is the principal branch (or principal value) of . … ►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. … ►which sometimes needs to be used in §15.4. … ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that is equal to the first terms of the Maclaurin series for .18: 5.15 Polygamma Functions
§5.15 Polygamma Functions
►The functions , , are called the polygamma functions. …Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►As in …For see §24.2(i). …19: 5.12 Beta Function
§5.12 Beta Function
… ►Euler’s Beta Integral
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5.12.2
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