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1: 23.15 Definitions
§23.15 Definitions
… ►A modular function is a function of that is meromorphic in the half-plane , and has the property that for all , or for all belonging to a subgroup of SL, …(Some references refer to as the level). …If, in addition, as , then is called a cusp form. … ►2: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of , , and , and also to complex values. … ►§8.17(ii) Hypergeometric Representations
… ►§8.17(iv) Recurrence Relations
… ►§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties
…3: 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
…4: 8 Incomplete Gamma and Related
Functions
Chapter 8 Incomplete Gamma and Related Functions
…5: 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. |
6: 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, …7: 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
…8: 15.2 Definitions and Analytical Properties
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