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1: 31.1 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 , , , and the polynomial .
…Sometimes the parameters are suppressed.
, | real variables. |
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2: 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). | |
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3: 5.2 Definitions
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§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(ii) Euler’s Constant
… ►§5.2(iii) Pochhammer’s Symbol
…4: 11.9 Lommel Functions
§11.9 Lommel Functions
… ►The inhomogeneous Bessel differential equation …where , are arbitrary constants, is the Lommel function defined by … ►For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). … ►5: 23.15 Definitions
§23.15 Definitions
… ►Elliptic Modular Function
… ►Klein’s Complete Invariant
… ►Dedekind’s Eta Function (or Dedekind Modular Function)
… ►6: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
… ► … ►§15.2(ii) Analytic Properties
… ►The same properties hold for , except that as a function of , in general has poles at . … ►For example, when , , and , is a polynomial: …7: 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
…8: 5.15 Polygamma Functions
§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). …9: 16.13 Appell Functions
§16.13 Appell Functions
►The following four functions of two real or complex variables and cannot be expressed as a product of two functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): … ►
16.13.2
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16.13.3
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