Weierstrass%0Aelliptic%20functions
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1: 23.2 Definitions and Periodic Properties
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§23.2(i) Lattices
… ► … ►§23.2(ii) Weierstrass Elliptic Functions
… ► ►§23.2(iii) Periodicity
…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: 23.15 Definitions
§23.15 Definitions
… ►In §§23.15–23.19, and denote the Jacobi modulus and complementary modulus, respectively, and () denotes the nome; compare §§20.1 and 22.1. … ►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, …If, as a function of , is analytic at , then is called a modular form. If, in addition, as , then is called a cusp form. …4: 5.12 Beta Function
§5.12 Beta Function
… ►In (5.12.1)–(5.12.4) it is assumed and . … ►In (5.12.8) the fractional powers have their principal values when and , and are continued via continuity. … ►When …where the contour starts from an arbitrary point in the interval , circles and then in the positive sense, circles and then in the negative sense, and returns to . …5: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
… ►When , is defined by analytic continuation. It is a meromorphic function with no zeros, and with simple poles of residue at . … ►6: 14.19 Toroidal (or Ring) Functions
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§14.19(i) Introduction
►When , , , and solutions of (14.2.2) are known as toroidal or ring functions. … ►With , … ►With , … ►With , …7: 15.2 Definitions and Analytical Properties
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(a)
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►As a multivalued function of , is analytic everywhere except for possible branch points at , , and .
The same properties hold for , except that as a function of , in general has poles at .
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►For example, when , , and , is a polynomial:
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§15.2(i) Gauss Series
… ►Converges absolutely when .
8: 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 . … ► … ►Functions and Derivatives
…9: 14.20 Conical (or Mehler) Functions
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►For and ,
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►uniformly for , where and are the modified Bessel functions (§10.25(ii)) and is an arbitrary constant such that .
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