general%0Aelliptic functions
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1: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
… ► … ►Polynomials
… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►§16.2(v) Behavior with Respect to Parameters
…2: 1.16 Distributions
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is called a distribution, or generalized function, if it is a continuous linear functional on , that is, it is a linear functional and for every in ,
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►More generally, for a nondecreasing function the corresponding Lebesgue–Stieltjes measure (see §1.4(v)) can be considered as a distribution:
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►More generally, if is an infinitely differentiable function, then
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►For ,
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►Friedman (1990) gives an overview of generalized functions and their relation to distributions.
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3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
►§35.8(i) Definition
… ►Convergence Properties
… ►§35.8(iv) General Properties
… ►Confluence
…4: 8.19 Generalized Exponential Integral
§8.19 Generalized Exponential Integral
►§8.19(i) Definition and Integral Representations
… ►§8.19(ii) Graphics
… ►§8.19(vi) Relation to Confluent Hypergeometric Function
… ►§8.19(xi) Further Generalizations
…5: 8.21 Generalized Sine and Cosine Integrals
§8.21 Generalized Sine and Cosine Integrals
… ►§8.21(iii) Integral Representations
… ►Spherical-Bessel-Function Expansions
… ►§8.21(vii) Auxiliary Functions
… ►§8.21(viii) Asymptotic Expansions
…6: 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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7: 23.15 Definitions
§23.15 Definitions
►§23.15(i) General Modular Functions
… ►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. …8: 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 , …9: 14.20 Conical (or Mehler) Functions
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►Lastly, for the range , is a real-valued solution of (14.20.1); in terms of (which are complex-valued in general):
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§14.20(vi) Generalized Mehler–Fock Transformation
… ►§14.20(viii) Asymptotic Approximations: Large ,
… ►§14.20(ix) Asymptotic Approximations: Large ,
…10: 15.2 Definitions and Analytical Properties
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