Jacobi%20epsilon%20function
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1: 22.16 Related Functions
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§22.16(i) Jacobi’s Amplitude () Function
… ►§22.16(ii) Jacobi’s Epsilon Function
►Integral Representations
… ►Quasi-Addition and Quasi-Periodic Formulas
… ►§22.16(iii) Jacobi’s Zeta Function
…2: 18.3 Definitions
§18.3 Definitions
►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of for Jacobi polynomials, in powers of for the other cases). … ►Jacobi on Other Intervals
►For a finite system of Jacobi polynomials is orthogonal on with weight function . …3: 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 . … ►For fixed , each is an entire function of with period ; is odd in and the others are even. For fixed , each of , , , and is an analytic function of for , with a natural boundary , and correspondingly, an analytic function of for with a natural boundary . … ►For , the -zeros of , , are , , , respectively.4: 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. … ►Elliptic Modular Function
… ►Dedekind’s Eta Function (or Dedekind Modular Function)
… ►5: 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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complex parameters. |
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: 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). …8: 5.2 Definitions
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§5.2(i) Gamma and Psi Functions
►Euler’s Integral
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5.2.1
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►It is a meromorphic function with no zeros, and with simple poles of residue at .
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5.2.2
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