About the Project

Jacobi%20epsilon%20function

AdvancedHelp

(0.006 seconds)

1—10 of 968 matching pages

1: 22.16 Related Functions
§22.16(i) Jacobi’s Amplitude ( am ) 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 x 1 for Jacobi polynomials, in powers of x for the other cases). …
Jacobi on Other Intervals
For 1 β > α > 1 a finite system of Jacobi polynomials P n ( α , β ) ( x ) is orthogonal on ( 1 , ) with weight function w ( x ) = ( x 1 ) α ( x + 1 ) β . …
3: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
Corresponding expansions for θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , can be found by differentiating (20.2.1)–(20.2.4) with respect to z . … For fixed τ , each θ j ( z | τ ) is an entire function of z with period 2 π ; θ 1 ( z | τ ) is odd in z and the others are even. For fixed z , each of θ 1 ( z | τ ) / sin z , θ 2 ( z | τ ) / cos z , θ 3 ( z | τ ) , and θ 4 ( z | τ ) is an analytic function of τ for τ > 0 , with a natural boundary τ = 0 , and correspondingly, an analytic function of q for | q | < 1 with a natural boundary | q | = 1 . … For m , n , the z -zeros of θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , are ( m + n τ ) π , ( m + 1 2 + n τ ) π , ( m + 1 2 + ( n + 1 2 ) τ ) π , ( m + ( n + 1 2 ) τ ) π respectively.
4: 23.15 Definitions
§23.15 Definitions
In §§23.1523.19, k and k ( ) denote the Jacobi modulus and complementary modulus, respectively, and q = e i π τ ( τ > 0 ) 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
(For other notation see Notation for the Special Functions.)
x , y real variables.
q , α , β , γ , δ , ϵ , ν complex parameters.
The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) 𝐻𝑓 m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) 𝐻𝑓 m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) . …Sometimes the parameters are suppressed.
6: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
k nonnegative integer, except in §9.9(iii).
The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: Ai ( x ) and Bi ( x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 1 / 3 π Ai ( 3 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( x ) , e ~ 0 ( x ) = π Gi ( x ) (Tumarkin (1959)).
7: 5.15 Polygamma Functions
§5.15 Polygamma Functions
The functions ψ ( n ) ( z ) , n = 1 , 2 , , are called the polygamma functions. In particular, ψ ( z ) is the trigamma function; ψ ′′ , ψ ( 3 ) , ψ ( 4 ) are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … For B 2 k see §24.2(i). …
8: 5.2 Definitions
§5.2(i) Gamma and Psi Functions
Euler’s Integral
5.2.1 Γ ( z ) = 0 e t t z 1 d t , z > 0 .
It is a meromorphic function with no zeros, and with simple poles of residue ( 1 ) n / n ! at z = n . …
5.2.2 ψ ( z ) = Γ ( z ) / Γ ( z ) , z 0 , 1 , 2 , .
9: 9.12 Scorer Functions
§9.12 Scorer Functions
where …
§9.12(ii) Graphs
Functions and Derivatives
10: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
§14.19(ii) Hypergeometric Representations
§14.19(iv) Sums
§14.19(v) Whipple’s Formula for Toroidal Functions