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scaled spheroidal wave functions

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1: 30.1 Special Notation
The main functions treated in this chapter are the eigenvalues λ n m ( γ 2 ) and the spheroidal wave functions Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) , Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) , and S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for Ps , Qs , Ps , Qs , respectively.
Other Notations
Flammer (1957) and Abramowitz and Stegun (1964) use λ m n ( γ ) for λ n m ( γ 2 ) + γ 2 , R m n ( j ) ( γ , z ) for S n m ( j ) ( z , γ ) , and …
2: 30.11 Radial Spheroidal Wave Functions
§30.11 Radial Spheroidal Wave Functions
§30.11(i) Definitions
Connection Formulas
§30.11(ii) Graphics
§30.11(iv) Wronskian
3: 30.2 Differential Equations
§30.2(i) Spheroidal Differential Equation
In applications involving prolate spheroidal coordinates γ 2 is positive, in applications involving oblate spheroidal coordinates γ 2 is negative; see §§30.13, 30.14. … The Liouville normal form of equation (30.2.1) is …
§30.2(iii) Special Cases
4: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
Except where indicated otherwise principal branches of F ( a , b ; c ; z ) and F ( a , b ; c ; z ) are assumed throughout the DLMF. … The principal branch of F ( a , b ; c ; z ) is an entire function of a , b , and c . …As a multivalued function of z , F ( a , b ; c ; z ) is analytic everywhere except for possible branch points at z = 0 , 1 , and . … (Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does F ( a , b ; c ; z ) , which is analytic at c = 0 , - 1 , - 2 , .) …
5: 14.19 Toroidal (or Ring) Functions
§14.19 Toroidal (or Ring) Functions
§14.19(i) Introduction
where the constant c is a scaling factor. … With F as in §14.3 and ξ > 0 , …
§14.19(v) Whipple’s Formula for Toroidal Functions
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: 31.1 Special Notation
(For other notation see Notation for the Special Functions.)
x , y

real variables.

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.
8: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
Elliptic Modular Function
Dedekind’s Eta Function (or Dedekind Modular Function)
9: 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). …
10: 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 , .