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scaled spheroidal wave functions

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1: 30.1 Special Notation
β–ΊThe main functions treated in this chapter are the eigenvalues Ξ» n m ⁑ ( Ξ³ 2 ) and the spheroidal wave functions π–―π—Œ n m ⁑ ( x , Ξ³ 2 ) , π–°π—Œ n m ⁑ ( x , Ξ³ 2 ) , 𝑃𝑠 n m ⁑ ( z , Ξ³ 2 ) , 𝑄𝑠 n m ⁑ ( z , Ξ³ 2 ) , and S n m ⁒ ( j ) ⁑ ( z , Ξ³ ) , j = 1 , 2 , 3 , 4 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for π–―π—Œ , π–°π—Œ , 𝑃𝑠 , 𝑄𝑠 , respectively. β–Ί
Other Notations
β–ΊFlammer (1957) and Abramowitz and Stegun (1964) use Ξ» m ⁒ n ⁒ ( Ξ³ ) for Ξ» n m ⁑ ( Ξ³ 2 ) + Ξ³ 2 , R m ⁒ n ( j ) ⁑ ( Ξ³ , z ) for S n m ⁒ ( j ) ⁑ ( z , Ξ³ ) , and …
2: 30.11 Radial Spheroidal Wave Functions
§30.11 Radial Spheroidal Wave Functions
β–Ί
§30.11(i) Definitions
β–Ί
Connection Formulas
β–Ί
§30.11(ii) Graphics
β–Ί
§30.11(iv) Wronskian
3: 30.2 Differential Equations
β–Ί
§30.2(i) Spheroidal Differential Equation
β–ΊIn applications involving prolate spheroidal coordinates Ξ³ 2 is positive, in applications involving oblate spheroidal coordinates Ξ³ 2 is negative; see §§30.13, 30.14. … β–ΊThe Liouville normal form of equation (30.2.1) is … β–Ί
§30.2(iii) Special Cases
4: 15.2 Definitions and Analytical Properties
β–Ί
§15.2(i) Gauss Series
β–ΊExcept where indicated otherwise principal branches of F ⁑ ( a , b ; c ; z ) and 𝐅 ⁑ ( a , b ; c ; z ) are assumed throughout the DLMF. … β–ΊThe 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 . … β–Ί(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
β–Ί
§14.19(i) Introduction
β–Ίwhere the constant c is a scaling factor. … β–ΊWith 𝐅 as in §14.3 and ΞΎ > 0 , … β–Ί
§14.19(v) Whipple’s Formula for Toroidal Functions
6: 9.1 Special Notation
β–Ί(For other notation see Notation for the Special Functions.) β–Ί β–Ίβ–Ί
k nonnegative integer, except in §9.9(iii).
β–ΊThe 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). β–ΊOther 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 ) = Ο€ ⁒ Bi ⁑ ( x ) , V ⁑ ( x ) = Ο€ ⁒ Ai ⁑ ( x ) (Fock (1945)); A ⁑ ( x ) = 3 1 / 3 ⁒ Ο€ ⁒ Ai ⁑ ( 3 1 / 3 ⁒ x ) (SzegΕ‘ (1967, §1.81)); e 0 ⁑ ( x ) = Ο€ ⁒ Hi ⁑ ( x ) , e ~ 0 ⁒ ( x ) = Ο€ ⁒ Gi ⁑ ( x ) (Tumarkin (1959)).
7: 31.1 Special Notation
β–Ί(For other notation see Notation for the Special Functions.) β–Ί β–Ίβ–Ί
x , y real variables.
β–ΊThe main functions treated in this chapter are H ⁒ β„“ ⁑ ( a , q ; Ξ± , Ξ² , Ξ³ , Ξ΄ ; z ) , ( s 1 , s 2 ) ⁒ 𝐻𝑓 m ⁑ ( a , q m ; Ξ± , Ξ² , Ξ³ , Ξ΄ ; z ) , ( s 1 , s 2 ) ⁒ 𝐻𝑓 m Ξ½ ⁑ ( a , q m ; Ξ± , Ξ² , Ξ³ , Ξ΄ ; z ) , and the polynomial 𝐻𝑝 n , m ⁑ ( a , q n , m ; n , Ξ² , Ξ³ , Ξ΄ ; z ) . …Sometimes the parameters are suppressed.
8: 23.15 Definitions
§23.15 Definitions
β–Ί
§23.15(i) General Modular Functions
β–Ί
Elliptic Modular Function
β–Ί
Dedekind’s Eta Function (or Dedekind Modular Function)
β–Ί
9: 5.15 Polygamma Functions
§5.15 Polygamma Functions
β–ΊThe 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. … β–ΊFor B 2 ⁒ k see §24.2(i). …
10: 5.2 Definitions
β–Ί
§5.2(i) Gamma and Psi Functions
β–Ί
Euler’s Integral
β–Ί
5.2.1 Ξ“ ⁑ ( z ) = 0 e t ⁒ t z 1 ⁒ d t , ⁑ z > 0 .
β–ΊIt is a meromorphic function with no zeros, and with simple poles of residue ( 1 ) n / n ! at z = n . … β–Ί
5.2.2 ψ ⁑ ( z ) = Ξ“ ⁑ ( z ) / Ξ“ ⁑ ( z ) , z 0 , 1 , 2 , .