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21: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
§4.23(i) General Definitions
Other Inverse Functions
§4.23(viii) Gudermannian Function
The inverse Gudermannian function is given by …
22: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
§8.17(ii) Hypergeometric Representations
§8.17(iii) Integral Representation
§8.17(iv) Recurrence Relations
§8.17(vi) Sums
23: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
24: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. …
Polynomials
§16.2(v) Behavior with Respect to Parameters
25: 12.14 The Function W ( a , x )
§12.14 The Function W ( a , x )
§12.14(vii) Relations to Other Functions
Bessel Functions
Confluent Hypergeometric Functions
§12.14(x) Modulus and Phase Functions
26: 30.1 Special Notation
(For other notation see Notation for the Special Functions.) … 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
27: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
§25.11(i) Definition
The Riemann zeta function is a special case: …
§25.11(ii) Graphics
§25.11(vi) Derivatives
28: 14.1 Special Notation
§14.1 Special Notation
(For other notation see Notation for the Special Functions.) … Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. The main functions treated in this chapter are the Legendre functions P ν ( x ) , Q ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions P ν μ ( x ) , Q ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , Q ν μ ( z ) ; conical functions P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) , Q ^ - 1 2 + i τ μ ( x ) , P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) (also known as Mehler functions). …
29: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
Each of these inverse functions is multivalued. The principal values satisfy …
30: 35.1 Special Notation
(For other notation see Notation for the Special Functions.) …
a , b

complex variables.

The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( T ) and (of the second kind) B ν ( T ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; T ) or F 1 1 ( a b ; T ) and (of the second kind) Ψ ( a ; b ; T ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; T ) or F 1 2 ( a 1 , a 2 b ; T ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; T ) or F q p ( a 1 , , a p b 1 , , b q ; T ) . An alternative notation for the multivariate gamma function is Π m ( a ) = Γ m ( a + 1 2 ( m + 1 ) ) (Herz (1955, p. 480)). Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( T ) = A ν ( T ) / A ν ( 0 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | S , T ) = | T | ν B ν ( S T ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( T ) = | T | ν B ν ( S T ) (Faraut and Korányi (1994, pp. 357–358)).