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1: 25.7 Integrals
For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).
2: 19.13 Integrals of Elliptic Integrals
For definite and indefinite integrals of complete elliptic integrals see Byrd and Friedman (1971, pp. 610–612, 615), Prudnikov et al. (1990, §§1.11, 2.16), Glasser (1976), Bushell (1987), and Cvijović and Klinowski (1999). For definite and indefinite integrals of incomplete elliptic integrals see Byrd and Friedman (1971, pp. 613, 616), Prudnikov et al. (1990, §§1.10.2, 2.15.2), and Cvijović and Klinowski (1994). …
3: 10.71 Integrals
§10.71(ii) Definite Integrals
4: 14.17 Integrals
§14.17(ii) Barnes’ Integral
§14.17(iii) Orthogonality Properties
§14.17(iv) Definite Integrals of Products
5: 4.40 Integrals
§4.40(iii) Definite Integrals
Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).
6: 35.1 Special Notation
a , b

complex variables.

Ω

space of positive-definite real symmetric matrices.

X > T

X - T is positive definite. Similarly, T < X is equivalent.

7: 4.26 Integrals
§4.26(iii) Definite Integrals
Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2000, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).
8: 9.11 Products
§9.11(iii) Integral Representations
§9.11(v) Definite Integrals
For further definite integrals see Prudnikov et al. (1990, §1.8.2), Laurenzi (1993), Reid (1995, 1997a, 1997b), and Vallée and Soares (2010, Chapters 3, 4).
9: 35.5 Bessel Functions of Matrix Argument
35.5.3 B ν ( T ) = Ω etr ( - ( T X + X - 1 ) ) | X | ν - 1 2 ( m + 1 ) d X , ν , T Ω .
35.5.5 0 < X < T A ν 1 ( S 1 X ) | X | ν 1 A ν 2 ( S 2 ( T - X ) ) | T - X | ν 2 d X = | T | ν 1 + ν 2 + 1 2 ( m + 1 ) A ν 1 + ν 2 + 1 2 ( m + 1 ) ( ( S 1 + S 2 ) T ) , ν j , ( ν j ) > - 1 , j = 1 , 2 ; S 1 , S 2 𝒮 ; T Ω .
10: 35.3 Multivariate Gamma and Beta Functions
35.3.2 Γ m ( s 1 , , s m ) = Ω etr ( - X ) | X | s m - 1 2 ( m + 1 ) j = 1 m - 1 | ( X ) j | s j - s j + 1 d X , s j , ( s j ) > 1 2 ( j - 1 ) , j = 1 , , m .