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21: 12.5 Integral Representations
§12.5 Integral Representations
§12.5(i) Integrals Along the Real Line
§12.5(ii) Contour Integrals
§12.5(iii) Mellin–Barnes Integrals
§12.5(iv) Compendia
22: 5.9 Integral Representations
§5.9 Integral Representations
Binet’s Formula
§5.9(ii) Psi Function, Euler’s Constant, and Derivatives
23: 9.5 Integral Representations
§9.5 Integral Representations
§9.5(i) Real Variable
§9.5(ii) Complex Variable
24: 33.7 Integral Representations
§33.7 Integral Representations
25: 14.12 Integral Representations
§14.12 Integral Representations
§14.12(i) - 1 < x < 1
14.12.3 Q ν μ ( cos θ ) = π 1 / 2 Γ ( ν + μ + 1 ) ( sin θ ) μ 2 μ + 1 Γ ( μ + 1 2 ) Γ ( ν - μ + 1 ) ( 0 ( sinh t ) 2 μ ( cos θ + i sin θ cosh t ) ν + μ + 1 d t + 0 ( sinh t ) 2 μ ( cos θ - i sin θ cosh t ) ν + μ + 1 d t ) , 0 < θ < π , μ > - 1 2 , ν ± μ > - 1 .
§14.12(ii) 1 < x <
For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.
26: 1.17 Integral and Series Representations of the Dirac Delta
§1.17(ii) Integral Representations
Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation
Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))
Coulomb Functions (§33.14(iv))
Airy Functions (§9.2)
27: 15.6 Integral Representations
§15.6 Integral Representations
The function F ( a , b ; c ; z ) (not F ( a , b ; c ; z ) ) has the following integral representations: …
See accompanying text
Figure 15.6.1: t -plane. … Magnify
28: 10.32 Integral Representations
§10.32 Integral Representations
§10.32(i) Integrals along the Real Line
§10.32(ii) Contour Integrals
§10.32(iii) Products
§10.32(iv) Compendia
29: 35.9 Applications
For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). …
30: 9.11 Products
§9.11(iii) Integral Representations
For an integral representation of the Dirac delta involving a product of two Ai functions see §1.17(ii). For further integral representations see Reid (1995, 1997a, 1997b). …