integrals of Bessel and Hankel functions

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1: 10.76 Approximations

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§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions

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2: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

3: 10.74 Methods of Computation

… ►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). …

4: 10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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Schläfli–Sommerfeld Integrals

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$H_{\nu }^{(2)}\left(z\right)=-{\displaystyle \frac{1}{\pi \mathrm{i}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }-\pi \mathrm{i}}}{\mathrm{e}}^{z\sinh t-\nu t}dt.$

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§10.9(iv) Compendia

►For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

5: 10.77 Software

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§10.77(ii) Bessel Functions–Real Argument and Integer or Half-Integer Order (including Spherical Bessel Functions)

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§10.77(iii) Bessel Functions–Real Order and Argument

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§10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)

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§10.77(ix) Integrals of Bessel Functions

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§10.77(x) Zeros of Bessel Functions

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6: 10.22 Integrals

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Products

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Trigonometric Arguments

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Convolutions

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Fractional Integral

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§10.22(v) Hankel Transform

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7: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►In this way there is less usage of many-valued functions. … ► ►

§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

8: 9.17 Methods of Computation

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§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). … ►

§9.17(iv) Via Bessel Functions

►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to $\mathrm{Ai}\left(z\right)$, $\mathrm{Bi}\left(z\right)$, and their derivatives. …

9: 10.23 Sums

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§10.23(i) Multiplication Theorem

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§10.23(ii) Addition Theorems

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Fourier–Bessel Expansion

… ►For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).

10: 10.43 Integrals

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§10.43(i) Indefinite Integrals

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§10.43(iii) Fractional Integrals

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§10.43(v) Kontorovich–Lebedev Transform

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