Kelvin-function analogs

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1: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

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2: 10.61 Definitions and Basic Properties

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§10.61(v) Orders $\pm \frac{1}{2}$

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3: 28.27 Addition Theorems

… ►They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

4: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices. … ►The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.

5: 17.17 Physical Applications

… ►They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. …

6: 25.17 Physical Applications

… ►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. …

7: 20.11 Generalizations and Analogs

§20.11 Generalizations and Analogs

… ►It is a discrete analog of theta functions. If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): …This is the discrete analog of the Poisson identity (§1.8(iv)). …

8: 5.16 Sums

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).

9: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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$q$-Analog of Bailey’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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$q$-Analog of Gauss’s ${}_2{}^{}F_1^{}\left(-1\right)$ Sum

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F. H. Jackson’s Terminating $q$-Analog of Dixon’s Sum

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$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

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Gasper–Rahman $q$-Analogs of the Karlsson–Minton Sums

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10: 7.22 Methods of Computation

… ►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. …