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1: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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27.2.3
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27.2.4
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27.2.14
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2: 25.12 Polylogarithms
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►Other notations and names for include (Kölbig et al. (1970)), Spence function (’t Hooft and Veltman (1979)), and (Maximon (2003)).
►In the complex plane has a branch point at .
The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i).
The remainder of the equations in this subsection apply to principal branches.
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►For other values of , is defined by analytic continuation.
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3: 10.75 Tables
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Kerimov and Skorokhodov (1985a) tabulates 5 (nonreal) complex conjugate pairs of zeros of the principal branches of and for , 8D.
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Parnes (1972) tabulates all zeros of the principal value of , for , 9D.
Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of , for , 29S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
4: 20 Theta Functions
Chapter 20 Theta Functions
…5: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…6: 6.16 Mathematical Applications
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►Hence if and , then the limiting value of overshoots by approximately 18%.
Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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6.16.5
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7: Bibliography C
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The hypergeometric function and the -function near their branch points.
Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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Zeros of the Hankel function of real order out of the principal Riemann sheet.
J. Comput. Appl. Math. 37 (1-3), pp. 89–99.
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8: 25.6 Integer Arguments
9: 32.8 Rational Solutions
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– possess hierarchies of rational solutions for special values of the parameters which are generated from “seed solutions” using the Bäcklund transformations and often can be expressed in the form of determinants.
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32.8.3
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32.8.5
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32.8.9
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