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1: 8.21 Generalized Sine and Cosine Integrals
Furthermore, si ( a , z ) and ci ( a , z ) are entire functions of a , and Si ( a , z ) and Ci ( a , z ) are meromorphic functions of a with simple poles at a = 1 , 3 , 5 , and a = 0 , 2 , 4 , , respectively. … When ph z = 0 (and when a 1 , 3 , 5 , , in the case of Si ( a , z ) , or a 0 , 2 , 4 , , in the case of Ci ( a , z ) ) the principal values of si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector | ph z | π the principal values are defined by analytic continuation from ph z = 0 ; compare §4.2(i). … For the functions on the right-hand sides of (8.21.10) and (8.21.11) see §6.2(ii). … When | ph z | < 1 2 π , …
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3: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . …To compute a particular value p ( n ) it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
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6: 27.17 Other Applications
Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. …Apostol and Zuckerman (1951) uses congruences to construct magic squares. … Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.
7: 21 Multidimensional Theta Functions
8: 31 Heun Functions
9: 5.21 Methods of Computation
Or we can use forward recurrence, with an initial value obtained e. …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …
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