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11: 27.10 Periodic Number-Theoretic Functions
§27.10 Periodic Number-Theoretic Functions
►If is a fixed positive integer, then a number-theoretic function is periodic (mod ) if … ►Every function periodic (mod ) can be expressed as a finite Fourier series of the form …where is also periodic (mod ), and is given by … ►is a periodic function of and has the finite Fourier-series expansion …12: 22.4 Periods, Poles, and Zeros
§22.4 Periods, Poles, and Zeros
►§22.4(i) Distribution
… ►Table 22.4.2 displays the periods and zeros of the functions in the -plane in a similar manner to Table 22.4.1. … ►Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. … ►§22.4(iii) Translation by Half or Quarter Periods
…13: 4.14 Definitions and Periodicity
§4.14 Definitions and Periodicity
…14: 20.2 Definitions and Periodic Properties
§20.2 Definitions and Periodic Properties
… ►§20.2(ii) Periodicity and Quasi-Periodicity
►For fixed , each is an entire function of with period ; is odd in and the others are even. … ►The theta functions are quasi-periodic on the lattice: … ►§20.2(iii) Translation of the Argument by Half-Periods
…15: 23.7 Quarter Periods
§23.7 Quarter Periods
…16: 28.29 Definitions and Basic Properties
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►
is the minimum period of .
…
►where the function is -periodic.
…
►The solutions of period
or are exceptional in the following sense.
If (28.29.1) has a periodic solution with minimum period
, , then all solutions are periodic with period
.
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17: 28.30 Expansions in Series of Eigenfunctions
…
►Let , , be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let , , be the eigenfunctions, that is, an orthonormal set of -periodic solutions; thus
…Then every continuous -periodic function whose second derivative is square-integrable over the interval can be expanded in a uniformly and absolutely convergent series
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18: Diego Dominici
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►He was elected as Program Director for the period 2011–2016 and served as OPSF-Talk moderator from 2010–2022 with Bonita Saunders, and co-editor for OPSF-Net from 2006–2015 with Martin Muldoon.
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19: 25.2 Definition and Expansions
20: 22.19 Physical Applications
…
►The period is .
…
►Figure 22.19.1 shows the nature of the solutions of (22.19.3) by graphing for both , as in Figure 22.16.1, and , where it is periodic.
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►As from below the period diverges since are points of unstable equilibrium.
…
►Such oscillations, of period
, with modulus are given by:
…As from below the period diverges since is a point of unstable equlilibrium.
…