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1: 25.13 Periodic Zeta Function
§25.13 Periodic Zeta Function
The notation F ( x , s ) is used for the polylogarithm Li s ( e 2 π i x ) with x real:
25.13.1 F ( x , s ) n = 1 e 2 π i n x n s ,
Also,
25.13.2 F ( x , s ) = Γ ( 1 - s ) ( 2 π ) 1 - s ( e π i ( 1 - s ) / 2 ζ ( 1 - s , x ) + e π i ( s - 1 ) / 2 ζ ( 1 - s , 1 - x ) ) , 0 < x < 1 , s > 1 ,
2: 27.10 Periodic Number-Theoretic Functions
§27.10 Periodic Number-Theoretic Functions
If k is a fixed positive integer, then a number-theoretic function f is periodic (mod k ) if … Every function periodic (mod k ) can be expressed as a finite Fourier series of the form …where g ( m ) is also periodic (mod k ), and is given by … is a periodic function of n ( mod k ) and has the finite Fourier-series expansion …
3: 25.1 Special Notation
k , m , n

nonnegative integers.

B ~ n ( x )

periodic Bernoulli function B n ( x - x ) .

The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
4: 4.28 Definitions and Periodicity
§4.28 Definitions and Periodicity
Periodicity and Zeros
The functions sinh z and cosh z have period 2 π i , and tanh z has period π i . …
5: 4.14 Definitions and Periodicity
§4.14 Definitions and Periodicity
6: 21.8 Abelian Functions
An Abelian function is a 2 g -fold periodic, meromorphic function of g complex variables. …For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
7: 24.2 Definitions and Generating Functions
§24.2(iii) Periodic Bernoulli and Euler Functions
8: 29.19 Physical Applications
Simply-periodic Lamé functions ( ν noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
9: 28.30 Expansions in Series of Eigenfunctions
Then every continuous 2 π -periodic function f ( x ) whose second derivative is square-integrable over the interval [ 0 , 2 π ] can be expanded in a uniformly and absolutely convergent series …
10: 22.4 Periods, Poles, and Zeros
§22.4(i) Distribution
For each Jacobian function, Table 22.4.1 gives its periods in the z -plane in the left column, and the position of one of its poles in the second row. … Table 22.4.2 displays the periods and zeros of the functions in the z -plane in a similar manner to Table 22.4.1. …
Figure 22.4.1: z -plane. …
§22.4(iii) Translation by Half or Quarter Periods