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11: 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
12: 29.10 Lamé Functions with Imaginary Periods
§29.10 Lamé Functions with Imaginary Periods
The first and the fourth functions have period 2 i K ; the second and the third have period 4 i K . …
13: 20.2 Definitions and Periodic Properties
§20.2(ii) Periodicity and Quasi-Periodicity
For fixed τ , each θ j ( z | τ ) is an entire function of z with period 2 π ; θ 1 ( z | τ ) is odd in z and the others are even. … The theta functions are quasi-periodic on the lattice: …
§20.2(iii) Translation of the Argument by Half-Periods
14: 23.7 Quarter Periods
§23.7 Quarter Periods
15: 28.11 Expansions in Series of Mathieu Functions
Let f ( z ) be a 2 π -periodic function that is analytic in an open doubly-infinite strip S that contains the real axis, and q be a normal value (§28.7). …
16: 28.12 Definitions and Basic Properties
28.12.6 me ν ( z + π , q ) = e π i ν me ν ( z , q ) ,
17: 23.2 Definitions and Periodic Properties
§23.2(iii) Periodicity
Hence ( z ) is an elliptic function, that is, ( z ) is meromorphic and periodic on a lattice; equivalently, ( z ) is meromorphic and has two periods whose ratio is not real. … The function ζ ( z ) is quasi-periodic: for j = 1 , 2 , 3 , … For further quasi-periodic properties of the σ -function see Lawden (1989, §6.2).
18: 22.2 Definitions
As a function of z , with fixed k , each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … …
19: 29.3 Definitions and Basic Properties
Table 29.3.1: Eigenvalues of Lamé’s equation.
eigenvalue h parity period
They are called Lamé functions with real periods and of order ν , or more simply, Lamé functions. …
Table 29.3.2: Lamé functions.
boundary conditions
eigenvalue
h
eigenfunction
w ( z )
parity of
w ( z )
parity of
w ( z K )
period of
w ( z )
20: 4.2 Definitions
§4.2(iii) The Exponential Function