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with imaginary periods

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11: 28.12 Definitions and Basic Properties
28.12.6 me ν ( z + π , q ) = e π i ν me ν ( z , q ) ,
12: About Color Map
For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of π / 2 correponding to the real and imaginary axes, at more immediately recognizable colors. …
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Figure 3: Continuous phase mapping
13: 29.12 Definitions
Table 29.12.1: Lamé polynomials.
ν
eigenvalue
h
eigenfunction
w ( z )
polynomial
form
real
period
imag.
period
parity of
w ( z )
parity of
w ( z K )
parity of
w ( z K i K )
14: 21.3 Symmetry and Quasi-Periodicity
21.3.3 θ ( 𝐳 + 𝐦 1 + 𝛀 𝐦 2 | 𝛀 ) = e 2 π i ( 1 2 𝐦 2 𝛀 𝐦 2 + 𝐦 2 𝐳 ) θ ( 𝐳 | 𝛀 ) ,
15: 25.1 Special Notation
k , m , n nonnegative integers.
s = σ + i t complex variable.
z = x + i y complex variable.
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 ) .
16: 28.29 Definitions and Basic Properties
π is the minimum period of Q . … where the function P ν ( z ) is π -periodic. … The solutions of period π or 2 π are exceptional in the following sense. If (28.29.1) has a periodic solution with minimum period n π , n = 3 , 4 , , then all solutions are periodic with period n π . …
17: 20.2 Definitions and Periodic Properties
§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
18: 29.17 Other Solutions
29.17.1 F ( z ) = E ( z ) i K z d u ( E ( u ) ) 2 .
They are algebraic functions of sn ( z , k ) , cn ( z , k ) , and dn ( z , k ) , and have primitive period 8 K . … Lamé–Wangerin functions are solutions of (29.2.1) with the property that ( sn ( z , k ) ) 1 / 2 w ( z ) is bounded on the line segment from i K to 2 K + i K . …
19: 28.33 Physical Applications
  • Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

  • with W ( x , y , t ) = e i ω t V ( x , y ) , reduces to (28.32.2) with k 2 = ω 2 ρ / τ . …The separated solutions V n ( ξ , η ) must be 2 π -periodic in η , and have the form … If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as cos ( 2 ω t ) . … For points ( q , a ) that are at intersections of with the characteristic curves a = a n ( q ) or a = b n ( q ) , a periodic solution is possible. …
    20: 1.17 Integral and Series Representations of the Dirac Delta
    1.17.9 ( 1 2 π e i ( x a ) t d t ) ϕ ( x ) d x = ϕ ( a ) .
    1.17.12 δ ( x a ) = 1 2 π e i ( x a ) t d t .
    The sum k = e i k ( x a ) does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that …provided that ϕ ( x ) is continuous and of period 2 π ; see §1.8(ii). …
    1.17.21 δ ( x a ) = 1 2 π k = e i k ( x a ) .