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translation by half-periods

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1: 20.2 Definitions and Periodic Properties
§20.2(iii) Translation of the Argument by Half-Periods
2: 22.4 Periods, Poles, and Zeros
§22.4 Periods, Poles, and Zeros
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. …(b) The difference between p and the nearest q is a half-period of p q ( z , k ) . This half-period will be plus or minus a member of the triple K , i K , K + i K ; the other two members of this triple are quarter periods of p q ( z , k ) .
§22.4(iii) Translation by Half or Quarter Periods
3: 21.2 Definitions
It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: …
21.2.7 θ [ 0 0 ] ( z | Ω ) = θ ( z | Ω ) .
Characteristics whose elements are either 0 or 1 2 are called half-period characteristics. For given Ω , there are 2 2 g g -dimensional Riemann theta functions with half-period characteristics. …