Doubly periodic square array of SRRs are used to illustrate the methodology.
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This function will then be necessarily doubly periodic.
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The totality of meromorphic doubly periodic functions with given periods defines an algebraic function field associated to that curve.
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Stagnation zones theorems are closely related to pre-Liouville theorem about conversion of the entire doubly periodic function into the identical constant.
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The doubly periodic function is thus a two-dimensional extension of the simpler singly periodic function, which repeats itself in a single dimension.
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Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic.
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As an arbitrary mapping from pairs of reals ( or complex numbers ) to reals, a doubly periodic function can be constructed with little effort.
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If a doubly periodic function is also a poles in other words, a meromorphic function then a lot of information about such a function can be obtained by applying some basic theorems from complex analysis.
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Certain elliptic functions ( doubly periodic functions ) denoted cm and sm satisfying the identity cm ( " z " ) 3 + sm ( " z " ) 3 = 1 are known as Dixon's elliptic functions.
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The Weierstrass theory also includes the Weierstrass zeta function, which is an indefinite integral of & # 8472; and not doubly periodic, and a theta function called the Weierstrass sigma function, of which his zeta-function is the log-derivative.
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