About the Cover: Meromorphic Functions with Doubly Periodic Phase
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About the Cover: Meromorphic Functions with Doubly Periodic Phase Gunter Semmler1 · Elias Wegert1
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract The phase plot of the function depicted on the cover of this volume is doubly periodic. In this expository paper, we discuss a canonical representation of all functions with doubly periodic phase (argument) in terms of the Weierstrass σ function. In particular, we point out that the zeros and poles of such a function in a fundamental domain can be prescribed arbitrarily, with the only restriction that their total numbers (counting multiplicities) must coincide. Keywords Meromorphic function · Phase plot · Elliptic function · Weierstrass σ -function Mathematics Subject Classification 30D30 · 30A99
1 Introduction Inspired by the visualization of complex functions in terms of their phase portraits (see [8,9]), we became interested in the class of meromorphic functions f in the complex plane having doubly periodic phase P f := f /| f |. A hasty conjecture about the general form of such phase-elliptic functions made in our paper [9] was proven to be wrong by Stefan [6], who also obtained the correct representation of these functions in terms
Communicated by Mario Bonk.
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Elias Wegert [email protected] Gunter Semmler [email protected]
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Institute of Applied Analysis, TU Bergakademie Freiberg, 09596 Freiberg, Germany
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G. Semmler, E. Wegert
of the Weierstrass σ -function. As pointed out by a referee of the paper at hand, phaseelliptic functions form a subclass of the so-called elliptic functions of the second kind, which were introduced and studied by Charles Hermite as early as in the 1870s ( [2], [3, Vol. 3, p. 266 ff], see also Klein and Sommerfeld [4, p. 415, p. 422]). A comprehensive description of this class, as well as of the even more general class of elliptic functions of the third kind was given by Robert Fricke [1, pp. 217–222]. Though the characterization of phase-elliptic functions could be derived directly from these classical investigations, we basically follow Stefan’s elementary approach, aiming at a more lucid and explicit form of the results. Using the transformation rule (1) of the Weierstrass σ -function as the only prerequisite beyond a standard course in complex analysis, we shall prove that a phase-elliptic function is uniquely determined (up to a constant factor) by its zeros a1 , . . . , an and poles b1 , . . . , bn in a fundamental parallelogram (repeated according to their multiplicity), as well as two integers n 1 , n 2 which are related to an exponential factor. Moreover, the zeros, the poles, and the two integers can be prescribed arbitrarily, with the only required restriction that the numbers of zeros and poles must be equal.
2 Elliptic Functions To compare the results, we first consider elliptic functions with pairs of fundamental periods ω1 , ω2 ∈ C which are linearly independent over R. Denoting :={n 1 ω1 + n 2 ω2 : n 1 , n 2 ∈ Z} the lattice generated by ω1 and ω2 , the Weierstrass σ -
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