4,136 research outputs found
Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations
Students of elementary complex analysis usually begin by seeing the derivation of the Cauchy--Riemann equations. A topic of interest to both the development of the theory and its applications is the reconstruction of a holomorphic function from its real part, or the extraction of the imaginary part from the real part, or vice versa. Usually this takes place by solving the partial differential system embodied by the Cauchy-Riemann equations. Here I show in general how this may be accomplished by purely algebraic means. Several examples are given, for functions with increasing levels of complexity. The development of these ideas within the Mathematica software system is also presented. This approach could easily serve as an alternative in the early development of complex variable theory
Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains
We study the integrable structure of the Dirichlet boundary problem in two
dimensions and extend the approach to the case of planar multiply-connected
domains. The solution to the Dirichlet boundary problem in multiply-connected
case is given through a quasiclassical tau-function, which generalizes the
tau-function of the dispersionless Toda hierarchy. It is shown to obey an
infinite hierarchy of Hirota-like equations which directly follow from
properties of the Dirichlet Green function and from the Fay identities. The
relation to multi-support solutions of matrix models is briefly discussed.Comment: 41 pages, 5 figures, LaTeX; some revision of exposition, misprints
corrected, the version to appear in Commun. Math. Phy
Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem
A new approach to the analytic theory of difference equations with rational
and elliptic coefficients is proposed. It is based on the construction of
canonical meromorphic solutions which are analytical along "thick paths". The
concept of such solutions leads to a notion of local monodromies of difference
equations. It is shown that in the continuous limit they converge to the
monodromy matrices of differential equations. New type of isomonodromic
deformations of difference equations with elliptic coefficients changing the
periods of elliptic curves is constructed.Comment: 38 pages, no figures; typos remove
Zeros of weakly holomorphic modular forms of levels 2 and 3
Let be the space of weakly holomorphic modular forms for
that are holomorphic at all cusps except possibly at . We
study a canonical basis for and and prove that
almost all modular forms in this basis have the property that the majority of
their zeros in a fundamental domain lie on a lower boundary arc of the
fundamental domain.Comment: Added a reference, corrected typo
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