31,800 research outputs found
Complex Analysis of Real Functions IV: Non-Integrable Real Functions
In the context of the complex-analytic structure within the unit disk
centered at the origin of the complex plane, that was presented in a previous
paper, we show that a certain class of non-integrable real functions can be
represented within that same structure. In previous papers it was shown that
essentially all integrable real functions, as well as all singular Schwartz
distributions, can be represented within that same complex-analytic structure.
The large class of non-integrable real functions which we analyze here can
therefore be represented side by side with those other real objects, thus
allowing all these objects to be treated in a unified way.Comment: 21 pgs. Small formatting corrections and bibliography updat
Complex Analysis of Real Functions III: Extended Fourier Theory
In the context of the complex-analytic structure within the unit disk
centered at the origin of the complex plane, that was presented in a previous
paper, we show that the complete Fourier theory of integrable real functions is
contained within that structure, that is, within the structure of the space of
inner analytic functions on the open unit disk. We then extend the Fourier
theory beyond the realm of integrable real functions, to include for example
singular Schwartz distributions, and possibly other objects.Comment: 23 pgs. Small formatting corrections and bibliography updat
Complex Analysis of Real Functions VII: A Simple Extension of the Cauchy-Goursat Theorem
In the context of the complex-analytic structure within the open unit disk,
that was established in a previous paper, here we establish a simple
generalization of the Cauchy-Goursat theorem of complex analytic functions. We
do this first for the case of inner analytic functions, and then generalize the
result to all analytic functions. We thus show that the Cauchy-Goursat theorem
holds even if the complex function has isolated singularities located on the
integration contour, so long as these are all integrable ones.Comment: 17 pages, 6 figures. Small formatting corrections and bibliography
updat
Complex Analysis of Real Functions V: The Dirichlet Problem on the Plane
In the context of the correspondence between real functions on the unit
circle and inner analytic functions within the open unit disk, that was
presented in previous papers, we show that the constructions used to establish
that correspondence lead to very general proofs of existence of solutions of
the Dirichlet problem on the plane. At first, this establishes the existence of
solutions for almost arbitrary integrable real functions on the unit circle,
including functions which are discontinuous and unbounded. The proof of
existence is then generalized to a large class of non-integrable real functions
on the unit circle. Further, the proof of existence is generalized to real
functions on a large class of other boundaries on the plane, by means of
conformal transformations.Comment: 26 pgs. Small formatting corrections and bibliography updat
Fourier Theory on the Complex Plane I: Conjugate Pairs of Fourier Series and Inner Analytic Functions
A correspondence between arbitrary Fourier series and certain analytic
functions on the unit disk of the complex plane is established. The expression
of the Fourier coefficients is derived from the structure of complex analysis.
The orthogonality and completeness relations of the Fourier basis are derived
in the same way. It is shown that the limiting function of any Fourier series
is also the limit to the unit circle of an analytic function in the open unit
disk. An alternative way to recover the original real functions from the
Fourier coefficients, which works even when the Fourier series are divergent,
is thus presented. The convergence issues are discussed up to a certain point.
Other possible uses of the correspondence established are pointed out.Comment: 44 pages, including 19 pages of appendices with explicit calculations
and examples, 2 figures; fixed a few typos and made a few improvements;
updated cross-references; made a few further improvements in the tex
Fourier Theory on the Complex Plane IV: Representability of Real Functions by their Fourier Coefficients
The results presented in this paper are refinements of some results presented
in a previous paper. Three such refined results are presented. The first one
relaxes one of the basic hypotheses assumed in the previous paper, and thus
extends the results obtained there to a wider class of real functions. The
other two relate to a closer examination of the issue of the representability
of real functions by their Fourier coefficients. As was shown in the previous
paper, in many cases one can recover the real function from its Fourier
coefficients even if the corresponding Fourier series diverges almost
everywhere. In such cases we say that the real function is still representable
by its Fourier coefficients. Here we establish a very weak condition on the
Fourier coefficients that ensures the representability of the function by those
coefficients. In addition to this, we show that any real function that is
absolutely integrable can be recovered almost everywhere from, and hence is
representable by, its Fourier coefficients, regardless of whether or not its
Fourier series converges. Interestingly, this also provides proof for a
conjecture proposed in the previous paper.Comment: 13 pages, including 3 pages of appendices; there was some expansion
of the content in this version; a few improvements in the text and on some
equations were also made; improved the treatment of the concept of
integration in the tex
Hadronic Decays
A review of current experimental results on exclusive hadronic decays of
bottom mesons to a single or double charmed final state is presented. We
concentrate on branching fraction measurements conducted at colliders
at the and at the resonance. The experimental results
reported are then used in tests of theoretical model predictions, the
determination of the QCD parameters and and tests of
factorizationComment: 17 pages, 8 figures, To be published in Proceedings of the IInd
International Conference on B Physics and CP Violations Honlulu, Hawaii,
March 199
Complex Analysis of Real Functions II: Singular Schwartz Distributions
In the context of the complex-analytic structure within the unit disk
centered at the origin of the complex plane, that was presented in a previous
paper, we show that singular Schwartz distributions can be represented within
that same structure, so long as one defines the limits involved in an
appropriate way. In that previous paper it was shown that essentially all
integrable real functions can be represented within the complex-analytic
structure. The infinite collection of singular objects which we analyze here
can thus be represented side by side with those real functions, thus allowing
all these objects to be treated in a unified way.Comment: 23 pgs. Small formatting corrections and bibliography updat
Factorization and Color-Suppression in Hadronic B->D^(*)npi Decays
The factorization hypothesis and color-suppression are investigated by
analyzing the largest, to date, sample of B mesons. In all, 20 hadronic
two-body decay modes are reconstructed using 2.04 fb^-1 of data collected with
the CLEOII detector. We measure the branching fraction of five class I and five
class III decay modes and set upper limits on branching fractions of ten class
II decays. The branching fraction measurements are used to determine the BSW
parameters a_1, and a_2/a_1. In addition, we measure the fraction of B-> D^{*+}
rho^- decays which decay longitudinally polarized. The results are found to be
consistent with factorization and color-suppression.Comment: Tex, 5 pages, 2 postscript figures plus additional tex macros
submitted as a gziped tar file. A postscript version is available from
http://www.phys.hawaii.edu/~jorge/postscript/vietnam.ps This replacement
version corrects errors in the original preprint and updates the URL link
above. The corrections are to the Reported measurement numbers of the
polarization in D*+/rho- from B -> D*rho- decays and the BSW parameters. The
error in the polarization measurement was not caught in time before
publication in the Proceeding of the IInd Recontres du Vietnam (1995). It has
been corrected her
Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions
A complex-analytic structure within the unit disk of the complex plane is
presented. It can be used to represent and analyze a large class of real
functions. It is shown that any integrable real function can be obtained by
means of the restriction of an analytic function to the unit circle, including
functions which are non-differentiable, discontinuous or unbounded. An explicit
construction of the analytic functions from the corresponding real functions is
given. The complex-analytic structure can be understood as an universal
regulator for analytic operations on real functions.Comment: 26 pgs. Small formatting corrections and bibliography updat
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