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

Complex Analysis of Real Functions VI: On the Convergence of Fourier Series

We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms of Dirichlet integrals. The expression for the remainder is written in terms of a similar type of integral. Since the asymptotic limit of the remainder being zero is a necessary and sufficient condition for the convergence of the series, this same condition on the asymptotic behavior of the corresponding integrals constitutes such a necessary and sufficient condition.Comment: 26 pgs. Small formatting corrections and bibliography updat

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

b→cb \to c 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 $e^+e^-$ colliders at the $\Upsilon(4s)$ and at the $Z^0$ resonance. The experimental results reported are then used in tests of theoretical model predictions, the determination of the QCD parameters $a_1$ and $a_2/a_1$ 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

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