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

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

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

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 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 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 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 II: Weak Convergence, Classification and Factorization of Singularities

The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of convergence of the Taylor series of the inner analytic function is the open unit disk. An essentially complete classification, in terms of the singularity structure of the corresponding inner analytic functions, of the modes of convergence of a large class of DP Fourier series, is established. Given a weakly convergent Fourier series of a DP real function, it is shown how to generate from it other expressions involving trigonometric series, that converge to that same function, but with much better convergence characteristics. This is done by a procedure of factoring out the singularities of the corresponding inner analytic function, and works even for divergent Fourier series. This can be interpreted as a resummation technique, which is firmly anchored by the underlying analytic structure.Comment: 43 pages, including 20 pages of appendices with explicit calculations and examples; updated cross-references; made a few improvements in the text and in one equation; updated the reference

Fourier Theory on the Complex Plane V: Arbitrary-Parity Real Functions, Singular Generalized Functions and Locally Non-Integrable Functions

A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set of all integrable real functions is then extended to include a set of singular "generalized functions" by the side of the integrable real functions. A general definition of these generalized functions is proposed and explored. The generalized functions are introduced loosely in the spirit of the Schwartz theory of distributions, and include the Dirac delta "function" and its derivatives of all orders. The inner analytic functions corresponding to this infinite set of singular real objects are given by means of a recursion relation. The set of inner analytic functions is then further extended to include a certain class of non-integrable real functions. The concept of integral-differential chains is used to help to integrate both the normal functions and the singular generalized functions seamlessly into a single structure. It does the same for the class of non-integrable real functions just mentioned. This extended set of generalized functions also includes arbitrary real linear combinations of all these real objects. An interesting connection with the Dirichlet problem on the unit disk is established and explored.Comment: 32 pages, including 3 pages of appendice

Real Functions for Physics

A new classification of real functions and other related real objects defined within a compact interval is proposed. The scope of the classification includes normal real functions and distributions in the sense of Schwartz, referred to jointly as "generalized functions". This classification is defined in terms of the behavior of these generalized functions under the action of a linear low pass-filter, which can be understood as an integral operator acting in the space of generalized functions. The classification criterion defines a class of generalized functions which we will name "combed functions", leaving out a complementary class of "ragged functions". While the classification as combed functions leaves out many pathological objects, it includes in the same footing such diverse objects as real analytic functions, the Dirac delta "function", and its derivatives of arbitrarily high orders, as well as many others in between these two extremes. We argue that the set of combed functions is sufficient for all the needs of physics, as tools for the description of nature. This includes the whole of classical physics and all the observable quantities in quantum mechanics and quantum field theory. The focusing of attention on this smaller set of generalized functions greatly simplifies the mathematical arguments needed to deal with them.Comment: 15 pages, including 1 page of appendices, and 1 figur