A presentation of this theory in Hölder classes of functions is to be found in a monograph of one of its creators, N. Noether first introduced the concept of an index and proved the theorems 1—3 above by applying the method of left regularization. They are highly effective in solving boundary problems occurring in the theory of functions of a complex variable, potential theory, the theory of elasticity, and the theory of fluid mechanics. Petersburg in 1873; however, this research remained unnoticed. Singular integral equations play important roles in physics and theoretical mechanics, particularly in the areas of elasticity, aerodynamics, and unsteady aerofoil theory. However, the exact scale is unknown and classified.
In this case the symbol in 4 denotes the sum of the increments of the function between brackets after a circuit of each of the contours. Russian names have been printed in Russian letters in the authors index, in order to overcome any possible confusion arising from transliteration. This high-level treatment by a noted mathematician considers one-dimensional singular integral equations involving Cauchy principal values. Some basic problems of the mathematical theory of elastisity. If , then equation 3 is solvable in if and only if satisfies the condition When these conditions hold, 3 has a unique -solution, given by the formula 6 with.
In preparing this translation for publication certain minor modifications and additions have been introduced into the original Russian text, in order to increase its readibility and usefulness. In the theory of multi-dimensional singular integral equations, an important role is played by the notion of a symbol cf. It is only necessary to bear in mind that, in the case of general singular integral equations, and are both non-zero in general, in contrast to the case of characteristic singular integral equations, when one of them must be zero. He got also awarded for the launch of the world's first artificial satellite into space in 1961 on which development he had contributed as well. The chapters and their subsections of the Russian edition have been renamed parts and chapters respectively and the last have been numbered consecutively.
He was one of the first to apply the theory of functions of complex variables to the problems of elasticity theories, proposing a number of techniques that have been successfully implemented in numerous areas of mathematics, theoretical physics and mechanics. Like the Noether theorems, the formulas 6 and 8 remain valid in the case when consists of a finite number of smooth mutually-disjoint closed contours. In particular, the former has been combined with the list of references of the original text, in order to enable the reader to find quickly all information on anyone reference in which he may be especially interested. For example, the theories of one-dimensional and multi-dimensional singular integral equations, both in the formulation of definitive results and in the methods used to establish them, differ significantly from one another. Thus it is assumed that 13 always holds. Intended for graduate students, applied and pure mathematicians, engineers, physicists, and researchers in a variety of scientific and industrial fields, this text is accessible to students acquainted with the basic theory of functions of a complex variable and the theory of Fredholm integral equations.
I Fundamental Propkrtibs of Cauchy Integrals. Let be a complex-valued continuous function that does not vanish on an oriented closed simple smooth contour , and let 4 where denotes the increment of the function between brackets after a single circuit of in the positive direction. In particular, is the adjoint of. His first scientific magazine was published already earlier in 1915 containing a number of issues on elasticity theory. Thus, instead of the first person, the third person has been used throughout; wherever possible footnotes have been included with the main text. It can be shown see that under certain conditions they also remain valid in the case of a piecewise-smooth contour that is, when is the union of a finite number of smooth open arcs, which are mutually-disjoint except for their end points. Muskhelishvili's place in the world of science and mathematics.
If the equations and are equivalent for each , then is called a left equivalent regularizer of. A special case, a singular integral equation with Cauchy kernel, was considered much earlier in the doctoral thesis of Yu. As a rule, the limit in 12 does not exist when the following condition is violated: 13 where is the unit sphere with centre at the origin. An important class of one-dimensional singular integral equations are those with a Cauchy kernel: 1 where , , , are known functions, is a Fredholm kernel see , is the desired function, is a planar curve, and the improper integral is to be understood as a Cauchy principal value, i. Hubert on boundary value problems. Singular integral equations have been investigated in detail in the space of continuing functions Φ and in the space of square-integrable functions. It is defined in terms of the functions and , and from a given symbol the original singular operator can be recovered up to a completely-continuous term.
Moscow, 1960, 142—143 jointly with I. Georgian Metniereba, Tbilisi, 1991, 60 pp. The homogeneous singular integral equations and have a finite number of linearly independent solutions. The great significance, both theoretical and practical, of singular integral equations became especially apparent towards the end of the 1930s in connection with the solution of certain very important problems in the mechanics of a solid medium the theory of elasticity, hydro- and aeromechanics, and others and theoretical physics. This monograph also stimulated scientific investigations in certain other directions, for example in the theory of singular integral equations not satisfying the Hausdorff normality condition, singular integral equations with non-diagonal singularities with displacements , Wiener—Hopf equations, multi-dimensional singular integral equations, etc. Let and be the numbers of linearly independent solutions of the homogeneous equations and , respectively. This high-level treatment by a noted mathematician considers one-dimensional singular integral equations involving Cauchy principal values.
Systems of singular integral equations. One says that a function defined on is almost-everywhere finite, measurable, integrable, etc. Depending on the dimension of the manifold over which the integrals are taken, one distinguishes one-dimensional and multi-dimensional singular integral equations. Thus, instead of the first person, the third person has been used throughout; wherever possible footnotes have been included with the main text. Berlin, Verlag Chemie, 1938, S. Contributions in honor of the seventieth birthday of academician N.
An equation containing the unknown function under the integral sign of an improper integral in the sense of Cauchy cf. He completed a series of research, experimental and theoretical work in different areas of applied mathematics, physics and mechanics, which all had great practical importance and decisive impact on the development of a range of military hardware during and after the war. He graduated from local grammar school in 1909 and afterwards from the Physics and Mathematics Faculty of in 1914. Russian names have been printed in Russian letters in the authors index, in order to overcome any possible confusion arising from transliteration. If belongs to for some admissible value and knowledge of the numerical value of is not required, then one writes , or even if it is clear from the context which contour is meant.
It has been shown that under certain restrictions 11 admits a regularization in the space , , if and only if the absolute value of its symbol has a positive lower bound, and in this case the Fredholm theorems hold. In preparing this translation for publication certain minor modifications and additions have been introduced into the original Russian text, in order to increase its readibility and usefulness. Reduction to an integral equation. Third revised and augmented ed. These are equations of the form 11 where is a domain in the Euclidean space ,. Privalov were of great importance in the development of the theory of singular integral equations.