## Integral Equations of First KindThis book studies classes of linear integral equations of the first kind most often met in applications. Since the general theory of integral equations of the first kind has not been formed yet, the book considers the equations whose solutions either are estimated in quadratures or can be reduced to well-investigated classes of integral equations of the second kind.In this book the theory of integral equations of the first kind is constructed by using the methods of the theory of functions both of real and complex variables. Special attention is paid to the inversion formulas of model equations most often met in physics, mechanics, astrophysics, chemical physics etc. The general theory of linear equations including the Fredholm, the Noether, the Hausdorff theorems, the Hilbert-Schmidt theorem, the Picard theorem and the application of this theory to the solution of boundary problems are given in this book. The book studies the equations of the first kind with the Schwarz Kernel, the Poisson and the Neumann kernels; the Volterra integral equations of the first kind, the Abel equations and some generalizations, one-dimensional and many-dimensional analogues of the Cauchy type integral and some of their applications. |

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### Contents

General Remarks on Linear Integral Equations of | 27 |

The Picard Theorem of Solvability of a Class of Integral | 51 |

Integral Equations of the First Kind with Kernels | 81 |

Integral Equations of the First Kind with the Kernels | 103 |

Some Other Classes of Integral Equation of the First Kind | 125 |

The Abel Integral Equation and Some of Its Generalizations | 181 |

A TwoDimensional Analogue of the Cauchy Type Integral | 215 |

References | 259 |

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analytic function arbitrary arctg assumptions boundary condition Cauchy principal value Cauchy problem Cauchy type integral circle completely continuous complex variable const continuously extendible converges corresponding defined denote density Dirichlet problem due to equation eigenvalue equality equation 1.2 equation 3.1 exists expression Fourier coefficients Fredholm integral equation function u(x half-plane harmonic function Hilbert-Schmidt theorem Hoelder continuous holomorphic homogeneous equation infinity kernel K(x linear equations linearly independent linearly independent solutions Lyapunov mapping Math matrix metric spaces necessary and sufficient Neumann problem normally solvable obtain operator orthogonal coordinates Picard theorem piecewise smooth plane of complex positive number potential respect right-hand side satisfies the boundary satisfies the condition Schwarz kernel second kind Section segment simple layer singular integral equations solution of equation space square summable function theory transforms Two-Dimensional Analogue unique valid vector P(x yields