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Handbook of mathematics for engineers and scienteists part 124

Handbook of mathematics for engineers and scienteists part 124. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration 829 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration 16.4.1. Fredholm Integral Equations of the Second Kind. Resolvent 16.4.1-1. Some definitions. The eigenfunctions of a Fredholm integral equation. Linear integral equations of the second kind with constant limits of integration have the general form y x - A J K x t y t dt f x 16.4.1.1 where y x is the unknown function a x b K x t is the kernel of the integral equation and f x is a given function which is called the right-hand side of equation 16.4.1.1 . For convenience of analysis a number A is traditionally singled out in equation 16.4.1.1 which is called the parameter of integral equation. The classes of functions and kernels under consideration were defined above in Paragraphs 16.3.1-1 and 16.3.1-2. Note that equations of the form 16.4.1.1 with constant limits of integration and with Fredholm kernels or kernels with weak singularity are called Fredholm equations of the second kind and equations with weak singularity of the second kind respectively. Equation 16.4.1.1 is said to be homogeneous if f x 0 and nonhomogeneous otherwise. A number A is called a characteristic value of the integral equation 16.4.1.1 if there exist nontrivial solutions of the corresponding homogeneous equation. The nontrivial solutions themselves are called the eigenfunctions of the integral equation corresponding to the characteristic value A. If A is a characteristic value the number 1 A is called an eigenvalue of the integral equation 16.4.1.1 . A value of the parameter A is said to be regular if for this value the above homogeneous equation has only the trivial solution. Sometimes the characteristic values and the eigenfunctions of a Fredholm integral equation are called the characteristic values and the eigenfunctions of the kernel K x t . The kernel K x t of the integral equation 16.4.1.1 is called a .