Integral Equations

A similar statement is valid for the Fourier–Chebyshev series in polynomials U n : if f(x) hasp continuous derivatives in the segment [−1, 1] and p ≥ 2, thenN ∣ f(x) − ∑b n U n (x)∣ ≤n=0C , x ∈ [−1, 1]. (334)Np−3/2We will often use the following three relationships involving the Chebyshev polynomials:∫1−11x − y T dxn(x) √ = πU 1 − x2n−1(y), −1 < y < 1, n ≥ 1; (335)∫1−11x − y U n−1(x) √ 1 − x 2 dx = −πT n (y), −1 < y < 1, n ≥ 1; (336)∫1−1{1ln|x − y| T dx π ln 2, n = 0;n(x) √ = 1 − x2 π T n (y)/n, n ≥ 1.(337)The integrals in (335) and (336) are improper integrals in the sense of Cauchy (a detailedanalysis of such integrals is performed in [8, 9]). The integral in (337) is a standard convergentimproper integral.17 Solution to integral equations with a logarithmic singulatityof the kernel17.1 Integral equations with a logarithmic singulatity of the kernelMany boundary value problems of mathematical physics are reduced to the integral equationswith a logarithmic singulatity of the kerneland(L + K)ϕ :=Lϕ :=∫1−1(ln∫1−1ln1|x − y| ϕ(x) dx√1 − x21|x − y| + K(x, y)) ϕ(x) dx√1 − x2= f(y), −1 < y < 1 (338)= f(y), −1 < y < 1. (339)In equations (338) and (339) ϕ(x) is an unknown function and the right-hand side f(y) is agiven (continuous) function. The functions ln (1/|x − y|) in (338) and ln (1/|x − y|) + K(x, y)in (339) are the kernels of the integral equations; K(x, y) is a given function which is assumedto be continuous (has no singularity) at x = y. The kernels go to infinity at x = y andtherefore referred to as functions with a (logarithmic) singulatity, and equations (338) and(339) are called integral equations with a logarithmic singulatity of the kernel. In equations(338) and (339) the weight function 1/ √ 1 − x 2 is separated explicitly. This function describes63