Some boundary value problems of \(x^{k}\)-analytic functions with weight in boundary conditions (Q2767618)

Let \(G\) be a domain from the right half-plane \(\text{Re}(z)>0\) \((z=x+iy)\) bounded by a segment \([a,b]\) of the imaginary axis and a simple smooth contour \(C\) that connects points \(a\) and \(b\). Let \(a_1, b_1\) be points of the imaginary axis and let \(a<a_1<b_1<b\). Let \(f(z)=u(x,y)+iv(x,y)\) be an analytic in \(\overline G\) function which satisfies the condition \(\text{Im} f(z)|_{x=0}=0\) \((a<y<a_1\), \(b_1<y<b)\). The author obtains an integral representation of an \(x^{k}\)-analytic in \(G\) function \(\tilde f(z)=\tilde u(x,y)+i\tilde v(x,y)\) via the boundary values of \(f(z)\) in the case when the corresponding system of differential equations becomes degenerate on a part of the imaginary axis. This integral representation is used for the solution of two boundary value problems for the half-disk with weight in the boundary conditions.