Adjoint integral equation

Summary

Let an integral equation of the second kind have kernel $K(x, s)$ and parameter $\lambda$. Its adjoint is the equation of the same kind with the kernel $\overline{K(s, x)}$ and the parameter $\bar\lambda$. Its integral operator is the adjoint of the original integral operator for the $L^2$ inner product. The Fredholm alternative uses it: the original equation is solvable exactly when its right side is orthogonal to the solutions of the homogeneous adjoint equation. For a real kernel the adjoint kernel is the transposed kernel $K(s, x)$.