Actually, I have made a lot efforts. The first identity has been proved by myself. But as for the second identities, I have been thinking for a long time and I still can not figure it out.
Prove two integral identities?
1. The following integral identity holds
\dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{\sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a\sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right]
Hints: this...
The matrix \mathbf{B}satifies the following Lyapunov equation
\begin{gathered}\mathbf{A}^{T}\mathbf{B}\end{gathered}+\mathbf{BA}=-\mathbf{I}
prove that necessary and sufficient condition generating a symmetric and positive determined \mathbf{B}is that all of the eigen values of...
an integral equation Abel and L operator??
1. The Abel operator
The general Abel integral equation
\begin{gathered}\intop_{x}^{a}\dfrac{F(y)dy}{\left(y^{2}-x^{2}\right)^{\frac{1+u}{2}}}=f(x)\end{gathered}
has the solution
\begin{gathered}F(r)=-\dfrac{2\cos\frac{\pi...