Recent content by firenze

Here is my try: Choose a sequence y_n \in [0,+\infty ) such that y_n \to y (\neq 0). Define the function g_n(x)=y_n \arctan x e^{-xy_n}, then its limit is g(x)=y\arctan x e^{-xy}. Note that |g_n(x)| \leq |y_n\arctan x|, it follows g_n is integrable. Hence by dominated convergence thm we have...

Homework Statement f: [0,+\infty) \to \mathbb{R}: y \mapsto \int_0^{+\infty} y \arctan x \exp(-xy)\,dx. Show that this function is continuous in y if y \neq 0 and discontinuous if y = 0 Homework Equations The Attempt at a Solution I just can't get started, any hint?

Find the two integrals: \int_0^{\infty}\frac{e^{-\alpha x^2}}{x^2+1}\sin(\alpha x) \, dx \int_0^{\infty}e^{-\beta^2t}\cos(\beta x) \, d\beta Any hint?

Thanks:smile:

For a linearized system I have eigenvalues \lambda_1, \lambda_2 = a \pm bi \;(a>0) and \lambda_3 < 0 , then it should be an unstable spiral point. As t \to +\infty the trajectory will lie in the plane which is parallel with the plane spanned by eigenvectors v_1,v_2 corresponding to \lambda_1...