A system of two functions

• May 7th 2011, 11:44 AM
bkarpuz
A system of two functions
Dear MHF members,

The problem is as follows.
Let $u(x):=\int_{0}^{\infty}\dfrac{\mathrm{e}^{-t}\sin(tx)}{\sqrt{t}}\mathrm{d}t$ and $v(x):=\int_{0}^{\infty}\dfrac{\mathrm{e}^{-t}\cos(tx)}{\sqrt{t}}\mathrm{d}t$.
Show that $u$ and $v$ are solutions of a system of first order.
Hence find explicit forms of $u$ and $v$.

In the following link you may see that $u$ and $v$ are functions in $C^{1}$.
http://www.mathhelpforum.com/math-he...tml#post647936

Thanks.
bkarpuz