I need to find maximum value of the following complex valued function \begin{align*} \underset{-\pi\le x\le \pi}{\mathrm{max}}\left\vert x e^{-\mathrm{i}k x^2}\right\vert,\quad k\in \mathbb{R}^{+}. \end{align*} I do not know how to proceed.
3
-
$\begingroup$ This is a higly oscillating function. I need a general expression in the form of "k". The parameter "k" controls oscillations. $\endgroup$– skorpionSep 6, 2017 at 14:34
-
1$\begingroup$ Remember that $\left| e^{it}\right|=1\quad \forall t\in\mathbb{R}.$ $\endgroup$– M. StrochykSep 6, 2017 at 14:35
-
$\begingroup$ Thanks. If its that easy, I will not ask here. Thats little tricky. $\endgroup$– skorpionSep 6, 2017 at 14:39
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
We know that
$$\forall t\in \Bbb R\;\;\; e^{it}=\cos (t)+i\sin (t) $$ and
$$|e^{it}|=\sqrt {\cos^2 (t)+\sin^2 (t)}=1$$
thus $$|xe^{-ikx^2}|=|x| |e^{-ikx^2}|=|x|$$
the max is $\pi $.
answered Sep 6, 2017 at 14:35
-
$\begingroup$ Thanks. I did this way but thats not true. If you plot the function, you will know. As the oscillations increase so is the amplitude. The end points do not provide max value. $\endgroup$– skorpionSep 6, 2017 at 14:37
-
$\begingroup$ @skorpion can you please provide an example where $$\left| x\exp\left(-ikx^2\right)\right| \ne |x|?$$ $\endgroup$– gt6989bSep 6, 2017 at 14:40