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.

  • $\begingroup$ This is a higly oscillating function. I need a general expression in the form of "k". The parameter "k" controls oscillations. $\endgroup$
    – skorpion
    Sep 6, 2017 at 14:34
  • 1
    $\begingroup$ Remember that $\left| e^{it}\right|=1\quad \forall t\in\mathbb{R}.$ $\endgroup$ Sep 6, 2017 at 14:35
  • $\begingroup$ Thanks. If its that easy, I will not ask here. Thats little tricky. $\endgroup$
    – skorpion
    Sep 6, 2017 at 14:39

1 Answer 1


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 $.

  • $\begingroup$ 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. $\endgroup$
    – skorpion
    Sep 6, 2017 at 14:37
  • $\begingroup$ @skorpion can you please provide an example where $$\left| x\exp\left(-ikx^2\right)\right| \ne |x|?$$ $\endgroup$
    – gt6989b
    Sep 6, 2017 at 14:40

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