4
$\begingroup$

From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved.

My first thought was to draw a long strip and integrate over it. By Cauchy's theorem the integral is zero, and I can compare the contributions. I'd like it if, for well-chosen such strips, the contribution on the sides were entirely imaginary and that on the top/bottom were entirely real, or vice-versa. But so far, no luck.

$\endgroup$
3
$\begingroup$

Take a rectangle with vertices $\pm R, \pm R + iY$, for an arbitrary but fixed $Y$.

On the edges between $R$ and $R+iY$ resp $-R$ and $-R+iY$, the integrand can be estimated

$$\bigl\lvert e^{iz}e^{-z^2} \bigr\rvert \leqslant e^{\lvert Y\rvert}e^{Y^2-R^2},$$

so the contribution of these integrals is dominated by

$$\lvert Y\rvert e^{\lvert Y\rvert + Y^2 - R^2} \xrightarrow{R\to +\infty} 0,$$

so since by Cauchy's integral theorem the integral over the rectangle is $0$, the proposition follows.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.