Home > Complex questions > Complex #27: integrating over a long rectangle

## Complex #27: integrating over a long rectangle

Problem 27: Show that $\int_{\gamma} e^{iz}e^{-z^2}dz$ has the same value on every straight line path $\gamma$ (oriented in the $+x$ direction) parallel to the real axis.

Proof: Consider the rectangle $R=[-M,M]\times [0,Y]$, where $Y$ is some given number and $M$ is a very large number. Since the integrand is entire, the integral over the boundary of $R$ is zero by the Cauchy Integral Theorem, and as $M\to\infty$, the horizontal pieces correspond to $\int_{\gamma} e^{iz}e^{-z^2}dz$ with $\gamma$ being the line $Im (z)=0$ or the line $Im(z)=Y$, with one being negative and the other positive. Thus, the result is proved if we can show that the contributions from the vertical pieces go to zero as $M$ goes to $\infty$.

Observe that the absolute value of the integrand at $z=x+iy$ is $e^{-x^2+y^2-y}$, which on each vertical boundary piece is $e^{-M^2+y^2-y}$ for $0\le y\le Y$. Integrating this over $y$ will give a value bounded by a constant times $e^{-M^2}$, which clearly goes to zero as $M\to\infty$.

QED.

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