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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