Application of Cauchy's Theorem

pdestud

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Sep 14, 2019
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Apply Cauchy's theorem to the function
f(z) = e^{-z^2}
and the sector of the circle
z = R
bounded by a section of the real axis and a linear segment making the angle
\frac{\pi}{4}
with the real axis.

1. Show that the integral over the circular boundary tends to zero as R tends to infinity. Is it because [MATH] \left| e^{-z^2} \right| \leq 1 [/MATH]?

2. Show that

1568490332054.png

I can't seem to solve it. Looking it up, the websites refer to Fresnel but I haven't learned anything about it so I can't use that method.
 
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