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## Homework Statement

show that

[itex]\int^{∞}_{0}\frac{sin^{2}x}{x^{2}}dx= \frac{\pi}{2}[/itex]

## Homework Equations

consider

[itex]\oint_{C}\frac{1-e^{i2z}}{z^{2}}dz[/itex]

where C is a semi circle of radius R, about 0,0 with an indent (another semi circle) excluding 0,0.

## The Attempt at a Solution

curve splits into

C1, line segment from -R to -ε

C2, semi circle z=εe

^{iθ}θ goes from ∏ to 0

C3, line segment from ε to R

C4, semi circle z=Re

^{iθ}θ goes from 0 to ∏

function is holomorphic in/on C. so integral =0

for C4, applying limit R > infinity integral = 0

C3. really stuck here

I sub in the value of z

to get

[itex]\int^{0}_{∏}\frac{1-e^{i2e^{iθ}}}{ε^{2}e^{2iθ}}εe^{iθ}[/itex]

but I can't take the limit ε -> 0 of this since theres an ε on the denominator