- #1
lostminty
- 82
- 0
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=εeiθ θ goes from ∏ to 0
C3, line segment from ε to R
C4, semi circle z=Reiθ 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 there's an ε on the denominator