Thursday, January 14, 2016

calculus - Confirm that intinfty0t1sintdt=pi/2




Confirm that 0t1sintdt=π/2.



The guide book I am using gives the following help:




Consider γz1eizdz, where for $0



Exercise IV.4.20. For r with $0

Using the hint, I know that γz1eizdz=0 for the Cauchy theorem, with which [s,r]z1eizdz+γrz1eizdz+[r,s]z1eizdzγsz1eizdz=0, but I do not know what else to do here, could someone help me please? Thank you very much.


Answer



Define a path in the Complex Plane: enter image description here



Now consider Ceizzdz=arceizzdz+Arceizzdz+rReizzdz+Rreizzdz




By parametizing the integrals over the arcs, and letting r0 for arc and R for Arc, we see that arci0πdθ and Arc0.



So we have Ceizzdz=PVeizzdzπi

where PV denotes the Cauchy Principal Value.



Since the contour does not enclose any poles, the entire contour integral is 0.



So 0=PVeizzdzπi

PVeizzdz=πi
Note that due to Euler's Formula Im(PVeizzdz)=sin(z)zdz

And so sin(z)zdz=π

Lastly since sin(z)z is an even function:

0sin(z)zdz=π2


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