Sunday, 28 June 2015

integration - Show that intlimitsinfty0fracdttecostsinsint=fracpi2(e1)




How do you show that




0dttecos(t)sin(sin(t))=π2(e1)




I managed to get the left-hand side to equal the imaginary part ofI=0dtteeit

But I’m not very sure what to do next. I’m thinking of a substitute teit, but I’m not very sure how to evaluate the limit as t. I also tried contour integration, but I’m not exactly sure what contour to draw.


Answer




ecostsinsint=Imexp(eit)=Imn0enitn!=n1sin(nt)n!


and since for any a>0 we have +0sin(at)tdt=π2 it follows that
+0ecostsinsintdtt=π2n11n!=π2(e1),

pretty simple.






I have a counter-proposal:
+0(ecostsinsint)2dtt2=π2m,n1min(m,n)m!n!=π2I1(2)+πe(e1)2πe10I1(2x)ex2dx.


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