Nov 1, 2011 #2 lurflurf Homework Helper 2,459 158 2i sin(x)=eix-e-ix and (n+1)undu=dun+1 with u=eix is (n+1)einxdeix=dei(n+1)x
Nov 2, 2011 #3 asmani 105 0 Sorry, I can't see how to use these facts. Can you give any further hint, please? Besides, what contour should be chosen? Thanks Last edited: Nov 2, 2011
Sorry, I can't see how to use these facts. Can you give any further hint, please? Besides, what contour should be chosen? Thanks
Nov 2, 2011 #4 lurflurf Homework Helper 2,459 158 contour is unit circle let z=ei x dx=dz/(i z) sin(x)=((z-1/z)/(2i)) sin2n(x)=((z-1/z)/(2i))2n [tex]\int_0^\pi \sin^{2n}(x) dx=\frac{1}{2}\int_{-\pi}^\pi \sin^{2n}(x) dx=\oint_{|z|=1}\left( \frac{z-\frac{1}{z}}{2i}\right)^{2n}\frac{dz}{2i z}[/tex]
contour is unit circle let z=ei x dx=dz/(i z) sin(x)=((z-1/z)/(2i)) sin2n(x)=((z-1/z)/(2i))2n [tex]\int_0^\pi \sin^{2n}(x) dx=\frac{1}{2}\int_{-\pi}^\pi \sin^{2n}(x) dx=\oint_{|z|=1}\left( \frac{z-\frac{1}{z}}{2i}\right)^{2n}\frac{dz}{2i z}[/tex]