Question

Complex Variables Question, one sec while I type it up. Im totally stumped >.<

Answer 1

\[\int\limits_{-\pi}^{\pi}\cos^{911}(\cos t+i\sin t)\cos t dt\]i need to evaluate this integral. I'll type what ive attempted in a sec.

Answer 2

Let:\[I=\int\limits_{-\pi}^{\pi}\cos^{911}(\cos t+i\sin t)\sin t dt\]and\[J=\int\limits_{-\pi}^{\pi}\cos^{911}(\cos t+i\sin t)\cos t dt\]Then it turns out that:\[-I+iJ=0\]since \[-I+iJ=\int\limits_{\gamma}\cos^{911}(z)dz\]where gamma is the unit circle. Im pretty sure I cant conclude that I and J must be 0 from just that, but I cant come up with another relation between the two. any suggestions are greatly appreciated.

Answer 3

Ive also determined that:\[\int\limits_{-\pi}^{\pi}\cos^{911}(\cos t+i\sin t) dt=2\pi\]by looking at the problem:\[\int\limits_{-\pi}^{\pi}\frac{\cos^{911}(z)}{z}dz=\frac{2\pi i}{0!}\cos^{911}(0)=2\pi i\]and then using\[z=\cos t+i\sin t\Longrightarrow dz=(-\sin t+i \cos t )dt\] to simplify.

Answer 4

oops, on that cos^911(z)/z, there should be no limits, its a line integral over the path gamma, the unit circle again.

Answer 5

the bad thing is that my professor has this habit of typing out problems wrong. But I wont know that for sure until later today after the assignment is due. I dont think that cost is supposed to be there at the end...