Tag Archives: unit

Creating Difficult integrals by the residue theorem

Theorem Let \( f \) be analytic function in the unit circle \( |z|\leq 1 \)  such that \( f\neq 0\) . Then $$\int^{2\pi}_0f(e^{it})\,dt =2\pi \, f(0) $$ $$\textit{proof}$$ Since the function \(f \) is analytic in and on the … Continue reading

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Integrating along the unit circle

Prove that $$\int^{2\pi}_0e^{\cos \theta}\cos(n\theta -\sin \theta)\,d \theta=\frac{2\pi}{n!}$$ $$\textit{proof}$$   Consider the following function $$f(z)=e^{z^{-1}}z^{n-1}$$ Now we integrate the function along a circle of radius 1 The contour encloses a pole at \(z = 0\) $$\oint_{|z|=1}e^{z^{-1}}z^{n-1} dz=2\pi i\mathrm{Res}(f(z),0) $$ Now we … Continue reading

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