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 Integrating a cosine log integral around a semicircle contour
 Creating Difficult integrals by the residue theorem
 Proving a trigonometric integral by integrating around an ellipse in the complex plain
 Integrating a fraction of exponential and trignometric using rectangular contour
 Integrating around a triangular contour for Fresnel integral
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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
Posted in Contour Integration
Tagged circle, contour, difficult, impossible, Integral, residue, theorem, unit
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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 $$f(z)=e^{z^{1}}z^{n1}$$ Now we integrate the function along a circle of radius 1 The contour encloses a pole at \(z = 0\) $$\oint_{z=1}e^{z^{1}}z^{n1} dz=2\pi i\mathrm{Res}(f(z),0) $$ Now we … Continue reading
Posted in Contour Integration
Tagged analysis, circle, complex, contour, integration, residue, theorem, unit
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