Q.1 (e) Prove that log(x+iy/x-iy)=2itan⁻¹(y/x) .

Let x+iy=re^iθ where r=√(x²+y²),\ θ=tan⁻¹y/x . Then x-iy=re^(-iθ) . log(x+iy/x-iy)=logfracre^iθre^(-iθ)=log e^2iθ=2iθ (principal value) =2itan⁻¹(y/x) (proved).

Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\).

Q.1 (e) Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\).

Doubtly Asked 25 Oct 2026 mu engineering Save WhatsApp Report

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Let \(x+iy=re^{i\theta}\) where \(r=\sqrt{x^2+y^2},\ \theta=\tan^{-1}\frac yx\). Then \(x-iy=re^{-i\theta}\).

\(\log\left(\frac{x+iy}{x-iy}\right)=\log\frac{re^{i\theta}}{re^{-i\theta}}=\log e^{2i\theta}=2i\theta\) (principal value)

\(=2i\tan^{-1}\left(\frac yx\right)\) (proved).

Doubtly AI · 25 Oct 2026 AI-verified · Oct 2026 Report

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Prove that \(\log\left(\frac{x+iy}{x-iy}\right)=2i\tan^{-1}\left(\frac yx\right)\).

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