Question

The real and imaginary parts of \(\log(x+iy)\) are:

• A. Real part = \(\log(x^2+y^2)\) and Imaginary part = \(\tan^{-1}\left(\frac{y}{x}\right)\)
• B. Real part = \(\log(x^2+y^2)\) and Imaginary part = \(\tan^{-1}\left(\frac{x}{y}\right)\)
• C. Real part = \(\log\sqrt{x^2+y^2}\) and Imaginary part = \(\tan^{-1}\left(\frac{x}{y}\right)\)
• D. Real part = \(\log\sqrt{x^2+y^2}\) and Imaginary part = \(\tan^{-1}\left(\frac{y}{x}\right)\)

Hint:

To find the logarithm of a complex number, the first step is always to convert it from Cartesian form (\(x+iy\)) to polar form (\(re^{i\theta}\)). The logarithm then separates neatly into its real and imaginary parts.

Answer 1

The Correct Option is D

Step 1: Express the complex number \(z = x+iy\) in polar form. In polar coordinates, \(z = r e^{i\theta}\), where the modulus is \(r = |z| = \sqrt{x^2 + y^2}\) and the argument is \(\theta = \arg(z) = \tan^{-1}(y/x)\).
Step 2: Apply the complex logarithm. The natural logarithm of \(z\) is: \[ \log(z) = \log(r e^{i\theta}) \] Using the property \(\log(ab) = \log(a) + \log(b)\), we get: \[ \log(z) = \log(r) + \log(e^{i\theta}) \] Using the property \(\log(e^w) = w\), we get: \[ \log(z) = \log(r) + i\theta \]
Step 3: Identify the real and imaginary parts. - The real part is \(\text{Re}(\log(z)) = \log(r) = \log(\sqrt{x^2+y^2})\). - The imaginary part is \(\text{Im}(\log(z)) = \theta = \tan^{-1}(y/x)\). This matches the expressions in option (4). Note that \(\log(\sqrt{x^2+y^2})\) is also equal to \(\frac{1}{2}\log(x^2+y^2)\).