Question

If \(f(z) = \frac{1}{2} \log_e(x^2 + y^2) + i \tan^{-1} \left( \frac{y}{x} \right)\) be an analytic function, then \( \alpha \) is ___ .

• A. 1
• B. -1
• C. 2
• D. -2

Hint:

In complex analysis, for a function to be analytic, it must satisfy the Cauchy-Riemann equations, which help in determining such conditions for the solution.

Answer 1

The Correct Option is B

For the function \( f(z) = \frac{1}{2} \log_e(x^2 + y^2) + i \tan^{-1} \left( \frac{y}{x} \right) \) to be analytic, it must satisfy the Cauchy-Riemann equations. These equations provide the necessary conditions for a function to be analytic (holomorphic) in the complex plane. In this case, the value of \( \alpha \), which ensures the function satisfies these conditions, is **\( \alpha = -1 \)**. The function involves both a logarithmic and inverse trigonometric term, both of which have known conditions for analyticity. These conditions determine that \( \alpha = -1 \).