Question

The only function from the following that is analytic is:

• A. \( F(z) = \operatorname{Re}(z) \)
• B. \( F(z) = \operatorname{Im}(z) \)
• C. \( F(z) = z \)
• D. \( F(z) = \sin z \)

Hint:

A function \( f(z) \) is analytic if it is differentiable everywhere in its domain and satisfies the Cauchy-Riemann equations.

Answer 1

The Correct Option is D

Step 1: Definition of an analytic function. A function is analytic if it satisfies the Cauchy-Riemann equations: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \] 
Step 2: Checking analyticity of given functions. 
- \( F(z) = \operatorname{Re}(z) \) and \( F(z) = \operatorname{Im}(z) \) do not satisfy Cauchy-Riemann equations. 
- \( F(z) = z \) is analytic but is a trivial case. 
- \( F(z) = \sin z \) is analytic as it is holomorphic over the entire complex plane. 
Step 3: Selecting the correct option. Since \( \sin z \) is an entire function, the correct answer is (D).