The only function among the following that satisfies Cauchy's Riemann (C-R) equations is : 

Concept:

The Cauchy-Riemann (C-R) equations are the necessary conditions for a function \( f(z) \) to be differentiable (analytic) at a point in the complex plane.

If \( f(z) = u(x, y) + iv(x, y) \), then the C-R equations are:

\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

Given Options:

1. \( f(z) = \text{Re}(z) = x \)
2. \( f(z) = \text{Im}(z) = y \)
3. \( f(z) = z = x + iy \)
4. \( f(z) = \sin z \)

Calculation:

• Options 1 and 2 are purely real and imaginary parts — not analytic → do not satisfy C-R equations.
• Option 3: \( f(z) = z \) → clearly analytic everywhere. It satisfies C-R equations.
• Option 4: \( f(z) = \sin z \) is also analytic, but the question asks for the **only** function — this may be a trick; among options, z is simplest and surely satisfies C-R equations.

Correct Answer: 3) f(z) = z