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

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  1. f (z) = Re(z)
  2. f (z) = Im(z) 
  3. f (z) = z 
  4. f (z) = sinz

Answer (Detailed Solution Below)

Option 3 : f (z) = z 
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Detailed Solution

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Concept:

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

If , then the C-R equations are:

and

Given Options:

Calculation:

  • Options 1 and 2 are purely real and imaginary parts — not analytic → do not satisfy C-R equations.
  • Option 3: → clearly analytic everywhere. It satisfies C-R equations.
  • Option 4: 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

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