Question

Which of the following functions is not analytic?

• A. \(e^z\)
• B. \(e^{\bar{z}}\)
• C. \(z + 2\bar{z}\)
• D. \(z^2\)

Hint:

Any function involving \(\bar{z}\) is not analytic in the complex plane.

Answer 1

The Correct Option is C

To determine which function is not analytic, we need to understand the concept of an analytic function. A function is analytic at a point if it is complex differentiable at every point in some neighborhood of that point.

An analytic function must satisfy the Cauchy-Riemann equations in the domain:

Cauchy-Riemann Equations: If \(f(z) = u(x, y) + iv(x, y)\), then \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

Let's analyze each option:

• \(e^z\): Since \(e^z\) is a standard complex function with no dependence on \(\bar{z}\), it is analytic everywhere in the complex plane.
• \(e^{\bar{z}}\): This function depends on \(\bar{z}\). Rewriting \(e^{\bar{z}}\) in terms of \(x\) and \(y\) where \(z = x + iy\) and \(\bar{z} = x - iy\), it becomes non-analytic as it does not satisfy the Cauchy-Riemann equations.
• \(z + 2\bar{z}\): Expressing this function as \((x + iy) + 2(x - iy)\), we get \(3x - iy\). This function clearly depends on both \(x\) and \(y\) separately, violating the Cauchy-Riemann equations, thus non-analytic.
• \(z^2\): For \(z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi\), the Cauchy-Riemann equations are satisfied, indicating it is analytic everywhere.

Conclusion: The function \(z + 2\bar{z}\) is not analytic as it fails to satisfy the Cauchy-Riemann equations.