Question

Let \( f: \mathbb{C} \to \mathbb{C} \) be defined by \( f(z) = |z|^2 - 5\bar{z} + 2 \).
Which one of the following is TRUE?

• A. \( f \) is differentiable at \( z = 5i \).
• B. \( f \) is differentiable at \( z = 5 \).
• C. \( f \) is differentiable at \( z = -5i \).
• D. \( f \) is differentiable at \( z = -5 \).

Hint:

Split f into real and imaginary parts and solve the Cauchy-Riemann equations for x and y.

Answer 1

The Correct Option is B

Here is a quicker route using Wirtinger derivatives instead of splitting into real and imaginary parts by hand. Recall $|z|^2 = z\bar z$, so we can write $f(z) = z\bar z - 5\bar z + 2$ directly as a function of $z$ and $\bar z$ treated as independent variables.

For a function built from $z$ and $\bar z$, complex differentiability at a point, in the sense tested by the Cauchy-Riemann equations, happens exactly where $\dfrac{\partial f}{\partial \bar z} = 0$ at that point, as long as $f$ is smooth as a function of $x,y$, which is true here since it is a polynomial. Compute this Wirtinger derivative, treating $z$ as a constant while differentiating with respect to $\bar z$:

\[ \frac{\partial f}{\partial \bar z} = z - 5 \]

So $f$ is differentiable exactly where $z - 5 = 0$, that is, only at the single point $z = 5$. Now check each option against this:

1. (A) z = 5i: Here $z - 5 = 5i - 5 \neq 0$, so $f$ is not differentiable here.
2. (B) z = 5: Here $z - 5 = 0$, so this is exactly the point of differentiability.
3. (C) z = -5i: Here $z - 5 = -5i - 5 \neq 0$, not differentiable.
4. (D) z = -5: Here $z - 5 = -10 \neq 0$, not differentiable.

Only $z = 5$ makes the Wirtinger derivative $\partial f/\partial \bar z$ vanish, so $f$ is complex differentiable at that single point and nowhere else nearby.

Let's summarize:

• Writing $f$ using $z$ and $\bar z$ turns the Cauchy-Riemann test into the single condition $\partial f/\partial \bar z = 0$.
• Here that condition reduces to the linear equation $z - 5 = 0$, giving one isolated point of differentiability.
• That point is $z = 5$, matching option (B) and confirming the same answer found using the standard real-variable Cauchy-Riemann equations.

\[ \boxed{z = 5} \]