Part BCSIR NET December 2024harmonic-modulus-squared-forces-constant

Harmonic modulus squared forces constant

Let 𝔻 = {z = x + iy $\in C$ : |z| < 1} be the open unit disc and f : 𝔻 $\to C a$ holomorphic function such that f(0) = 0. Let $ψ (z) =$ |f(z)| $^{2}$ , and $\partial^{2} ψ / \partial x^{2} + \partial^{2} ψ / \partial y^{2} \equiv 0$ . Which of the following statements is FALSE?

A.f can be extended to $C$ as an entire function.
B.f must have infinitely many zeros in 𝔻.
C.f is not a polynomial.✓
D.exp(f) cannot take every complex value.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Harmonic modulus squared forces constant

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