Part BCSIR NET December 2024agreement-on-a-line-propagates-by-reflection

Agreement on a line propagates by reflection

For z = x + iy $\in \mathbb{C}$ , let f(z) = u(x, y) + iv(x, y) define an entire function. Consider the function g(z) = u(x, −y) − iv(x, −y), for z = x + iy $\in \mathbb{C}$ . Suppose that v(x, 0) = 0 for all $x \in \mathbb{R}$ . Define E = { $z \in \mathbb{C}$ | f(z) = g(z)}. Which of the following statements is true?

A.E is the real axis.
B.E is the imaginary axis.
C.E contains an open subset of $\mathbb{C}$ , but $E \ne \mathbb{C}$ .
D. $E = \mathbb{C}$ ✓ Correct

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, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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Agreement on a line propagates by reflection

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