Part CCSIR NET June 2025e-to-the-1-over-z-has-an-essential-singularity-so-non-constant-does-not-mean-pole

E to the 1 over z has an essential singularity so non constant does not mean pole

Let f be an entire function. Which of the following statements are true?

A.If f(z) = f(z + 1) for all $z \in C$ then f is a constant function.
B.If f(z) = f(z + 1) = f(z + i) for all $z \in C$ then f is a constant function.✓
C.If f(1/z) has a removable singularity at 0 then f is a constant function.✓
D.If f is a non-constant function then f(1/z) has a pole at 0.

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.

Drill statements like this

E to the 1 over z has an essential singularity so non constant does not mean pole

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