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Part BCSIR NET December 2024room-inside-the-domain-shrinks-the-derivative

Room inside the domain shrinks the derivative

Let ℍ = {z = x + iy ∈C\in \mathbb{C} | y > 0} and f : ℍ →C\to \mathbb{C} be a non-constant holomorphic function satisfying |f(z)| < 1 for all z ∈ ℍ. Which of the following statements is true?

  1. A.lim⁡(y→+∞)f′(\lim(y\to+\infty) f'(iy) = 0✓ Correct
  2. B.lim⁡(y→+∞)f′(\lim(y\to+\infty) f'(iy) is a complex number with absolute value 1.
  3. C.lim⁡(y→+∞)\lim(y\to+\infty) |f′(iy)| =+∞= +\infty
  4. D.lim⁡(y→+∞)f′(\lim(y\to+\infty) f'(iy) is not a real number.

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Cauchy's theorem applies to any domain without singularities of f

More on this topic

The chapter behind this: Cauchy's theorem and integral formula — the toolkit — free to read

From Cauchy Theory › Cauchy's theorem and integral formula

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