GATE EE 2015 Set 2 — Question 11
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Engineering Mathematics → Complex Variables → Analytic Functions
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Question
Given , where are complex valued functions of a complex variable . Which one of the following statements is TRUE?
Correct answer
(B) If g(z) and h(z) are differentiable at z₀, then f(z) is also differentiable at z₀.
Solution
Statement (B) is true because the sum of two differentiable functions is always differentiable.
Statement (A) is false: is differentiable, but and are not.
Statement (C) is false: Continuity does not imply differentiability (e.g., ).
Statement (D) is false: is differentiable, but its real part is not differentiable as a complex function because it doesn't satisfy the Cauchy-Riemann equations.
Statement (A) is false: is differentiable, but and are not.
Statement (C) is false: Continuity does not imply differentiability (e.g., ).
Statement (D) is false: is differentiable, but its real part is not differentiable as a complex function because it doesn't satisfy the Cauchy-Riemann equations.
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