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CSIR NET Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free to read right now. Pick your subject below.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

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Tagged “complex”

“Bounded real part ⇒ entire function is constant — fails if only |f| is bounded on a half-plane”
“Zeros of a non-constant analytic function can accumulate inside the domain”
“|f| bounded near an isolated singularity ⇒ pole”

False — f(z) = sin(z)/z at 0

“The Cauchy–Riemann equations at a point imply complex differentiability there”

False — f(z)=(zˉ)2/zf(z) = ({\bar{z}})^{2}/z for z ≠ 0, f(0) = 0

“A harmonic function on any domain has a harmonic conjugate”
“Zeros of a non-constant holomorphic function cannot accumulate”

False — sin(1/z) on C\mathbb{C}∖{0}

“C∞C^\infty implies analytic”
“If f is holomorphic on the domain enclosed by γ\gamma except possibly at isolated points, and ∮γf=0\oint_\gamma f = 0, then f has no singularity inside”
“Cauchy's theorem applies to any domain without singularities of f”

False — f(z) = 1/z on the annulus ½ < |z| < 2

“A bounded holomorphic function on an unbounded domain is constant”

False — f(z) = eᶻ on {Re z < 0}

“An entire function omitting one value is constant”
“|f| attains its minimum on the boundary for holomorphic f”
“If |f| is bounded near an isolated singularity, the singularity is a pole”

False — f(z) = sin z / z at 0

“A function has a unique Laurent expansion about a point”
“∮γf=0\oint_\gamma f = 0 implies f is holomorphic inside γ\gamma”
“A small indentation around a simple pole contributes 2πi⋅2\pi{}i\cdotRes”

False — The indentation at 0 when computing ∫0∞sin⁡x/x\int_{0}^\infty \sin x/x dx

“f′(z) ≠ 0 everywhere implies f is injective”

False — f(z) = eᶻ on C\mathbb{C}

“C\mathbb{C} and the unit disc are biholomorphic (both are simply connected)”

False — Liouville's theorem