Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

All ODE PDE abelian analyticity anova baire calculus of variations canonical forms cauchy-riemann compactness completeness complex conformal connectedness continuity contour integration convergence differentiation distributions fields fixed point galois green groups harmonic injectivity integral equations integration laurent limsup linear algebra liouville lp markov measure mechanics metric spaces multivariable multivariate numerical p-groups permutations polynomials power series probability quadratic forms real analysis regression residues rings sampling separability separation sequences series singularities stability statistics sylow testing topology uniform convergence wronskian zeros

#1 · Complex Analysis, Algebra & Topology › Cauchy Theory

“Bounded real part ⇒ entire function is constant — fails if only |f| is bounded on a half-plane” — false

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complexliouville

#2 · Complex Analysis, Algebra & Topology › Singularities and Residues

“Zeros of a non-constant analytic function can accumulate inside the domain” — false

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complexzeros

#3 · Complex Analysis, Algebra & Topology › Singularities and Residues

“|f| bounded near an isolated singularity ⇒ pole” — false

Counterexample: f(z) = sin(z)/z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complexsingularities

#4 · Complex Analysis, Algebra & Topology › Analytic Functions

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

Counterexample: f(z) = (z̄)²/z for z ≠ 0, f(0) = 0

CR hold at 0, but the difference quotient along z = t(1+i) differs from the one along the real axis.

complexcauchy-riemann

#5 · Complex Analysis, Algebra & Topology › Analytic Functions

“A harmonic function on any domain has a harmonic conjugate” — false

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complexharmonic

#6 · Complex Analysis, Algebra & Topology › Analytic Functions

“Zeros of a non-constant holomorphic function cannot accumulate” — false

Counterexample: sin(1/z) on ℂ∖{0}

Zeros $1/ (nπ)$ accumulate at 0, which lies outside the domain. Inside the domain zeros are always isolated.

complexzeros

#7 · Complex Analysis, Algebra & Topology › Analytic Functions

“C^∞ implies analytic” — false

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complexanalyticity

#8 · Complex Analysis, Algebra & Topology › Cauchy Theory

“If f is holomorphic on the domain enclosed by γ except possibly at isolated points, and ∮_γ f = 0, then f has no singularity inside” — false

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complexcontour integration

#9 · Complex Analysis, Algebra & Topology › Cauchy Theory

“Cauchy's theorem applies to any domain without singularities of f” — false

Counterexample: f(z) = 1/z on the annulus ½ < |z| < 2

f is holomorphic there but $\oint_{∣ z ∣ = 1}$ dz $/ z = 2 π i \neq = 0$ — the annulus is not simply connected.

complex

#10 · Complex Analysis, Algebra & Topology › Cauchy Theory

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

Counterexample: f(z) = eᶻ on {Re z < 0}

|eᶻ| < 1 there, but f is not constant. Liouville needs the whole plane.

complexliouville

#11 · Complex Analysis, Algebra & Topology › Cauchy Theory

“An entire function omitting one value is constant” — false

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complexliouville

#12 · Complex Analysis, Algebra & Topology › Cauchy Theory

“|f| attains its minimum on the boundary for holomorphic f” — false

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complex

#13 · Complex Analysis, Algebra & Topology › Singularities and Residues

“If |f| is bounded near an isolated singularity, the singularity is a pole” — false

Counterexample: f(z) = sin z / z at 0

Bounded ⇒ removable (Riemann). Poles have |f| $\to \infty$ .

complexsingularities

#14 · Complex Analysis, Algebra & Topology › Singularities and Residues

“A function has a unique Laurent expansion about a point” — false

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complexlaurent

#15 · Complex Analysis, Algebra & Topology › Singularities and Residues

“∮_γ f = 0 implies f is holomorphic inside γ” — false

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complexresidues

#16 · Complex Analysis, Algebra & Topology › Singularities and Residues

“A small indentation around a simple pole contributes 2πi·Res” — false

Counterexample: The indentation at 0 when computing ∫₀^∞ sin x/x dx

A half-circle contributes $iπ \cdot$ Res; using $2 π i$ gives twice the right answer.

complexresidues

#17 · Complex Analysis, Algebra & Topology › Zeros and Mappings

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

Counterexample: f(z) = eᶻ on ℂ

f′ = eᶻ never vanishes, yet $f (z) = f (z + 2 π i)$ . Non-vanishing derivative gives only local injectivity.

complexinjectivity

#18 · Complex Analysis, Algebra & Topology › Zeros and Mappings

“ℂ and the unit disc are biholomorphic (both are simply connected)” — false

Counterexample: Liouville's theorem

A biholomorphism 𝔻 $\to C$ would invert to a bounded entire function. $C$ is the sole exception in the Riemann mapping theorem.

complexconformal