89 problems.
Compactness, connectedness, and functions of one real variable
- Limit of the averaging recurrence \(x_n=(x_{n-1}+x_{n-2})/2\) — Problem 1
- Continuous functions vanishing at \(\pm\infty\) are uniformly continuous — Problem 2
- A differentiable function with discontinuous derivative — Problem 3
- Uniform convergence of \(f\circ g_n\) for uniformly continuous \(f\) — Problem 4
- Uniform differentiability is equivalent to continuity of \(f'\) — Problem 5
- Distance between disjoint compact sets is attained — Problem 6
- Union of intersecting connected sets is connected — Problem 7
- Dini’s theorem for decreasing sequences on \([0,1]\) — Problem 8
- Splitting an uncountable subset of \([0,1]\) into two uncountable halves — Problem 9
Several variables
- Continuity and differentiability of \(xy/\sqrt{x^2+y^2}\) — Problem 1
- Real differentiability, partials without differentiability, and nowhere holomorphic maps \(\mathbb{R}^2\to\mathbb{R}^2\) — Problem 2; exact existing canonical owner
- Real differentiability, the inverse function theorem, and complex differentiability in \(\mathbb R^2\) — Problem 3
- Cauchy–Riemann equations in polar coordinates — Problem 4; exact existing canonical owner
- Real and complex implicit function theorems for \(9s^3-6st+t^2=0\) — Problem 5
- A \(C^1\) function with \(\|\nabla F(0,0)\|<1\) maps small disks into small intervals — Problem 6
- Implicit function theorem from the inverse function theorem — Problem 7
Conformal mapping
- Conformal map of the unit disk onto the upper half-plane — Problem 1
- Conformal map of a horizontal strip onto the unit disk — Problem 2
- Möbius map of the upper half-plane onto the disk and the image of a quadrant — Problem 3
- Conformal map of the right half-plane outside \(|z-i|\le 1\) onto the upper half-plane — Problem 4
- Conformal map of the disk minus a tangent disk onto the unit disk — Problem 5; exact existing canonical owner
- Conformal map from the intersection of \(|z-1|<2\) and \(|z+1|<2\) to the upper half-plane — Problem 6; exact existing canonical owner
- Conformal map of a non-concentric circular region onto an annulus — Problem 7
- Conformal map of the slit disk \(\mathbb D\setminus[0,1)\) onto the disk — Problem 8
- Conformal map of the slit plane \(\mathbb C\setminus(-\infty,0]\) onto the disk — Problem 9
- Conformal map of \(\mathbb C\setminus\{x\in\mathbb R:|x|\ge1\}\) onto the disk — Problem 10
- Conformal map of a slit lens onto the upper half-plane — Problem 11
- Möbius transformations preserving the extended real line — Problem 12
Integrals and Cauchy’s theorem
- Riemann integrability of \(g\) when \(|g(x)-g(y)|\le|f(x)-f(y)|\) — Problem 1
- Cauchy’s theorem for rectangles via Green’s theorem — Problem 2
- Locally uniform limits of analytic functions are analytic — Problem 3
- Limits of entire functions converging uniformly on line segments — Problem 4
- No polynomials converge uniformly to \(1/z\) on the unit circle — Problem 5
- Fourier transforms of compactly supported continuous functions are entire — Problem 6
- An entire function with \(|f(z)|\le|z|^{1/2}\) for large \(|z|\) is constant — Problem 7
- Cauchy-type integrals are analytic off the curve — Problem 8
- Continuous function analytic off the real line is entire — Problem 9; exact existing canonical owner
- Liouville’s theorem via Cauchy’s formula — Problem 10
Liouville, FTA, and power series
- Boundary behavior of power series with radius of convergence \(1\) — Problem 1
- Taylor coefficients of an entire function as contour integrals, uniform convergence on bounded sets, and uniform convergence on \(\mathbb{C}\) — Problem 2; exact existing canonical owner
- The ring of analytic functions on a domain has no zero divisors — Problem 3
- Which values \(f(1/n)\) an analytic function on the disk can take — Problem 4
- Fundamental theorem of algebra — Problem 5; exact existing canonical owner
- Entire functions with \(|f(z)|\ge|z|\) — Problem 6
- Absolute and uniform convergence of power series inside the disk of convergence — Problem 7
- Entire functions with \(f(z)/z^n\to0\) are polynomials of degree less than \(n\) — Problem 8
- Entire functions with \(|f(z)|\le\sqrt{|z|}\) for large \(|z|\) — Problem 9
- Uniform convergence of \(\sum\sin(nz)/2^n\) — Problem 10
Laurent expansions and singularities
- Laurent expansions of \(\frac{z+1}{z(z-1)}\) — Problem 1
- Laurent expansions of \(e^{1/z}\) and \(\cos(1/z)\) — Problem 2
- Laurent expansions of \(\frac{z+1}{z(z-1)^2}\) — Problem 3
- Laurent series and singularities of \(\frac{\sin^2 z}{z}\), \(ze^{1/z^2}\), and \(\frac{1}{z(4-z)}\) — Problem 4
- Entire functions with a pole at \(\infty\) are polynomials — Problem 5
- Functions on the Riemann sphere with a simple pole at \(2\) and a double pole at infinity — Problem 6; exact existing canonical owner
- Singularity at \(\infty\) of an entire function and its zeros — Problem 7
- Partial fraction expansion of \(\pi^2/\sin^2\pi z\) — Problem 8
Residues
- \(\int_0^\infty\frac{dx}{(1+x^2)(1+9x^2)}\) — Problem 1
- \(\int_0^\infty\frac{x\sin x}{x^2+a^2}\,dx\) — Problem 2; exact existing canonical owner
- \(\int_0^\infty\frac{\sqrt x}{(x+1)^2}\,dx\) — Problem 3
- \(\int_0^\infty\frac{\cos x-\cos 4x}{x^2}\,dx\) — Problem 4
- \(\int_0^\infty\frac{x^2}{(x^2+a^2)^2}\,dx\) — Problem 5
- The Dirichlet integral \(\int_0^\infty\sin x/x\,dx\) — Problem 6; exact existing canonical owner
- \(\int_0^\infty\frac{\sin x}{x(x^2+1)}\,dx\) — Problem 7
- \(\int_0^\infty\frac{\sqrt x}{1+x^2}\,dx\) — Problem 8
- \(\int_{-\infty}^\infty\frac{1+x^2}{1+x^4}\,dx\) — Problem 9
- \(\int_0^\infty\frac{\cos x}{(x^2+a^2)^2}\,dx\) — Problem 10
- \(\int_0^\infty\frac{\sin^3 x}{x^3}\,dx\) — Problem 11
- A contour integral equal to \(\frac{\sin n\theta}{\sin\theta}\) — Problem 12
- \(\int_0^{2\pi}\frac{d\theta}{(a+b\cos\theta)^2}\) — Problem 13
Rouché’s theorem
- Partial sums of the exponential series have no zeros in the unit disk — Problem 1
- Two roots of \(z^3+3z^2+bz+b^2\) in the unit disk — Problem 2
- Unique fixed point of \(ce^z\) on \(\{\operatorname{Re} z<1\}\) when \(|c|<1/3\) — Problem 3; exact existing canonical owner
- Roots of \(z^7-4z^3-1\) in the unit disk — Problem 4
- Solutions of \(e^z=az^n\) in the unit disk — Problem 5
- Local \(m\)-to-one behavior of an analytic function at a zero of order \(m\) — Problem 6
- Finite Blaschke products take each value in the disk \(n\) times — Problem 7
- Zeros of the exponential partial sums \(P_n\) and \(P_n-1\) in a disk — Problem 8
- Monic polynomials have modulus at least \(1\) somewhere on the unit circle — Problem 9
- Rouché’s theorem implies the Fundamental Theorem of Algebra — Problem 10; exact existing canonical owner
Schwarz lemma and reflection principle
- Analytic self-maps of the disk with unimodular boundary values are finite Blaschke products — Problem 1
- A bound on the difference quotient of a holomorphic function on a half-radius disc — Problem 2; exact existing canonical owner
- Injective conformal maps of \(\{\operatorname{Re} z>0,\ |z-1|>1\}\) onto the disc — Problem 3; exact existing canonical owner
- Density of \(f(\mathbb{H})\) for a holomorphic map of the half-plane real precisely on \(\mathbb{R}\) — Problem 4; exact existing canonical owner
- Sharp bound for \(|f'(0)|\) for maps of the disk into the right half-plane — Problem 5
- Proper self-maps of the disk with a single zero are \(\lambda z^k\) — Problem 6
- Schwarz reflection across the circle and functions real on the unit circle — Problem 7
- Subordination: \(g(|z|<r)\subset f(|z|<r)\) for univalent \(f\) — Problem 8
- Reflection principle for functions on the closed upper half-disk — Problem 9
- Schwarz–Pick inequality for the derivative — Problem 10