114 problems.
Fall 2009
- Parseval’s identity for power series and Liouville’s theorem
- Limit of integrals over large circular arcs when \(zf(z)\to k\) — Question 2
- Lagrange inversion for \(z-a-qf(z)=0\) via the residue theorem — Question 3
- \(\int_0^\infty\frac{x\sin x}{x^2+a^2}\,dx\)
- Cauchy–Riemann equations in polar coordinates
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx=\frac{\pi}{n\sin\frac{a\pi}{n}}\) for \(0<a<n\)
- Analyticity of the gamma function and the reflection formula
- Fundamental theorem of algebra via Rouché’s theorem — Question 8
- An entire function with \(|f(z)|\ge A|z|^N\) at infinity is a polynomial of degree at least \(N\)
- Entire univalent functions are affine/linear
- Fixed points of analytic self-maps of the disk — Question 11
- Conformal map of the disk minus a tangent disk onto the unit disk
- Entire functions omitting a bounded open set are constant — Question 13
- Entire functions with \(|f(z)|\le M|z|^2\) are quadratic polynomials — Question 14
- Termwise differentiation of locally uniformly absolutely convergent series of analytic functions — Question 15
- Goursat’s theorem with a bounded isolated singularity — Question 16
- Limit of integrals over small circular arcs when \((z-a)f(z)\to A\) — Question 17
- Uniform continuity of \(z^2\) on disks but not on \(\mathbb C\) — Question 18
- Hadamard’s example: ill-posedness of the Cauchy problem for Laplace’s equation — Question 19
Fall 2011
- Parseval’s identity for power series and Liouville’s theorem
- Limit of integrals over large circular arcs when \(zf(z)\to k\) — Question 2
- Lagrange inversion for \(z-a-qf(z)=0\) via the residue theorem — Question 3
- \(\int_0^\infty\frac{x\sin x}{x^2+a^2}\,dx\)
- Cauchy–Riemann equations in polar coordinates
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx=\frac{\pi}{n\sin\frac{a\pi}{n}}\) for \(0<a<n\)
- Analyticity of the gamma function and the reflection formula
- Fundamental theorem of algebra via Rouché’s theorem — Question 8
- An entire function with \(|f(z)|\ge A|z|^N\) at infinity is a polynomial of degree at least \(N\)
- Entire univalent functions are affine/linear
- Fixed points of analytic self-maps of the disk
- Conformal map of the disk minus a tangent disk onto the unit disk
- Entire functions omitting a bounded open set are constant — Question 13
- Entire functions omitting a bounded open set are constant — Question 14
- Parseval’s identity for power series and Liouville’s theorem — Question 15
- Entire functions with \(|f(z)|\le M|z|^2\) are quadratic polynomials — Question 16
- Termwise differentiation of locally uniformly absolutely convergent series of analytic functions — Question 17
- Goursat’s theorem with a bounded isolated singularity — Question 18
- Limit of integrals over small circular arcs when \((z-a)f(z)\to A\) — Question 19
- Uniform continuity of \(z^2\) on disks but not on \(\mathbb{C}\) — Question 20
- Hadamard’s example: ill-posedness of the Cauchy problem for Laplace’s equation — Question 20, parts (1) and (2)
Spring 2014
- \(\int_\gamma z^n\,dz\) over circles centered at the origin without Cauchy’s theorem — Question 1, part 1
- \(\int_\gamma z^n\,dz\) over circles not enclosing the origin without Cauchy’s theorem — Question 1, part 2
- \(\int_{|z|=r}\frac{dz}{(z-a)(z-b)}\) for \(|a|<r<|b|\) without Cauchy’s theorem — Question 1, part 3
- Parseval’s identity for power series and Liouville’s theorem — Question 2
- Limit of integrals over large circular arcs when \(zf(z)\to k\) — Question 3
- \(\int_0^\infty\frac{x\sin x}{x^2+a^2}\,dx\)
- Cauchy–Riemann equations in polar coordinates
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx=\frac{\pi}{n\sin\frac{a\pi}{n}}\) for \(0<a<n\)
- Analyticity of the gamma function and the reflection formula — Question 7
- Fundamental theorem of algebra via Rouché’s theorem — Question 8
- An entire function with \(|f(z)|\ge A|z|^N\) at infinity is a polynomial of degree at least \(N\)
- Entire univalent functions are affine/linear
- Fixed points of analytic self-maps of the disk
- Conformal map of the disk minus a tangent disk onto the unit disk
Fall 2015
- The ratio limit bounds the root limit — Question 1
- Blaschke factors are involutive automorphisms of the disk — Question 2
- \(2^{n-1}\prod_{k=1}^{n-1}\sin(k\pi/n)=n\), polar Cauchy–Riemann equations, and holomorphy of \(\log z\)
- Limit of integrals over vertical segments when \(f(x+iy)\to A\) uniformly — Question 4
- Cauchy’s integral formula for an exterior region — Question 5
- Liouville’s theorem via \(\int_{|z|=R}\frac{f(z)}{(z-a)(z-b)}\,dz\) — Question 6
- The Gaussian \(e^{-\pi x^2}\) is its own Fourier transform for complex frequencies — Question 7
- Taylor coefficient ratios detect a pole on the unit circle — Question 8
- Accumulating zeros force an essential singularity — Question 9
- Entire functions with \(f(z)\to\infty\) are polynomials — Question 10
- Laurent series expansions in annuli
- Analytic functions real on a closed curve are constant — Question 12
- Roots of \(z^4-6z+3\) in \(|z|<1\) and \(1<|z|<2\) — Question 13
- \(z^4+2z^3-2z+10\) has one root in each open quadrant
- Schwarz lemma for functions with positive real part — Question 15
- Convergence of the iterates \(\sum f^n\) of a non-rotation self-map of the disk fixing \(0\) — Question 16
- Area of the image of a disk under a univalent function — Question 17
- Area of the image of an annulus under a univalent Laurent series — Question 18
Spring 2015
- Termwise differentiation of locally uniformly absolutely convergent series of analytic functions — Question 1
- Locally uniform convergence of analytic functions via \(L^1\) convergence on circles — Question 2
- Analytic functions with equal modulus differ by a unimodular constant — Question 3
- Analytic functions unimodular on the unit circle are finite Blaschke products — Question 4
- \(z^{-m}\) is not a uniform limit of polynomials on an annulus — Question 5
- Laurent series of \(\frac1z+\frac{1}{z^2-1}\) in all annuli — Question 6
- Entire functions of polynomial growth are polynomials — Question 7
- An entire function with \(|f(z)|\ge A|z|^N\) at infinity is a polynomial of degree at least \(N\)
- Zeros of analytic functions accumulating at the boundary and the identity theorem — Question 9
- Uniqueness and the Schwarz reflection principle for harmonic functions — Question 10
- Sup norm on a smaller disk bounded by the \(L^1\) norm for holomorphic functions — Question 11
- Casorati–Weierstrass for accumulating poles — Question 12
- Residue computations: rational, trigonometric, and Fourier-type integrals — Question 13
- Residue computations: \(\frac{\sin x}{x}\), Mellin-type, and logarithmic integrals — Question 14
- \(P_n(z)=\sum_{k=1}^n kz^{k-1}\) has no zeros in \(|z|<r<1\) for large \(n\)
- Analytic functions mapping a closed curve into \(\mathbb R\) are constant — Question 16
- Partial fraction expansion of \(\pi^2/\sin^2\pi z\) — Question 17
- Univalent functions on a punctured plane are Möbius transformations — Question 18
- Convergence of the iterates \(\sum f^n\) of a non-rotation self-map of the disk fixing \(0\) — Question 19
- Harnack-type bounds for \(|f(z)|\) for self-maps of the disk — Question 20
- Conformal maps of crescents, lunes, and slit disks — Question 21
- Möbius transformations mapping one circle to another with two prescribed points — Question 22
- Vitali’s convergence theorem — Question 23
- Integrals of \(|\psi_\alpha'|^2\) and \(|\psi_\alpha'|\) over the disk for a Blaschke factor — Question 24
- The Joukowski map \(-\frac12(z+1/z)\) from the upper half-disk to the upper half-plane — Question 25
- Maximum modulus principle inside a closed contour — Question 26
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx\) and \(\int_0^\infty\frac{\log x}{(1+x^2)^2}\,dx\) — Question 27
- \(P_n(z)=\sum_{k=1}^n kz^{k-1}\) has no zeros in \(|z|<r<1\) for large \(n\)
- Sup norm on a smaller disk bounded by the \(L^1\) norm for holomorphic functions — Question 29
- Integrals of \(|\psi_\alpha'|^2\) and \(|\psi_\alpha'|\) over the disk for a Blaschke factor — Question 30
- Maximum modulus principle inside a closed contour — Question 31
- \(\int_0^\infty\frac{x^{a-1}}{1+x^n}\,dx\) and \(\int_0^\infty\frac{\log x}{(1+x^2)^2}\,dx\) — Question 32
- \(P_n(z)=\sum_{k=1}^n kz^{k-1}\) has no zeros in \(|z|<r<1\) for large \(n\)
- Sup norm on a smaller disk bounded by the \(L^1\) norm for holomorphic functions — Question 34
Fall 2016
- Harmonic conjugate of a \(C^3\) harmonic function on a disc
- Orthogonality of level curves of \(\operatorname{Re} f\) and \(\operatorname{Im} f\) — Question 2
- Complex line integrals and the ML estimate for \(\int_{|z|=R}\frac{\log z}{z^2}\,dz\) — Question 3
- Polynomial growth of entire functions and Liouville for bounded real part
- All roots of \(z^7-5z^3+12\) lie between \(|z|=1\) and \(|z|=2\)
- Residue at a pole of order \(m\) as \(\lim_{\tau\to a}\frac{d^{m-1}}{d\tau^{m-1}}\bigl((\tau-a)^m F(\tau)\bigr)\), and \(\oint_{|z|=4}\frac{e^\tau}{(\tau^2+\pi^2)^2}\,d\tau\)
- Conformal map of the upper half-plane onto a half-strip — Question 7
- Fourier transform of \(1/\cosh\pi x\) — Question 8