23 problems.
- Polynomial growth of entire functions, vanishing in a sector, products of distances on \(S^1\), and bounded real part
- A nowhere-vanishing holomorphic function of constant modulus on \(\partial D\) is constant
- Maximum principle for harmonic functions
- A holomorphic function with \(|f|=1\) on the unit circle, or \(|f|\geq 1\) with an interior point of modulus less than \(1\), covers the disc
- An analytic function real on a closed curve is constant
- Simple zeros of \(\sin(\pi z)\) at the integers, and \(\operatorname{Res}_{z=n}\frac{1}{\sin(\pi z)}\)
- The integral of \(1/(1+x^4)\) over \(\RR\) and the poles of \(1/(1+z^4)\)
- \(\int_{-\infty}^{\infty}\frac{x\sin x}{x^2+a^2}\,dx=\pi e^{-a}\) for \(a>0\)
- The Fourier transform of \((1+x^2)^{-2}\)
- The integral of \((1+x^2)^{-n-1}\) over \(\RR\)
- \(\int_0^{2\pi}\frac{d\theta}{(a+\cos\theta)^2}=\frac{2\pi a}{(a^2-1)^{3/2}}\) for \(a>1\)
- \(\int_0^{2\pi}\frac{d\theta}{a+b\cos\theta}=\frac{2\pi}{\sqrt{a^2-b^2}}\) for \(a>|b|\)
- \(\int_0^1\log(\sin\pi x)\,dx=-\log 2\)
- \(\int_0^\infty\frac{\log x}{x^2+a^2}\,dx=\frac{\pi}{2a}\log a\) for \(a>0\)
- The integral of the log of the chord length over the circle
- \(\int_0^\infty\frac{x^{a-1}}{1+x^3}\,dx=\frac{\pi}{3}\csc\frac{\pi a}{3}\)
- \(\int_0^{\pi/2}\frac{d\theta}{a+\sin^2\theta}=\frac{\pi}{2\sqrt{a(a+1)}}\) for \(a>0\)
- An isolated singularity at \(0\) is essential if a nonconstant analytic \(f\) on \(|z|>0\) has zeros accumulating at \(0\)
- \(z^4+2z^3-2z+10\) has one root in each open quadrant
- Injective entire functions are of the form \(az+b\)
- Fundamental theorem of algebra via Rouché’s theorem and the maximum modulus principle
- Number of roots of \(4z^4-6z+3\) in \(|z|<1\) and \(1<|z|<2\)
- \(z\tan z-a\) has only real roots for \(a>0\)