6720-2025-05

1. (10 points) Prove that if both \(f(z)\) and \(\overline {f(z)}\) are holomorphic in a simply connected domain then \(f\) is a constant function in that domain.
2. (10 points) Identify, with justification, the branch points of the function \[ f(z)=\frac {1}{\sqrt {(z+1)(z-i)}}. \] Propose a corresponding branch cut.

Show that the function \(p(z)=z^4+2z^2-z+1\) has exactly one zero in each of the four quadrants.

Use residue calculus to show that \[ \int _0^\infty \frac {\log x}{x^3-1}\,dx=\frac {4\pi ^2}{27}. \] Hint: Use an appropriate branch of the function \((\log z)/(z^3-1)\) and a sector of aperture \(2\pi /3\).

Let \(c=\sqrt {2}-1\). Use the method of steepest descent to show that \[ \int _{-\infty }^{\infty } e^{ikx}(1+x^2)^{-k}\,dx \sim \sqrt {\frac {\pi (1-c)}{k}}\,e^{-ck}(2c)^{-k} \] as \(k\to \infty \). Hint: You do not need to work out the exact steepest descent contour, but give a sketch of what it looks like. You do not need to give full justification for the deformation from the original contour to the steepest descent contour, but at least write down an idea of why it should be true.