6720-2023-08

Complete the following:

1. (5 pts) Prove that if both \(f(z)\) and \(\overline {f(z)}\) are analytic in some domain, then \(f\) is the constant function in that domain.
2. (5 pts) Identify, with justification, the branch points of the function \[ f(z)=\frac {1}{\sqrt {z+1}} \] and propose a corresponding branch cut.

Prove the argument principle: Assume that \(f\) is a meromorphic function inside and on a simple closed contour \(C\) (oriented in the typical counter-clockwise fashion), that \(f\) has exactly \(N\) zeros (counting multiplicity) in this region all lying within \(C\), and that \(f\) has exactly \(M\) poles (counting order), all lying within \(C\). Prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)}\,dz=N-M. \]

Let \(f(z)\) be an entire function, and suppose there exists a non-negative constant \(C\) such that \(|f(z)|\le C|z|\) for every \(z\). Prove that \(f(z)=Az\), where \(A\) is a constant.

Evaluate the following integral by residue integration: \[ \int _0^\infty \frac {dx}{x^3+a^3},\qquad a>0. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Compute the following asymptotic expansions:

1. (5 pts) \(I(k)=\int _k^\infty e^{-t^2}\,dt\) as \(k\to 0^+\). You must compute the entire asymptotic expansion.
2. (5 pts) \(I(k)=\int _0^2 e^{-k(t^2-3t)}\,dt\) as \(k\to \infty \). You need only compute the leading order term in the expansion.