6720-2023-05

Complete the following:

1. (5 pts) Compute all values for \(\log (1-i)\).
2. (5 pts) Consider the function \(f(z)=z\log (z-2)\). Identify, with justification, the branch points of this function. In addition, choose a branch cut and identify the corresponding branch of \(f\).

Complete the following:

1. (5 pts) Compute the Laurent series expansion about the point \(z=0\) for \[ f(z)=\frac {1}{(z-i)(z+2)}, \] in the region \(1<|z|<2\).
2. (5 pts) Describe the singularities of the function \[ f(z)=\frac {z+1}{z\sin z}. \]

Complete the following:

1. (5 pts) Consider two entire functions with no zeros and having a ratio equal to unity at infinity. Use Liouville’s Theorem to show that they are in fact the same function. Justify all your steps, including satisfying the assumptions of Liouville’s theorem.
2. (5 pts) Let \(f(z)\) be analytic in and on a circle \(C\) with center \(w\) and radius \(\rho >0\). From the Cauchy Integral Formula show that \[ f(w)=\frac {1}{2\pi }\int _0^{2\pi } f\!\left (w+\rho e^{i\theta }\right )\,d\theta . \]

Evaluate the following integral: \[ I=\int _0^\infty \frac {\cos x}{x^2+1}\,dx. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Compute the following asymptotic expansions:

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