6720-2024-05

Complete the following:

1. (8 pts) Suppose that \(f\) is analytic in some domain with purely real values. Show that \(f\) is the constant function.
2. (12 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.

1. (10 pts) Compute the Laurent series expansion about the point \(z=0\) for \[ f(z)=\frac {i+2}{(z+i)(z-2)}, \] in the region \(1<|z|<2\).
2. (10 pts) Discuss the type of singularity (removable, pole and order, essential, branch, cluster). If the type is a pole give the strength of the pole, and give the nature (isolated or not). Include the point at infinity. \[ f(z)=\frac {e^{2z}-1}{z} \]

Let \(f\) be an entire function, with \(|f(z)|\le C|z|^2\) for all \(z\), where \(C\) is a constant. Show that \(f(z)=az^2+bz\) for some constants \(a,b\).

Show that \[ \int _0^\infty \frac {\cos 3x}{(x^2+1)^2}\,dx=\frac {\pi }{e^3}. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Compute the complete asymptotic expansion in powers of \(k\) of \[ I(k)=\int _0^\infty e^{-kt^2}\frac {\sin (t^4)}{t^{7/2}}\,dt,\qquad k\to \infty . \]