Exam
6720-2021-05
Summer 2021 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Analytic functions **(a)** (12 points) Let \(u(x,y)=\dfrac{y}{x^2+y^2}\) be the real part of an analytic function \(f(z)\). Find the imaginary part \(v(x,y)\) and the function \(f(z)\). **(b)** (13 points) Let \(f(z)\), \(g(z)\) be two entire functions with no zeroes such that \[ \lim_{z\to\infty}\frac{f(z)}{g(z)}=1. \] Use Liouville's Theorem to show that \(f(z)\equiv g(z)\) for all \(z\). **(a)**
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Question 2
Bilinear transformations Consider the transformation on the extended complex plane \(\mathbb{C}_\infty\) \[ f(z)=w=z+\frac{1}{z}. \] **(a)** (7 points) Find all possible fixed points. **(b)** (9 points) Show that the image of the points in the upper half \(z\)-plane that are exterior to the circle \(|z|=1\) corresponds to the entire upper half \(w\)-plane. **(c)** (9 points) Find the image of the unit circle. **(a)**
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Question 3
Series and singularities **(a)** (13 points) Expand the function \[ f(z)=\frac{z}{(z-2)(z+i)} \] in a Laurent series in powers of \(z\) in \(1<|z|<2\). **(b)** (12 points) 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. \[ f(z)=\frac{e^{2z}-1}{z^2} \] **(a)**
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Question 4
Residue theory Show that \[ \int_0^\infty \frac{\cos(2x)}{(1+x^2)^2}\,dx=\frac{3\pi e^{-2}}{4}. \] **(a)**
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Question 5
Asymptotic expansions Find the complete asymptotic expansion of \[ I(k)=\int_0^\infty e^{-kt}t^{-1/3}\cos t\,dt,\qquad k\to\infty. \]
Topics Covered by This Exam
Branch Points, Branch Cuts, and Logarithms
Conformal Mapping and Bilinear Transformations
Contour Integration and Residue Calculus
Entire Functions and Liouville's Theorem
Holomorphic Functions and Cauchy-Riemann Equations
Isolated Singularities and Residues
Laurent Series and Annuli
Real Integral Evaluation by Complex Methods
Watson's Lemma and Laplace's Method