Question

3. (20 points.) Consider the integral \begin{equation} I(a) = \frac{1}{2\pi} \int_0^{2\pi} \frac{d\theta}{1 - 2a \cos \theta + a^2}, \end{equation} (3) where $a$ is complex. (a) Substitute $z = e^{i\theta}$, such that \begin{equation} 2 \cos \theta = z + \frac{1}{z}, \end{equation} (4) and express the integral as a contour integral along the unit circle going counter- clockwise. Locate the poles. (b) Evaluate the residues and show that \begin{equation} I(a) = \begin{cases} \frac{1}{1 - a^2}, & \text{if } |a| < 1, \\\\frac{1}{a^2 - 1}, & \text{if } |a| > 1. \end{cases} \end{equation} (5) (c) Plot $I(a)$ for real values of $a$. Plot real and imaginary part of $I(a)$ for complex $a$. Argue that $I(1)$ is divergent.

          3. (20 points.) Consider the integral
\begin{equation} I(a) = \frac{1}{2\pi} \int_0^{2\pi} \frac{d\theta}{1 - 2a \cos \theta + a^2}, \end{equation} (3)
where $a$ is complex.
(a) Substitute $z = e^{i\theta}$, such that
\begin{equation} 2 \cos \theta = z + \frac{1}{z}, \end{equation} (4)
and express the integral as a contour integral along the unit circle going counter-
clockwise. Locate the poles.
(b) Evaluate the residues and show that
\begin{equation} I(a) = \begin{cases} \frac{1}{1 - a^2}, & \text{if } |a| < 1, \\\\frac{1}{a^2 - 1}, & \text{if } |a| > 1. \end{cases} \end{equation} (5)
(c) Plot $I(a)$ for real values of $a$. Plot real and imaginary part of $I(a)$ for complex $a$.
Argue that $I(1)$ is divergent.

$3. (20 points.) Consider the integral I(a) = (1)/(2π)∫0^2π(dθ)/(1 - 2a cosθ + a^2), (3) where a is complex. (a) Substitute z = e^iθ, such that 2 cosθ = z + (1)/(z), (4) and express the integral as a contour integral along the unit circle going counter- clockwise. Locate the poles. (b) Evaluate the residues and show that I(a) = (1)/(1 - a^2), if |a| < 1, frac1a^2 - 1, if |a| > 1. (5) (c) Plot I(a) for real values of a. Plot real and imaginary part of I(a) for complex a. Argue that I(1) is divergent.$

Added by Jose D.

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