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Week 11 Worksheet

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Infinite Real Integrals

    1. Explain why $\displaystyle\int\limits_{-\infty}^{\infty} \frac{1}{t^2+1}\intd t$ exists.
    2. Use Cauchy's Residue Theorem to evaluate $\displaystyle\int\limits_{-\infty}^{\infty} \frac{1}{t^2+1}\intd t$
    3. Evaluate $\displaystyle\int\limits_{-\infty}^{\infty} \frac{1}{t^2+1}\intd x$ using only calculus and limits.
    1. Use Cauchy's residue theorem to evaluate, the integral $\displaystyle\int\limits_{-\infty}^{\infty} \frac{\exp(2 it)}{t^2+1} \intd t$
    2. By taking real and imaginary parts, calculate $\displaystyle\int\limits_{-\infty}^{\infty} \frac{ \cos(2t)}{t^2+1} \intd t$ and $\displaystyle\int\limits_{-\infty}^{\infty} \frac{ \sin (2t)}{t^2+1} \intd t$
    3. How can you tell, without evaluating the integrals, that one of them integrals is zero?
  3. Why does the contour from the ``Infinite Real Integrals'' video fail when we try to integrate \[ \int\limits_{-\infty}^{\infty} \frac{\exp(-2 it)}{t^2+1} \intd t \] using our approach? By choosing a different contour, explain how one could evaluate this integral using Cauchy's Residue Theorem.
  4. Use Cauchy's Residue Theorem to evaluate the following real integrals.
    1. $\displaystyle\int\limits_{-\infty}^{\infty} \frac{1}{(t^2+1)(t^2+3)} \intd t$
    2. $\displaystyle\int\limits_{-\infty}^{\infty} \frac{1}{28+11t^2+t^4} \intd t$
  5. By considering the function \[ f(z) = \frac{\exp(iz)}{z^2+4z+5} \] integrated around a suitable contour, show that \[ \int\limits_{-\infty}^{\infty} \frac{\sin t}{t^2+4t+5} \intd t = \frac{-\pi\sin 2}{e}. \]
  6. This question is about the integral $\displaystyle\int\limits_0^{2\pi} \frac{1}{13+5\cos t} \intd t$
    1. Use the substitution $z=e^{it}$ to show that \[ \int\limits_0^{2\pi} \frac{1}{13+5\cos t} \intd t = \frac{2}{i} \int_{\gamma} \frac{1}{5z^2+26z+5} \intd z \] where $\gamma(t) = e^{it}$ on $[0,2\pi]$.
    2. Show that \[ f(z)=\dfrac{1}{5z^2+26z+5} \] has simple poles at $z=-5$ and $z=-1/5$. Show that $\Res(f,-1/5) = 1/24$.
    3. Use Cauchy's Residue Theorem to show that \[ \int\limits_0^{2\pi} \frac{1}{13+5\cos t} \intd t = \frac{\pi}{6} \]
  7. Use \[ \cos t = \dfrac{\exp(it) + \exp(-it)}{2} \] to convert the following real integrals into complex integrals around the unit circle in the complex plane. Then apply Cauchy's Residue Theorem to evaluate them.
    1. $\displaystyle\int\limits_0^{2\pi} 2(\cos t)^3 + 3 (\cos t)^2 \intd t$
    2. $\displaystyle\int\limits_0^{2\pi} \frac{1}{1+\cos^2 t} \intd t$

Summing Series

  1. Use the method from the ``Summing Series'' video to show that $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^4} = \dfrac{\pi^4}{90}$
  2. Why doesn't the method from the ``Summing Series'' video work on the series $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^3}$?