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Even more exercises

  1. Find the integral

    \begin{displaymath} \int_{1}^{\infty}dx \frac{\tan \left(a x^2- b x\right)}{x^3} \delta(x^2 -c); \end{displaymath}

    Here $\delta(f(x))$ is the `Dirac $\delta$' function, known to Mathematica . Simplify as much as possible.
  2. Use ContourPlot to look at the real and imaginary parts of the function

    \begin{displaymath} f = \log\frac{z^2-2}{z^2 +1} \end{displaymath}

    after the replacement $z\rightarrow x + I y$. Superimpose plots of the two parts and comment.
  3. Examine the package ReIm (part of Mathematica ). For the function in the previous part, find the real and imaginary parts assuming $x$ and $y$ are real.
  4. Find the frequency solutions $\omega^2$ for the set of linear equations

    \begin{displaymath} \left[\begin{array}{ll} \omega^2 - \frac{K+G}{M} & \frac{G}{... ...ft(\! \begin{array}{l} u_{0} v_{0}\end{array} \!\right) = 0. \end{displaymath}

    Now find the eigenvectors.
  5. Sums: Make a table (using Table) of the index $p$ and the value of the sum

    \begin{displaymath} 4\sum_{k=1}^{p} \frac{(-1)^{k+1}}{2k-1} \end{displaymath}

    for $p=1$ to $p=100$. Now try the same for $p=1$ to $p=10$ using the sum

    \begin{displaymath} 16^{-n} \left(-\frac{2}{8 n+4}-\frac{1}{8 n+5}-\frac{1}{8 n+6}+\frac{4}{8 n+1}\right) \end{displaymath}

  6. Plot as a three-dimensional contour plot the function $2x^2 - y^2 + z^3 - 2 x y + z x - z$ over the range $-2\leq x,y,z\leq +2$ for the contour value 0. Now plot its gradient over the same range and superimpose the two graphics [use Show[{g1,g2}] Anything worth noticing?
next up previous
Next: About this document ... Up: Mma Previous: More damn exercises
David Wood 2007-06-25