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More damn exercises

  1. Solve the ODE

    \begin{displaymath} \frac{d^{2} \psi}{dz^2} + \frac{2 m}{\hbar^2} (en - m g z)\psi(z) = 0 \end{displaymath}

    What is this equation anyway? Plot each solution. Which is physically relevant?
  2. Find the first 5 roots of the function AiryAi[x] .
  3. Investigate the package ReIm. Plot the function $f(M) = - a_{2} t M^2 +a_{4} M^4 - H M$ (for $a_{2}>0, a_{4}>0, H>0$) over a range of $M$ from negative to positive. Find analytic expressions for its minima (as a function of $M$) for $t>0$ and $t<0$. Are all real?
  4. Find the zeroes of $x^4+x^3-2 x^2 + 4 x-24$. Use Select and FreeQ to select out only real roots.
  5. Find the Laplace transform of $t^2 \left( \exp \lambda t\right) \sin \omega t$
  6. Find the Fourier transform of $e^{a x^2 - b x}$
  7. Dot the matrix

    \begin{displaymath} \begin{array}{ccc} a & b & b\\ b & a & b\\ b & b & a \end{array}\end{displaymath}

    into the (column) vector $(1,2,1)$
  8. Find the eigenvalues and eigenvectors of the previous matrix. Are the eigenvectors normalized?
  9. Evaluate the sum (for real x)

    \begin{displaymath} \sum_{p=0}^{n} e^{i p x} \end{displaymath}

  10. Plot the gradient of the scalar function $f(x,y,z) = 2x^2- 3y^2 + 2 x z$ over the range $-2\leq x,y,z\leq +2$.


David Wood 2007-06-25