problems for review

  1. Consider the language \(L\) consisting of all words in alphabet a,b,c having no subwords aaa. Find the generating function
    $$
    f(x)=\sum |L_n| z^n,
    $$
    where \(L_n\) is the set of words of lengths \(n\) in \(L\). Estimate how fast \(\# L_n\) grows.
  2. If \(\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\) is the standard surface area form for the unit sphere
    \(S\)centered at the origin, find
    $$
    \int_S x^4\omega.
    $$
  3. For \(a\geq 0\), find the image of the half-plane \(\Im z>a\) under the mapping \(z\mapsto z^2\).
  4. Find the line integral
    $$
    \int_\gamma \bar{z}^2 dz
    $$
    over the arc \(\gamma=R\exp(i\phi), \phi\in[0,\phi_*]\) (oriented by increasing \(\phi\)).
  5. Estimate (with reasonable precision, say 5%) the integral
    $$
    \int_0^1 x^{500}(1-x)^{1500} dx.
    $$
  6. Find (using residues) the integrals
    $$
    \int_{-\infty}^\infty \frac{e^{iks}}{1+s^4}ds,
    $$
  7. Find the integral of \(\exp(ax+by)\) over the area given by inequalities
    $$
    \begin{array}{rl}
    x&\geq 0\\
    y&\geq 0\\
    x+y&\geq 1\\
    x+y&\leq 2\\
    \end{array}
    $$
  8. Solve the quasilinear PDE
    $$
    u_t+xu_x=u,
    $$
    subject to the initial condition \(u(x,0)=\phi(x)\). Does the solution remain smooth at all times, if \(\phi\) is smooth?
  9. Consider the solution to the wave equation
    $$
    u_{tt}=u_{xx}
    $$
    on the unit interval with the initial conditions
    $$
    u(x,0)=\phi(x), u_t(x,0)=0,
    $$
    where the function \(\phi\) is sketched below, and the fixed boundary condition on the right end \(u_x(1,t)=0\), and free boundary condition on the left end, \(u_x(0,t)=0\).

    Sketch the wave at times \(t=.25, .5, .75, 1\).

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