# 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 $$t=.25, .5, .75, 1$$.