path: root/Problem Set 3.page

# Problems

1. The delta distribution $\delta(x)$ is a periodic ``generalized function'' defined by the property that
$$ \frac{1}{2\pi} \int_0^{2 \pi} \delta(x) f(x) dx = f(0) $$
Using this definition, compute the Fourier series of $\delta(x)$.  

2. Use the formula
$$ 1 + a + a^2 + \cdots + a^n = \frac{1 - a^{n+1}}{1-a} $$
to derive the following trigonometric identities:
$$ 1 + 2 \sum_{k = 0}^{N} cos(kx) = \sum_{k = -N}^{N} e^{ikx} = \frac{\sin(\frac{(2N+1)x}{2})}{\sin(\frac{x}{2})} $$
Graph the function on the right and see what it looks like as $N$ grows large. 

3. Solve the heat equation
$$ \frac{\partial f}{\partial t} = \kappa \frac{\partial^2 f}{\partial x^2} $$
with periodic boundary conditions, and initial conditions given by the delta distribution.  This is called the ``fundamental solution'' of the heat equation.  Show that the Fourier series defining $f(x,t)$ converges when $t$ is positive and diverges when $t$ is negative.  What happens at $t = 0$?

4. Show that the general periodic solution of the heat equation can be written as follows:
$$ u(x,t) = \int_0^{2\pi} f(x-y,t) u_0(y) dy $$
where $u_0(x)$ is the initial temperature distribution and $f(x,t)$ is the fundamental solution.  (Hint: Compute the Fourier coefficients of both sides of the equation).

5. The heat equation only makes sense when the heat conducting object is insulated, i.e. does not conduct heat to its surroundings.  To model an uninsulated wire whose surroundings are maintained at a constant temperature $u_0$ (or to model diffusion in a leaky pipe), the following equation can be used:
$$ \tau \frac{\partial u}{\partial t} - \lambda^2 \frac{\partial^2 u}{\partial x^2}  = (u - u_0) $$
Use Fourier series to solve this equation in the case of a circular wire.  How does the solution depend on the magnitudes of the positive constants $\kappa$ and $\tau$?  

6. The wave equation is a partial differential equation that models the propagation of disturbances in a medium (for example, the vibrations of a metal object that has been struck by a hammer).  In the case of a one-dimensional object it is given by:
$$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $$
Use Fourier series to solve the wave equation in the case of a vibrating ring.  Interpret the solution as a superposition of two waves traveling with a certain velocity around the ring (but in opposite directions).  At what velocity do they travel?

7. Write the Cauchy-Riemann equations in polar coordinates, i.e. express them as a relationship between $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$.

8. Let $f$ be a holomorphic function on an annulus.  Write
$$ f(r,\theta) = \sum_n a_n(r)e^{in\theta} $$
as a Fourier series whose coefficients depend on $r$.  Use the Cauchy-Riemann equations in polar coordinates to derive the Laurent series expansion of $f$ without using the map $z \mapsto e^{iz}$. 

9. Let $f(z)$ be a holomorphic function defined on a region that contains the disk of radius $1$.  Derive the following variant of the Cauchy integral formula:
$$ f(z) = \frac{1}{2\pi} \int_0^{2\pi} \frac{f(e^{i\theta})}{1 - e^{-i\theta}z} d\theta $$
Hint: Expand the right hand side using the formula for a geometric series:
$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n $$
Then check that the coefficients of the resulting power series are the same as the power series coefficients for $f(z)$.  Note that if you write $z = re^{i \psi}$ then this formula appears very similar to the general solution of the heat equation.

10. Let $f(z)$ be a holomorphic function defined on the entire complex plane, and let $\tau \in \mathbb{C}$ be a complex number that is not a multiple of $2\pi$.  Suppose that $f$ satisfies:
$$ f(z + 2\pi) = f(z) $$
$$ f(z + \tau) = f(z) $$
(Such a function is said to be doubly periodic).  Show that $f$ is constant.  Hint: Write down holomorphic Fourier series for $f(z)$ and $f(z+\tau)$, and compare their Fourier coefficients.

# Solutions