From 1eca287d3f83cf9133e42c644f9d809792f32cec Mon Sep 17 00:00:00 2001 From: luccul Date: Tue, 6 Jul 2010 05:00:00 +0000 Subject: forgot something apparently --- Problem Set 3.page | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/Problem Set 3.page b/Problem Set 3.page index 9045fa9..daae048 100644 --- a/Problem Set 3.page +++ b/Problem Set 3.page @@ -22,15 +22,15 @@ where $u_0(x)$ is the initial temperature distribution and $f(x,t)$ is the funda $$ \tau \frac{\partial u}{\partial t} - \lambda^2 \frac{\partial^2 u}{\partial x^2} = (u - u_0) $$ 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 propogation 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: +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 travelling with a certain velocity around the ring (but in opposite directions). At what velocity do they travel? +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 rederive the Laurent series expansion of $f$ without using the map $z \mapsto e^{iz}$. +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: $$ f(z) = \frac{1}{2\pi} \int_0^{2\pi} \frac{f(e^{i\theta})}{1 - e^{-i\theta}z} d\theta $$ @@ -38,7 +38,7 @@ 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 $\tau \in \C$ be a complex number that is not a multiple of $2\pi$. Suppose that $f$ satisfies: +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. -- cgit v1.2.3