forgot something apparently

1 files 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.