yet more tex

1 files changed, 3 insertions, 3 deletions

diff --git a/ClassJune26.page b/ClassJune26.page
index 0d8d588..b5454a4 100644
--- a/ClassJune26.page
+++ b/ClassJune26.page
@@ -162,8 +162,8 @@ and add them up just fine, so we can exponentiate complex values of
 $z$. We know what happens to real values, what happens to pure imaginary
 ones? Let $y\in\mathbb{R}$. Then  
 
-$$\begin{array}{ccc} 
-ee^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
+$$\begin{array}{ccl} 
+e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
  & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
  & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
  & = & \cos y+i\sin y\end{array}$$
@@ -219,7 +219,7 @@ means that $\left(\begin{array}{c}
 \frac{\partial u}{\partial y}\end{array}\right)$ is $\left(\begin{array}{c}
 \frac{\partial u}{\partial x}\\
 \frac{\partial v}{\partial x}\end{array}\right)$ rotated by $\pi/2$. If we write 
-$\[\left(\begin{array}{c}
+$\left(\begin{array}{c}
 a\\
 b\end{array}\right)=\left(\begin{array}{c}
 \frac{\partial u}{\partial x}\\