tex

1 files changed, 4 insertions, 3 deletions

diff --git a/ClassJune26.page b/ClassJune26.page
index 3724f4e..18f2292 100644
--- a/ClassJune26.page
+++ b/ClassJune26.page
@@ -161,11 +161,12 @@ We can raise complex numbers to powers, divide by the real denominators,
 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}{}ee^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
+
+$$\begin{array}{ccc} 
+ee^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
  & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
  & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
- & = & \cos y+i\sin y\end{array}$
-
+ & = & \cos y+i\sin y\end{array}$$
 
 Substituting $y=\pi$, we recover Euler's famous identity,
 $e^{i\pi}=-1.$