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authorjoshuab <>2010-06-29 15:24:17 +0000
committerbnewbold <bnewbold@adelie.robocracy.org>2010-06-29 15:24:17 +0000
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@@ -153,7 +153,7 @@ 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
$\begin{array}
-e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
+ e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots\\
& = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
& = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
& = & \cos y+i\sin y\end{array}$