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author | joshuab <> | 2010-06-29 15:23:39 +0000 |
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committer | bnewbold <bnewbold@adelie.robocracy.org> | 2010-06-29 15:23:39 +0000 |
commit | d4481a0d023cacf22a1dcf9a35324ba5a42c63f6 (patch) | |
tree | 656ad1bead36af92eceb5acbce9b3550a5b1868c /ClassJune26.page | |
parent | bce1315f07aa8377378cbd938761d9accaf64649 (diff) | |
download | afterklein-wiki-d4481a0d023cacf22a1dcf9a35324ba5a42c63f6.tar.gz afterklein-wiki-d4481a0d023cacf22a1dcf9a35324ba5a42c63f6.zip |
tex
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-rw-r--r-- | ClassJune26.page | 7 |
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diff --git a/ClassJune26.page b/ClassJune26.page index b28d40e..faa516c 100644 --- a/ClassJune26.page +++ b/ClassJune26.page @@ -152,9 +152,10 @@ 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 -$\begin{array}e^{iy} & = & 1+iy+\frac{(iy)^{2}}{2!}+\frac{(iy)^{3}}{3!}+\frac{(iy)^{4}}{4!}+\frac{(iy)^{5}}{5!}+\cdots - & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots - & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots) +$\begin{array} +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}$ |