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| author | bnewbold <bnewbold@alum.mit.edu> | 2010-06-30 06:16:45 +0000 |
|---|---|---|
| committer | bnewbold <bnewbold@adelie.robocracy.org> | 2010-06-30 06:16:45 +0000 |
| commit | 923d5b4198679221b1a1996747bfcc803352dd70 (patch) | |
| tree | 2fbc3b12657f43e751424fb91e9ba3ecd6e00626 | |
| parent | b4dc715e2d88e78600f6c2b48a1c6453166c5c31 (diff) | |
| download | afterklein-wiki-923d5b4198679221b1a1996747bfcc803352dd70.tar.gz afterklein-wiki-923d5b4198679221b1a1996747bfcc803352dd70.zip | |
tex fix?
| -rw-r--r-- | ClassJune26.page | 3 |
1 files changed, 1 insertions, 2 deletions
diff --git a/ClassJune26.page b/ClassJune26.page index 112eb6c..3724f4e 100644 --- a/ClassJune26.page +++ b/ClassJune26.page @@ -1,4 +1,3 @@ - Links to Josh's handwritten notes [link page 1](/L1p1.jpeg) @@ -162,7 +161,7 @@ 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}{}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}$ |