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cosine_function [2014/01/29 19:48] nikolaj |
cosine_function [2014/03/21 11:11] (current) |
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===== Cosine function ===== | ===== Cosine function ===== | ||
==== Function ==== | ==== Function ==== | ||
- | | @#FFBB00: $\mathrm{\cos}: \mathbb C\to\mathbb C$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\mathrm{\cos}: \mathbb C\to\mathbb C$ | |
- | | @#FFBB00: $\cos(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k)!}z^{2n} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $\cos(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k)!}z^{2n} $ | |
==== Discussion ==== | ==== Discussion ==== | ||
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^ $\cos(\theta) = \frac{1}{2}(\mathrm e^{i\theta}+\mathrm e^{-i\theta}) $ ^ | ^ $\cos(\theta) = \frac{1}{2}(\mathrm e^{i\theta}+\mathrm e^{-i\theta}) $ ^ | ||
- | i.e. if $\zeta(\theta):=\mathrm e^{i\theta}$, then $\frac{1}{2}(\zeta+\overline{\zeta})=\cos$. | + | i.e. if $\zeta:=\mathrm e^{i\theta}$, then $\zeta+\overline{\zeta}=2\cos(\theta)$. |
==== Parents ==== | ==== Parents ==== | ||
- | === Requirements === | + | === Context === |
[[Infinite sum of complex numbers]], | [[Infinite sum of complex numbers]], | ||
[[Factorial function]] | [[Factorial function]] | ||
=== Related === | === Related === | ||
[[Exponential function]], [[Sine function]] | [[Exponential function]], [[Sine function]] |