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'complex sine and cosine'
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| Title of object: |
complex sine and cosine |
| Canonical Name: |
ComplexSineAndCosine |
| Type: |
Definition |
| Created on: |
2004-10-21 18:49:22 |
| Modified on: |
2004-11-30 02:51:51 |
| Classification: |
msc:26A09 |
| Keywords: |
power series |
| Defines: |
complex sine, complex cosine, sine, cosine, goniometric formula |
Preamble:
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Content:
We define for all complex values of $z$:
\begin{itemize}
\item $\sin{z} := z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+-...$
\item $\cos{z} := 1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+-...$
\end{itemize}
Because these series converge for all real values of $z$, their radii of convergence are $\infty$, and therefore they also converge for all complex values of $z$ (by Abel's theorem). \,Thus they define holomorphic functions in the whole complex plane, i.e. entire functions.
Expanding the complex exponential functions $e^{iz}$ and $e^{-iz}$ to power series and separating the terms of even and odd degrees gives the generalized Euler's formulae
$$e^{iz} = \cos{z}+i\sin{z},\quad e^{-iz} = \cos{z}-i\sin{z}.$$
Adding, subtracting and multiplying these two formulae give respectively the two Euler's formulae
\begin{align}
\cos{z} = \frac{e^{iz}+e^{-iz}}{2},\quad\sin{z} = \frac{e^{iz}-e^{-iz}}{2i}
\end{align}
(which sometimes are used to define cosine and sine) and the ``fundamental formula of trigonometry''
$$\cos^2{z}+\sin^2{z} = 1.$$
As consequences of the generalized Euler's formulae one gets easily the addition formulas of sine and cosine:
$$\sin{(z_1+z_2)} = \sin{z_1}\cos{z_2}+\cos{z_1}\sin{z_2},$$
$$\cos{(z_1+z_2)} = \cos{z_1}\cos{z_2}-\sin{z_1}\sin{z_2};$$
so they are in $\mathbb{C}$ fully similar as in $\mathbb{R}$. \,It means that all {\em goniometric} formulae derived from these, such as
$$\sin{2z} = 2\sin{z}\cos{z},\,\,\,\sin{(\frac{\pi}{2}-z)} = \cos{z},
\,\,\,\sin^2{z} = \frac{1-\cos{2z}}{2},$$
have the old shape.
Another important consequence is that the functions $\sin$ and $\cos$ are periodic and have $2\pi$ as their
\PMlinkname{prime period}{ComplexExponentialFunction}:
$$\sin{(z+2\pi)} = \sin{z},\,\,\, \cos{(z+2\pi)} = \cos{z}\quad\forall z$$
The periodicity of the functions causes that their inverse functions, the {\em complex cyclometric functions}, are infinitely multivalued; they can be expressed via the complex logarithm and square root (see general power) as
$$\arcsin{z} = \frac{1}{i}\log(iz+\sqrt{1-z^2}),\,\,\,
\arccos{z} = \frac{1}{i}\log(z+i\sqrt{1-z^2}).$$
The derivatives of sine function and cosine function are obtained either from the series forms or from (1):
$$\frac{d}{dz}\sin{z} = \cos{z},\,\,\, \frac{d}{dz}\cos{z} = -\sin{z}$$
Cf. the \PMlinkname{higher derivatives}{HigherOrderDerivativesOfSineAndCosine}. |
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