PlanetMath: complex sine and cosine

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complex sine and cosine

(Definition)

We define for all complex values of :

$\displaystyle\sin{z} := z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+-\ldots$
$\displaystyle\cos{z} := 1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+-\ldots$

Because these series converge for all real values of

, their radii of convergence are $\infty$ , and therefore they converge for all complex values of

(by a known theorem of Abel; cf. the entry power series), too. Thus they define holomorphic functions in the whole complex plane, i.e. entire functions (to be more precise, entire transcendental functions). The series also show that sine is an odd function and cosine an even function.

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 formulas

$\displaystyle e^{iz} = \cos{z}+i\sin{z},\quad e^{-iz} = \cos{z}-i\sin{z}.$

Adding, subtracting and multiplying these two formulae give respectively the two Euler's formulae

$\displaystyle \cos{z} = \frac{e^{iz}+e^{-iz}}{2},\quad\sin{z} = \frac{e^{iz}-e^{-iz}}{2i}$

(1)

(which sometimes are used to define cosine and sine) and the “fundamental formula of trigonometry”

$\displaystyle \cos^2{z}+\sin^2{z} = 1.$

As consequences of the generalized Euler's formulae one gets easily the addition formulae of sine and cosine:

$\displaystyle \sin{(z_1+z_2)} = \sin{z_1}\cos{z_2}+\cos{z_1}\sin{z_2},$

$\displaystyle \cos{(z_1+z_2)} = \cos{z_1}\cos{z_2}-\sin{z_1}\sin{z_2};$

so they are in $\mathbb{C}$ fully similar as in $\mathbb{R}$ . It means that all goniometric formulae derived from these, such as

$\displaystyle \sin{2z} = 2\sin{z}\cos{z},\,\,\,\sin{(\pi-z)} = \sin{z}, \,\,\,\sin^2{z} = \frac{1-\cos{2z}}{2},$

have the old shape. See also the persistence of analytic relations.

The addition formulae may be written also as

$\displaystyle \sin{(x+iy)} = \sin{x}\cosh{y}+i\cos{x}\sinh{y},$

$\displaystyle \cos{(x+iy)} = \cos{x}\cosh{y}-i\sin{x}\sinh{y}$

which imply, when assumed that $x,\,y\in\mathbb{R}$ , the results

Re $\displaystyle (\sin(x+iy)) = \sin{x}\cosh{y},$ Im $\displaystyle (\sin(x+iy)) = \cos{x}\sinh{y},$

Re $\displaystyle (\cos(x+iy)) = \cos{x}\cosh{y},$ Im $\displaystyle (\cos(x+iy)) = -\sin{x}\sinh{y}.$

Thus we get the modulus estimation

$\begin{displaymath}\begin{array}{l} \vert\sin(x+iy)\vert = \sqrt{\sin^2{x}\cosh^... ...^2{x}\,\cdot 1 +\sinh^2{y}} \ge \vert\sinh{y}\vert, \end{array}\end{displaymath}$

which tends to infinity when

moves to infinity along any line non-parallel to the real axis. The modulus of $\cos(x+iy)$ behaves similarly.

Another important consequence of the addition formulae is that the functions $\sin$ and $\cos$ are periodic and have $2\pi$ as their prime period:

$\displaystyle \sin{(z+2\pi)} = \sin{z},\quad \cos{(z+2\pi)} = \cos{z}\quad\forall z$

The periodicity of the functions causes that their inverse functions, the complex cyclometric functions, are infinitely multivalued; they can be expressed via the complex logarithm and square root (see general power) as

$\displaystyle \arcsin{z} = \frac{1}{i}\log(iz+\sqrt{1-z^2}),\,\,\, \arccos{z} = \frac{1}{i}\log(z+i\sqrt{1-z^2}).$

The derivatives of sine function and cosine function are obtained either from the series forms or from (1):

$\displaystyle \frac{d}{dz}\sin{z} = \cos{z},\,\,\, \frac{d}{dz}\cos{z} = -\sin{z}$

Cf. the higher derivatives.

"complex sine and cosine" is owned by pahio.

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Also defines:

complex sine, complex cosine, sine, cosine, goniometric formula

Keywords:

power series

This object's parent.

Attachments:: complex tangent and cotangent (Definition) by pahio
values of complex cosine (Topic) by pahio

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Cross-references: derivatives, general power, square root, complex logarithm, cyclometric functions, inverse functions, periodicity, periodic, functions, real axis, line, infinity, modulus, imply, persistence of analytic relations, addition formulae, consequences, Euler's, Euler's formulas, degrees, odd, even, separating, complex exponential functions, even function, odd function, entire transcendental functions, entire functions, complex plane, holomorphic functions, power series, radii, real, converge, series, complex
There are 17 references to this entry.

This is version 26 of complex sine and cosine, born on 2004-10-21, modified 2008-01-24.
Object id is 6398, canonical name is ComplexSineAndCosine.
Accessed 30831 times total.

Classification:

AMS MSC:	30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
	30B10 (Functions of a complex variable :: Series expansions :: Power series )
	30D10 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Representations of entire functions by series and integrals)
	33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)

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