complex sine and cosine

We define for all complex values of z:

  • sinz:=z-z33!+z55!-z77!+-

  • cosz:= 1-z22!+z44!-z66!+-

Because these series converge for all real values of z, their radii of convergence are , and therefore they converge for all complex values of z (by a known of Abel; cf. the entry power seriesMathworldPlanetmath), too.  Thus they define holomorphic functionsMathworldPlanetmath in the whole complex plane, i.e. entire functionsMathworldPlanetmath (to be more precise, entire transcendental functions).  The series also show that sine is an odd functionMathworldPlanetmath and cosine an even function.

Expanding the complex exponential functions eiz and e-iz to power series and separating the of even and odd degrees gives the generalized Euler’s formulas


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

cosz=eiz+e-iz2,sinz=eiz-e-iz2i (1)

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

cos2z+sin2z= 1.

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


so they are in fully as in .  It means that all goniometric formulae derived from these, such as

sin2z= 2sinzcosz,sin(π-z)=sinz,sin2z=1-cos2z2,

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

The addition formulae may be written also as


which imply, when assumed that  x,y,  the results


Thus we get the modulus estimation


which tends to infinity when  z=x+iy  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 functionsMathworldPlanetmath sin and cos are periodic and have 2π as their prime periodPlanetmathPlanetmathPlanetmath (


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


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


Cf. the higher derivatives (

Title complex sine and cosine
Canonical name ComplexSineAndCosine
Date of creation 2013-03-22 14:45:25
Last modified on 2013-03-22 14:45:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 31
Author pahio (2872)
Entry type Definition
Classification msc 30D10
Classification msc 30B10
Classification msc 30A99
Classification msc 33B10
Related topic EulerRelation
Related topic CyclometricFunctions
Related topic ExampleOfTaylorPolynomialsForSinX
Related topic ComplexExponentialFunction
Related topic DefinitionsInTrigonometry
Related topic PersistenceOfAnalyticRelations
Related topic CosineAtMultiplesOfStraightAngle
Related topic HeavisideFormula
Related topic SomeValuesCharacterisingI
Related topic UniquenessOfFouri
Defines complex sine
Defines complex cosine
Defines sine
Defines cosine
Defines goniometric formulaPlanetmathPlanetmath