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单词 ComplexSineAndCosine
释义

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

eiz=cosz+isinz,e-iz=cosz-isinz.

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:

sin(z1+z2)=sinz1cosz2+cosz1sinz2,
cos(z1+z2)=cosz1cosz2-sinz1sinz2;

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

sin(x+iy)=sinxcoshy+icosxsinhy,
cos(x+iy)=cosxcoshy-isinxsinhy

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

Re(sin(x+iy))=sinxcoshy,Im(sin(x+iy))=cosxsinhy,
Re(cos(x+iy))=cosxcoshy,Im(cos(x+iy))=-sinxsinhy.

Thus we get the modulus estimation

|sin(x+iy)|=sin2xcosh2y+cos2xsinh2y=sin2xcosh2y+(1-sin2x)sinh2y=sin2x(cosh2y-sinh2y)+sinh2y=sin2x1+sinh2y|sinhy|,

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 theirprime periodPlanetmathPlanetmathPlanetmath (http://planetmath.org/ComplexExponentialFunction):

sin(z+2π)=sinz,cos(z+2π)=coszz

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

arcsinz=1ilog(iz+1-z2),arccosz=1ilog(z+i1-z2).

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

ddzsinz=cosz,ddzcosz=-sinz

Cf. the higher derivatives (http://planetmath.org/HigherOrderDerivativesOfSineAndCosine).

Titlecomplex sine and cosine
Canonical nameComplexSineAndCosine
Date of creation2013-03-22 14:45:25
Last modified on2013-03-22 14:45:25
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id31
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 30D10
Classificationmsc 30B10
Classificationmsc 30A99
Classificationmsc 33B10
Related topicEulerRelation
Related topicCyclometricFunctions
Related topicExampleOfTaylorPolynomialsForSinX
Related topicComplexExponentialFunction
Related topicDefinitionsInTrigonometry
Related topicPersistenceOfAnalyticRelations
Related topicCosineAtMultiplesOfStraightAngle
Related topicHeavisideFormula
Related topicSomeValuesCharacterisingI
Related topicUniquenessOfFouri
Definescomplex sine
Definescomplex cosine
Definessine
Definescosine
Definesgoniometric formulaPlanetmathPlanetmath
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更新时间:2025/5/4 23:00:24