complex sine and cosine

complex sine and cosine

We define for all

complex

values of

z

:

•

sin

z

:=

z

-

z

3

3

!

+

z

5

5

!

-

z

7

7

!

+

-

…

•

cos

z

:=

1

-

z

2

2

!

+

z

4

4

!

-

z

6

6

!

+

-

…

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 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

i

⁢

z

and

e

-

i

⁢

z

to power series and separating the of even and odd degrees gives the generalized Euler’s formulas

e

i

⁢

z

=

cos

⁡

z

+

i

⁢

sin

⁡

z

,

e

-

i

⁢

z

=

cos

⁡

z

-

i

⁢

sin

⁡

z

.

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

cos

⁡

z

=

e

i

⁢

z

+

e

-

i

⁢

z

2

,

sin

⁡

z

=

e

i

⁢

z

-

e

-

i

⁢

z

2

⁢

i

(1)

(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 formulae

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

ℂ

fully as in

ℝ

.  It means that all

goniometric

formulae derived from these, such as

sin

⁡

2

⁢

z

=

2

⁢

sin

⁡

z

⁢

cos

⁡

z

,

sin

⁡

(

π

-

z

)

=

sin

⁡

z

,

sin

2

⁡

z

=

1

-

cos

⁡

2

⁢

z

2

,

have the old shape.  See also the

persistence of analytic relations

.

The addition formulae may be written also as

sin

⁡

(

x

+

i

⁢

y

)

=

sin

⁡

x

⁢

cosh

⁡

y

+

i

⁢

cos

⁡

x

⁢

sinh

⁡

y

,

cos

⁡

(

x

+

i

⁢

y

)

=

cos

⁡

x

⁢

cosh

⁡

y

-

i

⁢

sin

⁡

x

⁢

sinh

⁡

y

which imply, when assumed that

x

,

y

∈

ℝ

,  the results

Re

⁢

(

sin

⁡

(

x

+

i

⁢

y

)

)

=

sin

⁡

x

⁢

cosh

⁡

y

,

Im

⁢

(

sin

⁡

(

x

+

i

⁢

y

)

)

=

cos

⁡

x

⁢

sinh

⁡

y

,

Re

⁢

(

cos

⁡

(

x

+

i

⁢

y

)

)

=

cos

⁡

x

⁢

cosh

⁡

y

,

Im

⁢

(

cos

⁡

(

x

+

i

⁢

y

)

)

=

-

sin

⁡

x

⁢

sinh

⁡

y

.

Thus we get the

modulus

estimation

|

sin

⁡

(

x

+

i

⁢

y

)

|

=

sin

2

⁡

x

⁢

cosh

2

⁡

y

+

cos

2

⁡

x

⁢

sinh

2

⁡

y

=

sin

2

⁡

x

⁢

cosh

2

⁡

y

+

(

1

-

sin

2

⁡

x

)

⁢

sinh

2

⁡

y

=

sin

2

⁡

x

⁢

(

cosh

2

⁡

y

-

sinh

2

⁡

y

)

+

sinh

2

⁡

y

=

sin

2

⁡

x

⋅

1

+

sinh

2

⁡

y

≥

|

sinh

⁡

y

|

,

which tends to

infinity

when

z

=

x

+

i

⁢

y

moves to infinity along any line non-parallel to the

real axis

.  The modulus of

cos

⁡

(

x

+

i

⁢

y

)

behaves similarly.

Another important consequence of the addition formulae is that the

functions

sin

and

cos

are

periodic

and have

2

⁢

π

as their

prime period

(

http://planetmath.org/ComplexExponentialFunction

):

sin

⁡

(

z

+

2

⁢

π

)

=

sin

⁡

z

,

cos

⁡

(

z

+

2

⁢

π

)

=

cos

⁡

z

∀

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

arcsin

⁡

z

=

1

i

⁢

log

⁡

(

i

⁢

z

+

1

-

z

2

)

,

arccos

⁡

z

=

1

i

⁢

log

⁡

(

z

+

i

⁢

1

-

z

2

)

.

The

derivatives

of

sine function

and

cosine function

are obtained either from the series forms or from (1):

d

d

⁢

z

⁢

sin

⁡

z

=

cos

⁡

z

,

d

d

⁢

z

⁢

cos

⁡

z

=

-

sin

⁡

z

Cf. the higher derivatives (

http://planetmath.org/HigherOrderDerivativesOfSineAndCosine

).

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 formula

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