complex tangent and cotangent

complex tangent and cotangent

The

tangent

and the

cotangent function

for

complex

values of the

z

are defined with the equations

tan

⁡

z

:=

sin

⁡

z

cos

⁡

z

,

cot

⁡

z

:=

cos

⁡

z

sin

⁡

z

.

Using the Euler’s formulae (

http://planetmath.org/ComplexSineAndCosine

), one also can define

tan

⁡

z

:=

-

i

⁢

e

i

⁢

z

-

e

-

i

⁢

z

e

i

⁢

z

+

e

-

i

⁢

z

,

cot

⁡

z

:=

i

⁢

e

i

⁢

z

+

e

-

i

⁢

z

e

i

⁢

z

-

e

-

i

⁢

z

.

(1)

The

subtraction formulae

of cosine and sine (

http://planetmath.org/ComplexSineAndCosine

) yield an additional between the

cotangent

and tangent:

cot

⁡

(

π

2

-

z

)

=

cos

⁡

(

π

2

-

z

)

sin

⁡

(

π

2

-

z

)

=

cos

⁡

π

2

⁢

cos

⁡

z

+

sin

⁡

π

2

⁢

sin

⁡

z

sin

⁡

π

2

⁢

cos

⁡

z

-

cos

⁡

π

2

⁢

sin

⁡

z

=

sin

⁡

z

cos

⁡

z

=

tan

⁡

z

.

Thus the properties of the tangent are easily derived from the corresponding properties of the cotangent.

Because of the identic equation

cos

2

⁡

z

+

sin

2

⁡

z

=

1

the cosine and sine do not vanish simultaneously, and so their

quotient

cot

⁡

z

is finite in all finite points

z

of the

complex plane

except in the zeros

z

=

n

⁢

π

(

n

=

0

,

±

1

,

±

2

,

…

) of

sin

⁡

z

, where

cot

⁡

z

becomes

infinite

.  We shall see that these multiples of

π

are

simple poles

of

cot

⁡

z

.

If one moves from

z

to

z

+

π

, then both

cos

⁡

z

and

sin

⁡

z

change their signs (cf.

antiperiodic function

), and therefore their quotient remains unchanged.  Accordingly,

π

is a period of

cot

⁡

z

.  But if

ω

is an arbitrary period of

cot

⁡

z

, we have

cot

⁡

(

z

+

ω

)

=

cot

⁡

z

,  and especially

z

=

0

gives

cot

⁡

ω

=

∞

;  then (1) says that

e

i

⁢

ω

=

e

-

i

⁢

ω

,  i.e.

e

2

⁢

i

⁢

ω

=

1

.  Since the

prime period

of the

complex exponential function

is

2

⁢

i

⁢

π

, the last equation is valid only for the values

ω

=

n

⁢

π

(

n

=

0

,

±

1

,

±

2

,

…

).  Thus we have shown that the prime period of

cot

⁡

z

is

π

.

We know that

sin

⁡

z

z

=

sin

⁡

z

-

sin

⁡

0

z

→

cos

⁡

0

=

1

as

z

→

0

;

therefore

z

⁢

cot

⁡

z

=

z

sin

⁡

z

⋅

cos

⁡

z

→

1

⋅

cos

⁡

0

=

1

as

z

→

0

.

This result, together with

cot

⁡

z

→

∞

as

z

→

0

,

means that

z

=

0

is a simple pole of

cot

⁡

z

.

Because of the

periodicity

,

cot

⁡

z

has the simple poles in the points

z

=

0

,

±

π

,

±

2

⁢

π

,

…

.  Since one has the

derivative

d

⁢

cot

⁡

z

d

⁢

z

=

-

1

sin

2

⁡

z

,

cot

⁡

z

is

holomorphic

in all finite points except those poles, which accumulate only to the point

z

=

∞

.  Thus the cotangent is a

meromorphic function

.  The same concerns naturally the

tangent function

.

As all meromorphic functions, the cotangent may be expressed as a series with the

partial fraction

(

http://planetmath.org/PartialFractionsOfExpressions

) terms of the form

a

j

⁢

k

(

z

-

p

j

)

k

, where

p

j

’s are the poles — see this entry (

http://planetmath.org/ExamplesOfInfiniteProducts

).

The real (

http://planetmath.org/CmplexFunction

) and

imaginary parts

of tangent and cotangent are seen from the formulae

tan

⁡

(

x

+

i

⁢

y

)

=

sin

⁡

x

⁢

cos

⁡

x

+

i

⁢

sinh

⁡

y

⁢

cosh

⁡

y

cos

2

⁡

x

+

sinh

2

⁡

y

,

cot

⁡

(

x

+

i

⁢

y

)

=

sin

⁡

x

⁢

cos

⁡

x

-

i

⁢

sinh

⁡

y

⁢

cosh

⁡

y

sin

2

⁡

x

+

sinh

2

⁡

y

,

which may be derived from (1) by substituting

z

:=

x

+

i

⁢

y

(

x

,

y

∈

ℝ

).

References

1

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava. Helsinki (1963).

Title

complex tangent and cotangent

Canonical name

ComplexTangentAndCotangent

Date of creation

2013-03-22 16:49:56

Last modified on

2013-03-22 16:49:56

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Classification

msc 30D10

Classification

msc 33B10

Related topic

ExamplesOfInfiniteProducts

Related topic

QuasiPeriodicFunction

Related topic

HyperbolicFunctions

Related topic

QuasiperiodicFunction

Generated on Fri Feb 9 20:13:15 2018 by

LaTeXML