values of complex cosine

values of complex cosine

Since the

complex cosine

function

z

↦

cos

⁡

z

has the

prime period

2

⁢

π

, the cosine attains all of its possible values in one of its

period strips

, for example in the period strip

{

z

∈

ℂ

⋮

-

π

≦

Re

(

z

)

<

π

}

.

(1)

For finding out which values the

cosine function

can attain in a period strip, we solve the equation

cos

⁡

z

=

w

,  where

w

is any

complex number

. Using Euler’s formula (

http://planetmath.org/ComplexSineAndCosine

)

cos

⁡

z

=

e

i

⁢

z

+

e

-

i

⁢

z

2

,

the equation may be written as

(

e

i

⁢

z

)

2

-

2

⁢

w

⁢

e

i

⁢

z

+

1

=

0

.

(2)

This is a

quadratic equation

in

e

i

⁢

z

, whence we obtain the two roots (

http://planetmath.org/Equation

)

e

i

⁢

z

=

w

±

w

2

-

1

.

The product of the roots is 1, and therefore the roots are distinct from zero for all values of

w

. If we set

w

+

w

2

-

1

=

r

e

i

⁢

φ

(

-

π

≦

φ

<

π

)

,

the other root is the

inverse number

w

-

w

2

-

1

=

1

r

⁢

e

-

i

⁢

φ

.

The solution of the equation

e

i

⁢

z

=

r

⁢

e

i

⁢

φ

is then obtained by taking the

complex logarithm

z

=

z

1

=

1

i

log

(

r

e

i

⁢

φ

)

=

φ

-

i

ln

r

+

n

⋅

2

π

(

n

∈

ℤ

)

,

and the other solution of (2) is

z

=

z

2

=

-

φ

+

i

ln

r

+

n

⋅

2

π

(

n

∈

ℤ

)

.

In the period strip (1), we have one solution

z

1

and one solution

z

2

, both obtained with the value

n

=

0

(except

z

2

in the case

φ

=

-

π

with

n

=

-

1

).  In (1), the points

z

1

and

z

2

are situated symmetrically with respect the origin.  In the cases

w

=

1

and

w

=

-

1

,  the equation (2) has double roots

z

=

0

and

z

=

-

π

,  respectively; then we may say that

z

1

and

z

2

coincide. Anyhow, we have the

Theorem.

In every period strip, cosine attains any

complex

value at two points.

Example.

The solution of the equation

cos

⁡

z

=

2

is obtained from

e

i

⁢

z

=

2

±

3

. In the period strip (1) we get

z

=

1

i

⁢

log

⁡

(

2

±

3

)

=

-

i

⁢

ln

⁡

(

2

±

3

)

+

0

⋅

2

⁢

π

.

Since

2

±

3

are inverse numbers of each other, we have as result the

purely imaginary numbers

z

=

±

i

⁢

ln

⁡

(

2

+

3

)

.

From

trigonometry

, we know that the real zeros of cosine are the odd multiples of

π

2

; from these points,

±

π

2

belong to the period strip (1). Thus

±

π

2

are the only points of (1) where the cosine vanishes. Therefore, according to the preceding theorem, the well-known points

(

2

n

+

1

)

π

2

(

n

=

0

,

±

1

,

±

2

,

…

)

are the only zeros of the cosine function on the whole

complex plane

.

The values of complex cosine function may be transferred to the complex sine function by means of the

complement formula

sin

⁡

z

=

cos

⁡

(

π

2

-

z

)

.

One can think all points of the

z

-plane to bear the corresponding value of cosine, and then one can translate the plane in the direction of the

real axis

the distance

π

2

; then the values of the sine have been placed to their correct . So one has transferred also the above properties of cosine to sine.

References

1

Ernst Lindelöf

:

Johdatus funktioteoriaan

. Second edition. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1936).

2

R. Nevanlinna & V. Paatero

:

Funktioteoria

. Kustannusosakeyhtiö Otava, Helsinki (1963).

Title

values of complex cosine

Canonical name

ValuesOfComplexCosine

Date of creation

2013-03-22 17:36:29

Last modified on

2013-03-22 17:36:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Topic

Classification

msc 33B10

Classification

msc 30A99

Related topic

RealPart

Related topic

PropertiesOfQuadraticEquation

Related topic

TakingSquareRootAlgebraically

Related topic

ComplexLogarithm

Defines

period strip

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