general power

general power

The

general power

z

μ

, where

z

(

≠

0

)

and

μ

are arbitrary

complex numbers

, is defined via the

complex exponential function

and

complex logarithm

(denoted here by “

log

”) of the by setting

z

μ

:=

e

μ

⁢

log

⁡

z

=

e

μ

⁢

(

ln

⁡

|

z

|

+

i

⁢

arg

⁡

z

)

.

The number

z

is the

base of the power

z

μ

and

μ

is its

exponent

.

Splitting the exponent

μ

=

α

+

i

⁢

β

in its real and

imaginary parts

one obtains

z

μ

=

e

α

⁢

ln

⁡

|

z

|

-

β

⁢

arg

⁡

z

⋅

e

i

⁢

(

β

⁢

ln

⁡

|

z

|

+

α

⁢

arg

⁡

z

)

,

and thus

|

z

μ

|

=

e

α

⁢

ln

⁡

|

z

|

-

β

⁢

arg

⁡

z

,

arg

⁡

z

μ

=

β

⁢

ln

⁡

|

z

|

+

α

⁢

arg

⁡

z

.

This shows that both the

modulus

and the argument (

http://planetmath.org/Complex

) of the general power are in general multivalued.  The modulus is unique only if

β

=

0

,  i.e. if the exponent

μ

=

α

is real; in this case we have

|

z

μ

|

=

|

z

|

μ

,

arg

⁡

z

μ

=

μ

⋅

arg

⁡

z

.

Let

β

≠

0

.  If one lets the point

z

go round the origin anticlockwise,

arg

⁡

z

gets an addition

2

⁢

π

and hence the

z

μ

has been multiplied by a having the modulus

e

-

2

⁢

π

⁢

β

≠

1

, and we may say that

z

μ

has come to a new

branch

.

Examples

1.

z

1

m

, where

m

is a positive integer, coincides with the

m

th

root (

http://planetmath.org/CalculatingTheNthRootsOfAComplexNumber

) of

z

.

2.

3

2

=

e

2

⁢

log

⁡

3

=

e

2

⁢

(

ln

⁡

3

+

2

⁢

n

⁢

π

⁢

i

)

=

9

⁢

(

e

2

⁢

π

⁢

i

)

2

⁢

n

=

9

∀

n

∈

ℤ

.

3.

i

i

=

e

i

⁢

log

⁡

i

=

e

i

⁢

(

ln

⁡

1

+

π

2

⁢

i

-

2

⁢

n

⁢

π

⁢

i

)

=

e

2

⁢

n

⁢

π

-

π

2

(with

n

=

0

,

±

1

,

±

2

,

…

);  all these values are positive real numbers, the simplest of them is

1

e

π

≈

0.20788

.

4.

(

-

1

)

i

=

e

(

2

⁢

n

+

1

)

⁢

π

(with

n

=

0

,

±

1

,

±

2

,

…

)  also are situated on the

positive real axis

.

5.

(

-

1

)

2

=

e

2

⁢

log

⁡

(

-

1

)

=

e

2

⁢

i

⁢

(

π

+

2

⁢

n

⁢

π

)

=

e

i

⁢

(

2

⁢

n

+

1

)

⁢

π

⁢

2

(with

n

=

0

,

±

1

,

±

2

,

…

);  all these are (meaning here that their imaginary parts are distinct from 0), situated on the circumference of the

unit circle

such that all points of the circumference are

accumulation points

of the

sequence

of the

(

-

1

)

2

(see this entry (

http://planetmath.org/SequenceAccumulatingEverywhereIn11

)).

6.

2

1

-

i

=

2

⁢

e

2

⁢

n

⁢

π

⁢

(

cos

⁡

ln

⁡

2

+

i

⁢

sin

⁡

ln

⁡

2

)

(with

n

=

0

,

±

1

,

±

2

,

…

), are situated on the half line beginning from the origin with the argument

ln

⁡

2

≈

0.69315

radians.

Title

general power

Canonical name

GeneralPower

Date of creation

2013-03-22 14:43:17

Last modified on

2013-03-22 14:43:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

31

Author

pahio (2872)

Entry type

Definition

Classification

msc 30D30

Synonym

complex power

Related topic

Logarithm

Related topic

ExponentialOperation

Related topic

GeneralizedBinomialCoefficients

Related topic

PuiseuxSeries

Related topic

PAdicExponentialAndPAdicLogarithm

Related topic

FractionPower

Related topic

SomeValuesCharacterisingI

Related topic

UsingResidueTheoremNearBranchPoint

Defines

base of the power

Defines

base

Defines

exponent

Defines

branch

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