1)
Computation of complex power
function
1.1
The complex power function is a multi-valued
function
1.2 Computation
of the roots of the complex value
1.3 Computation
of the complex z
and real a
1.4 The
power of a real number to a non-integer
power
1.5 Raising
a complex number to a complex power
2)
The rules for combining quantities
containing powers
3)
Derivatives
of complex power function
4)
Indefinite
integral of complex power function
1) Computation of the complex power
function
1.1
The complex power function is a multi-valued
function
Computation of complex
power function involves using the complex
Exponential and Logarithm functions.
For any z
and every a
where z
C.
Because Ln(z)
is a multi-valued function, the
function 
is
a multi-valued too. The principal branch
of the function is obtained by replacing
Ln(z) with the principal branch of the
logarithm.
The power a
may be an integer, real number, or complex
number.
A number other than "0"
taken to the power "0" is
defined to be 1, which follows from
the 
A number to the first power is, by definition,
equal to itself:
z1
= z.
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1.2 Computation of the roots
of the complex value
If a
= 
,
nN,
then
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1.3 Computation of the complex
z
and real a
For complex z
and real a,
where arg(z)
is the complex argument.
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1.4 The power of a real number
to a non-integer power
The power of a real number
to a non-integer power is not necessarily
itself a real number.
For example, 
is real only for x
0.
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1.5 Raising a complex number
to a complex power
A complex number may be
taken to the power of another complex
number. In particular, complex exponentiation
satisfies
where arg(z)
is the complex argument.
Written explicitly in
terms of real and imaginary parts,
Example of complex exponentiation:
The result of raising a complex number
to a complex power may be a real number.
Example
Let
us find a principle value of
=
0.20787957635
In fact, there is a family of values
k
such that 
is
real:
[Cos(k
ln
k)
+ iSin(k
ln
k)]
This will be real when Sin(k
ln
k)
= 0,
i.e., for
k
ln
k
= n
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The rules for combining quantities
containing powers
Some of the rules for
combining quantities containing powers
carry over from the real case. If c
and d are complex numbers,
and 
,
then
 ; |
(2.1)
|
 ; |
(2.2)
|
 ; |
(2.3)
|
 ,
where n
is real integer. |
(2.4) |
The identity (2.4) does
not hold if n
is replaced with an arbitrary complex
value.
Example:
,
k
is an integer.
,
k
is an integer.
Since these sets of solutions
are not equal, identity (2.4) does not
always hold.
The principal values of
and 
are the same.
The principal value of 
=
0.0432139
The principal value of 
=
0.0432139
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3) Derivatives of power function
3.1
For the proof, we use
the definition
where zC.
Then we have:
3.2
For the proof, we use
the definition 
where zC.
Then we have:
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4) Indefinite integral of power function
The indefinite integral:
by
Tetyana Butler
