In this page, I
introduce you how to solve the complex number power of complex number through a way that I
found.
(r is positive real number)
Next, I show you the real number power of complex
number and imaginary number power of complex number.
It is easy to think the real number power of complex number because there is a very useful formula. That is De Moivre's theorem, and it stated that
However,
thus,
First,
the complex number power of complex number means
.
In this case, of course, i
is imaginary unit. Also, a and b
are included in real number, like 1,0.77,-4.7..According
to Euler's formula, 
.
Therefore,all complex numbers are expressed as
.
Therefore,all complex numbers are expressed as(r is positive real number)
It is easy to think the real number power of complex number because there is a very useful formula. That is De Moivre's theorem, and it stated that
(m is real numbers)
Second, I tell you how to solve the imaginary number
power of complex number. According to , given 
, then 
.
Thus,
, given , then .
Thus,
Also,
therefore,
(
(n is whole numbers)
(n is whole numbers)
That is to say if a complex
number,which includes
a real and imaginary number, is exponentinated by an imaginary number,
the answer is not fixed one value.
However, generally, the n is dealt with as n=0
Finally, I will state about the complex number power of complex number.
First of all, I decide a complex number like below.
Finally, I will state about the complex number power of complex number.
First of all, I decide a complex number like below.
so,
as I shows above,
thus,
,so
the answer is below
,but the true answer is
because
does not have fixed one value!
Finally, I think mathematics is very beutiful and interesting, so if you have a time, please try to solve some mathematical problems.
because
does not have fixed one value!Finally, I think mathematics is very beutiful and interesting, so if you have a time, please try to solve some mathematical problems.
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