complex number raise to complex number?

That's actually defined in the complex plane:

say you have a complex number : z = a + ib

We can use the fact that:

z = e^ln(z)

If you have : z =(a + ib)^(c + id), then

e^ln(z) = e^(ln(a + ib)^(c + id)) = e^((c + id)*ln(a + ib))

We also know that the complex logarithm is described as followed:

ln(z) = ln|z| + i(Arg(z) + 2kpi), k = 0,1,2,3,4,...

and when we have k = 0, ln(z) = Ln(z), where Ln(z) is described as being the principal value of the logarithm.

Let's do an example:

z = i^i

z = e^(ln(i^i)) = e^(i*ln(i))

ln(i) = ln|i| + i(Arg(i) + 2kpi)

ln(i) = ln(1) + i(pi/2 + 2kpi)

ln(i) = 0 + i(pi/2 + 2kpi)

Well, i'm going to find the principal value, rather than the multivalue logarithm, i.e. i'm going to let k = 0.

The principal value of ln(i) is:

Ln(i) = i(pi/2)

Hence we now have:

i^i = e^(i*(ipi/2)) = e^(-pi/2)

Which is a real number

Note that if i didn't take the principal value of logarithm, my answer would not have been a real number.

Also, this is consistent with the fact that the real numbers are a subset of the complex numbers. You can define any real number with the method i used, as long as you use the principal value of the logarithm.

Here's a short explanation and the general, expanded expression for complex exponentiation. http://mathworld.wolfram.com/ComplexExponentiation...

You may also want to check the Wikipedia page on "exponentiation" where you can find a more detailed explanation.

i'm guessing you propose e^(a million + 2i) if so:: permit z = e^(a million + 2i) = e.e^(2i) subsequently z/e = cos(2) + i*sin(2) z = e cos2 + esin2 * i all simply by fact e^(i theta) = cos(theta) + i sin(theta)

not getting ur question...plz explain it