If n=complex number what values of n in i^n real?

[brandy]

would iⁿ = an infinite amount of real possibilites if n is complex. considering that 1+0i is still a complex number, or is that wrong?

 

[pbandjay]

For one example, iⁱ is real. In fact,

$$(it)^{it}=e^{-t\pi/2}[\cos(t\ln t) + i\sin(t\ln t)]$$

is real for t ln(t) = n pi, n integer.

 

[lurflurf]

i^n=exp(n log(i))
For the principle branch take log(i)=i*pi/2
i^n=exp(i*n*pi/2)
when will that be real?

 

[brandy]

i just asked if it would have an infinite amount of possibilites, obviously i have already thought about this and already know some examples of how it will be real. its a yes or no + justification response.

im using De Moirve's theorem and i can already prove that i^i is real. and hence i^ai is real even if its complex and if n=ai+c it will be real if c is an even number or o.

the other part of my question was, can i acurately say that a+0i is a complex number?

 

[slider142]

i just asked if it would have an infinite amount of possibilites, obviously i have already thought about this and already know some examples of how it will be real. its a yes or no + justification response.

im using De Moirve's theorem and i can already prove that i^i is real. and hence i^ai is real even if its complex and if n=ai+c it will be real if c is an even number or o.

the other part of my question was, can i acurately say that a+0i is a complex number?

Yes. The real numbers are a subset of the complex numbers.