Pretty much all operations with complex numbers are defined. Also, some stuff that you were told that was "undefined" can be defined using complex numbers. For example, the square root of a negative number is "not defined" in the set of real numbers, but it is in complex numbers. Numbers like ln(-1), sin(i), arcsin(2) are all defined in complex numbers.
I don't know why you automatically think it should be undefined...
First, you need to know Euler's identity, which says
e^(i*t) = cos(t) + i*sin(t).
Setting
t = pi/2 + 2pi*k
where k is any integer, we get
e^(i*(pi/2 + 2pi*k) ) = i
Taking both sides to the power of i, we get
e^(-pi/2 - 2pi*k) = i^i.
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Now, like a lot of "functions" in the complex numbers, we must choose a branch on which to evaluate. Different integers k will give you different values for i^i. One "natural" choice for k is k=0, which yields
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Pretty much all operations with complex numbers are defined. Also, some stuff that you were told that was "undefined" can be defined using complex numbers. For example, the square root of a negative number is "not defined" in the set of real numbers, but it is in complex numbers. Numbers like ln(-1), sin(i), arcsin(2) are all defined in complex numbers.
To compute i^i, note that:
i = e^[ln(i)], since e^[ln(x)] = x for all x.
In polar form, we can write:
i = cos π/2 + i sin π/2
= e^(iπ/2), by Euler's Formula.
Thus, ln(i) = iπ/2 and:
i^i = {e^[ln(i)]}^i
= e^[i*ln(i)]
= e^(i * iπ/2), since ln(i) = iπ/2
= e^(-π/2), since i^2 = -1
≈ 0.20788.
I hope this helps!
I don't know why you automatically think it should be undefined...
First, you need to know Euler's identity, which says
e^(i*t) = cos(t) + i*sin(t).
Setting
t = pi/2 + 2pi*k
where k is any integer, we get
e^(i*(pi/2 + 2pi*k) ) = i
Taking both sides to the power of i, we get
e^(-pi/2 - 2pi*k) = i^i.
---------
Now, like a lot of "functions" in the complex numbers, we must choose a branch on which to evaluate. Different integers k will give you different values for i^i. One "natural" choice for k is k=0, which yields
i^i = e^(-pi/2)
which is about 0.207879576.