I'm somewhat confused about complex numbers right now. Or rather about how WolframAlpha deals with them, which is presumably the correct way.
If I ask for solutions to the equation x^5 = 1, it gives me all the complex numbers with angles n * 72°, n = 0,1,2,3,4. (Left.) If I ask for solutions to the equation x^{5/2} = 1, it gives me the ones with n = 0, 2, 3. (Right)
So what exactly is the property that the points with angles 72° and 288° don't have?
It’s because the function which raises a complex number to the fifth power is a real function, whereas the function which raises a complex number to the 5/2 power is not, it’s a doubly valued multifunction. In all 5 of these cases, one of the two values of the multifunction is 1, but only in the three given solutions is 1 the *principal* value of the multifunction.
When you say double valued multifuction, you mean a function p : \C -> \powerset(\C) defined by p(c) = {[first square root of c^5], [second square root of c^5]}?
So what's the definition / consensus on what is the principal value?
Yeah that's one way to think about it (though the more proper and technical setting for this involves Riemann surfaces).
To define principal values, we need to talk about arguments, logarithms, and power functions.
The
argument of a complex number is its angle in the complex plane measured in radians from the positive real axis. Of course, this is not a single number, but a set of numbers, denoted arg z, since, e.g., 0 and 2pi refer to the same angle. So we single out one of these angles to be called the
principal argument, which (more or less arbitrarily) is the angle A in arg z satisfying -pi < A <= pi.
Then we note that we can write a complex number z in exponential notation, as z = r*e^(i*a), where r=|z| is the modulus and a is one of the arguments of z. Here we can choose any of the arguments of z, since the complex exponential function is periodic with period 2*pi*i (for the same reason that adding 2*pi to an angle gives the same angle). This means in particular that the complex exponential function is no longer one-to-one as the real exponential function is, and hence the inverse function (the logarithm) is not a real function.
But we decide that we want to talk about logarithms anyway, so we define it as a multifunction--the log of a nonzero complex number z is now the
set of complex numbers which exponentiate to z. It turns out that log z = log (re^ia) = ln|z| + i*arg z, where ln denotes the real logarithm of the positive real number |z| and arg z is again the full set of arguments of z. This is an infinite set, since arg z is an infinite set of angles differing by multiples of 2pi. If we insist on defining at a single-valued version of the log function, we by convention usually use the principle argument A, which gives the
principal branch of the log function, denoted Log. That is, Log z = ln|z| + i*A.
This now lets us define power functions. Given z and another complex number c, we can define z^c by z^c = e^(c*logz) = e^(c*ln|z| + i*c*arg z). Since log is multi-valued, the power function typically is as well, but we can single one of these out by using the principal branch of the log function, e^(c*Log z) = e^(c*ln|z| + i*c*A). It happens to be the case that the power function is infinitely-valued unless c is rational, in which case it has as many values as the denominator of c written in lowest terms. When c is an integer, the denominator is 1, so power functions with integer exponents are in fact all single-valued.
So for example, let's apply this to some of your numbers. The number z_1 at angle 72 degrees has principal argument of 2pi/5 radians, and a modulus of 1, so the principal value of z_1^(5/2) is
z_1^(5/2) = e^(5/2*ln(1) + i*(5/2)*(2pi/5)) = e^(i*pi)=-1.
The number z_3 at angle 3*72 = 216 degrees has argument of 6pi/5 radians, but its
principal argument is -4pi/5. Its modulus is also 1, so the principal value of z_3^(5/2) is
z_3^(5/2) = e^(5/2*ln(1) + i*(5/2)*(-4pi/5)) = e^(i*(-2pi))=1.
But remember that this whole principal business is rather arbitrary. Using the argument 12pi/5 for z_1 gives
z_1^(5/2) = e^(5/2*ln(1) + i*(5/2)*(12pi/5)) = e^(i*6pi)=1,
and using the argument 6pi/5 for z_3 gives
z_3^(5/2) = e^(5/2*ln(1) + i*(5/2)*(6pi/5)) = e^(i*3pi)=-1.
