So, what is the principal value of i^{3i}?

[hbweb500]

I am studying complex variables with Brown and Churchill. In it, they define the principal value of z^c, with both variables complex, to be e^{c\; \text{Log }z}, where \text{Log} is the principle value branch of the complex logarithm.

Now, suppose z = i and c = 3. We know that \text{Arg } i = \frac{\pi}{2}, so z^c = i^3 = e^{3 \pi / 2}. But is this really the principal value? Why don't we say e^{- \pi/2} is the principal value?

I ask because it seems like that is what the textbook does in one of its examples: it calculates
z^c to be something with an angle outside of -\pi < \theta \leq \pi, and just reduces it without explanation.

So, when finding the principle value of z^c after we have done the calculation, or is simply using the principle value branch logarithm enough?

 

[olivermsun]

Log already chooses a principal value by only taking i Arg z for the complex part. Hence it isn't necessary to introduce further conventions to get an principal value for z^c.

Anyhow, your exponential should have an i upstairs I think, so that it doesn't matter whether the exponent is -i pi/2 or i 3 pi/2.

Now if you were to take Arg of the exponential then I suppose you'd have to return a value in (-pi, pi].

 

[HallsofIvy]

Yes, you are missing an "i" in the numerator i^3= e^{3i\pi/2}= -i as you get by straight forward multiplication: i^3= (i^2)i= -i.

Doing it as e^{3 log(i)}, log(i)= i\pi/2+ 2k\pi i so that i^3= e^{3 log(i)}= e^{3i\pi/2+ 6ki\pi}= e^{3i\pi/2}e^{6ki\pi}. But e to any <b>even</b> multiple of i\pi is 1 so that <b>all</b> "branches" give the same thing. More generally, any complex number to a <b>positive integer</b> power is single valued.<br /> <br /> But, since you said "z^c with both variables complex", did you mean i^{3i}. In that case, i^{3i}= e^{3i log(i)} and now i= e^{i\pi/2} so that log(i)= i\pi/2+ 2k\pi as before, 3i \log(i)= -3\pi/2+ 5k\pi and, finally, i^{3i}= e^{-3\pi}e^{5k\pi}.<br /> <br /> Taking k= 0 gives i^{3i}= e^{-3\pi} which is the smallest positive value. Normally, the "principal value" of a calculation is the non-real value with smallest argument. When all values are real, it is the smallest positive value.