Raising to half power = PRINCIPAL square root?

1. Mar 17, 2014

perishingtardi

This may seem like a very elementary question...but here goes anyway.

When a positive number is raised to the power 1/2, I have always assumed that this is defined as the PRINCIPAL (positive) square root, e.g. $$7^{1/2} = \sqrt{7},$$. That is, it does not include both the positive and negative square rootsL $$7^{1/2} \neq -\sqrt{7} = -7^{1/2}.$$

In complex analysis, however, this doesn't seem to be the case? E.g. we write $$(-1)^{1/2} = \pm i.$$

Have I understood these conventions correctly? I have also been thinking about a similar situation: how in real analysis we think of every positive number as having a single natural logarithm, e.g. $$\ln 2 = 0.693\dotsc,$$ when in fact there are actually infinitely many:
$$\ln 2 = 0.693\dotsc + 2\pi n i \qquad (n=0,\pm1,\pm2,\dots).$$

2. Mar 17, 2014

micromass

In real analysis, you are right to take the principal square root.

In complex analysis, things are a bit more complicated. You have two options here: either you take a multivalues approach where complex exponentiation and logarithms yield not one value but several ones. The other approach is still to take a principal logarithm and a principal exponent. I think the latter approach is more popular in introductory complex analysis texts.