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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. [tex]7^{1/2} = \sqrt{7},[/tex]. That is, it does not include both the positive and negative square rootsL [tex]7^{1/2} \neq -\sqrt{7} = -7^{1/2}.[/tex]

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

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. [tex]\ln 2 = 0.693\dotsc,[/tex] when in fact there are actually infinitely many:

[tex]\ln 2 = 0.693\dotsc + 2\pi n i \qquad (n=0,\pm1,\pm2,\dots).[/tex]

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# Raising to half power = PRINCIPAL square root?

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