Complex exponent

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  • #1
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This is just a quick question which arose when doing an exercise, where you had to evaluate a complex exponent, az.
As you know you can easily generalize exponents to complex numbers using the fact that:
az = eln(a)[itex]\cdot z[/itex]
However, as you also know the function lnz is multivalued, i.e. ln(rexp(i[itex]\theta)) = ln(r) + i(\theta[/itex]+2k[itex]\pi[/itex]). Does that mean that the result for az should also be multivalued?
 

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  • #2
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Yes, the value [itex]a^z[/itex] is multivalued. However, if we take the principal branch of the logarithm, then we get only one value. This principal branch is what is often meant with [itex]a^z[/itex].
 

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