How to combine complex powers? It is not posssible to combine complex powers in the...

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How to combine complex powers?
It is not posssible to combine complex powers in the usual way:
There was mention of multi-valued functions, is there some theory that makes this all work out correctly?

Answer & Explanation



Expert2022-01-19Added 573 answers

Step 1 For complex numbers the general way of defining xy (with x nonzero) is as exp[(ly) where l is one of the logarithms of x. But x does not have a unique logarithm, it has many. If l is one logarithm of x then the general one is l+2nπi with n an integer. Hence the general value of xy is exp(ly+2πiny)=exp(ly)exp(2πiny) Before proceeding further note that if y is an integer then exp(2πiny)1 so that xn is uniquely determined. More generally if y is a rational with denominator d then exp(πiny) (and so xy) takes d different values. But if y is irrational, then xy takes infinitely many variables. In the problem in hand xy takes the values exp[(ly)exp(2πiny) as above, and xyz the values exp(lyz)exp(2πinyz) However in general each value of xy gives rise to infinitely many values of (xy)z, namely exp((ly+2πiny)z+2πimz) so (xy)z has a doubly-infinite family of values. In general, there will be values of (xy)z which are not values of xy so the paradox in the previous posting should be expected.


Skilled2022-01-19Added 457 answers

Step 1 I guess the problem in general arises because ab can have multiple values. We have log((xy)z)=z(log(xy)+2πim)=z(y(log(x)+2πin)+2πim) =zylog(x)+2πi(ny+m)z =log(xyze2πi(ny+m)z) It seems a correction factor of e2πi(ny+m)z is necessary. So all that can be said is that m, nZ:(xy)z=xyze2πi(ny+m)z In particular, we can say that the correction term is 1 if (ny+m)z is always an integer, which happens for example if we know y,zZ


Skilled2022-01-24Added 366 answers

The original question asked about a theory that would make this all work out correctly. The previous answers are all correct, but they did not point out the underlying, unifying factor which makes all these computations much simpler: the Unwinding number. The unwinding number K(z) is defined to be ln(exp(z))=z+2πiK(z) It is an integer which tells you how many times you need to unwind z before you get into the range where lnexp is the identity. This then allows a lot of identities to be fixed with relative ease. I also find the paper Reasoning about the Elementary Functions of Complex Analysis by Corless, Davenport, Jeffrey, Litt and Watt quite instructive in this domain.

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