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 Subject: maths stuff Category: Miscellaneous Asked by: mongolia-ga List Price: \$4.00 Posted: 12 Dec 2004 12:33 PST Expires: 11 Jan 2005 12:33 PST Question ID: 441682
 ```what is (1+i) raised to the power of (1+i) expressed as another complex number? Thank You Mongolia```
 ```Hi, mongolia-ga: The seemingly simple operation of raising a complex number to a complex power is not well-defined in general. This is a bit like the fact that a nonzero complex number has two square roots, three cube roots, etc., but in your case there turn out to be infinitely many "answers". One can raise a positive real number to a complex power, and the result is uniquely defined. One can raise a nonzero complex number to a (real) integer power, and the result is again uniquely defined. Here we have a nonzero (complex) Gaussian integer raised to a like power, but the result is not uniquely defined. To see why, let's write (1+i) as a power of e: 1 + i = e^( ln(2)/2 + i*(pi/4) ) But for any integer k, e^(i*(2pi*k)) = 1. In other words the exponential function (base e) has "imaginary" period 2pi*i, and any multiple of 2pi*i can be added to an exponent of e without changing the result. Therefore the general way of expressing 1 + i as a power of e is the following: 1 + i = e^( ln(2)/2 + i*(pi/4) + i*(2pi*k) ) for any integer k. Therefore when one attempts to raise both sides to the power (1+i): (1 + i)^(1 + i) = e^( (1+i)*(ln(2)/2 + i*(pi/4) + i*(2pi*k)) ) = e^( (ln(2)/2 - pi/4 - 2pi*k) + i*(ln(2)/2 + pi/4 + 2pi*k) ) = e^( (ln(2)/2 - pi/4 - 2pi*k) + i*(ln(2)/2 + pi/4) ) Even after disposing of the term i*(2pi*k) in the exponent, which would not affect the result of exponentiation, we still have the term -2pi*k involving an infinite number of possibilities for k. In other words there is no unique answer, but instead a countably infinite number of possible results. For the particular choice k = 0, the result would be: e^( (ln(2)/2 - pi/4) + i*(ln(2)/2 + pi/4) ) which is approximately: 0.27395725383... + i * 0.58370075876... Taken together all these possibilites form a doubly-infinite geometric progression whose common ratio is: e^(2pi) ~ 535.49165552... Plotted in the complex plane these points all lie on a ray extending from the origin in the first quadrant at angle ln(2)/2 + pi/4 radians, or 64.857204014... degrees, to the positive real axis. regards, mathtalk-ga```
 mongolia-ga rated this answer: and gave an additional tip of: \$1.00 ```Mathtalk although you fully answered my question your answer has begged some more questions. However I will submit these as new questions. Many Thanks Mongolia```