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November 28th, 2012, 12:02 PM   #1
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complex number calculation

Evaluate {(1+i)^(1-i)} and describe the set{1^x} when x is a real number, distinguish between the cases when x is rational and when x is rational. for now considering the complex number. i dont know how to start with,for firest part i just write it into e^((1-i)log(1+i)) then get the number with e to the power which inculding i , and the secound part , for 1=e^(i2npi) then 1^x is e^(2inxpi) then how to consider the case for rational and irrational here?????
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November 28th, 2012, 03:28 PM   #2
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Re: complex number calculation

When x is rational there are only a finite number of solutions, while for x irrational there are an infinite number of solutions.
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November 28th, 2012, 03:44 PM   #3
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Re: complex number calculation

Quote:
Originally Posted by mathman
When x is rational there are only a finite number of solutions, while for x irrational there are an infinite number of solutions.
when x is rational then it is p/q then thats the q roots of 1,but hows the idea for irrational. can u explain it with more details???? should it just consider that 1^x=exp(2ikpix) then x is irrational and i is infinite many values that i can chooes????
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November 29th, 2012, 01:41 PM   #4
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Re: complex number calculation

Every value of n gives a different angle. Proof by contradiction:

Assume two values (n and m) give the same angle, then xm - xn would be an integer k, so x = k/(n-m) which is rational.
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