james@umcp-cs.UUCP (02/10/84)
Easy, it is: __ -|| / / 2 e (which is exp(-pi/2)). The way to get this result is: Any complex number C may be written as exp(a+bi) for some real a,b. i Then exp(a+bi) = exp((a+bi) * i) = exp(ai - b). (Using exponent rule.) i So consider C = i. i = exp(0 + pi/2) ==> i = exp(0i - pi/2) = exp(-pi/2). The actual value is approximately .20787958 --Jim
rjnoe@ihlts.UUCP (Roger Noe @ N41:48:31, W88:07:13) (02/10/84)
No! There are an infinite number of values for i^i. Any complex number C can be written as (a+bi)=C for unique reals a and b, but it can also be written in polar form with the argument indeterminate except to multiples of 2*PI. Somewhere along the line you have to take the natural logarithm of i, which is (2n+.5)i*PI for all integers n. exp(-PI/2) is in fact the principal value of i^i but all its values are given by exp(-PI*(2n+.5)). Roger Noe ihnp4!ihlts!rjnoe
ags@pucc-i (Seaman) (02/10/84)
The value of i ** i is not well defined. Obviously, it must be equal to exp(i * log(i)), but there is a problem in defining log(i). Since exp((2*n+0.5)*i*PI) = i for any integer n, there are infinitely many candidates for log(i). It may seem reasonable to single out PI/2 as THE logarithm of i, but it is a well-known fact that the log function cannot be extended continuously to the entire complex plane, as a single-valued function. What is (-1) ** i? How about (-i) ** i? However you define the log function, you find yourself getting into trouble. -- Dave Seaman ..!pur-ee!pucc-i:ags "Against people who give vent to their loquacity by extraneous bombastic circumlocution."
rentsch@unc.UUCP (Tim Rentsch) (02/11/84)
Just to add fuel to the controversy -- the stuff with multiple values happens only if that nasty log function (which is multiple valued, and no mistake) gets involved. There are no such problems with exp(x) since it is defined by the power series exp(x) = sum from i = 0 to 1/0 of x**i/i! which converges for all x and does not require anything funny for multiple values (since integer exponentiation is quite well defined). What about the i**i case? Some clever person know how to define x**y without using log (and so clear up all this multiple valued nonsense)?