[net.math] what is i raised to the ith power?

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)?