Imaginary or real?

[alpertron]

I think the following message that I posted to USENET on November 1997 would be interesting for this thread. It was about computing i^i without computing e or pi.

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I don't know of a series for i^i, but below there is a continued
fraction expansion:

From the Handbook of Mathematical Functions (Milton Abramowitz and Irene
Stegun), formula 4.2.42:

                               
                           2  2   2  2   2  2
 2a*atn(1/z)         2a   a +1   a +2   a +3
e             = 1 + ----  -----  -----  ----- ...
                    z-a+   3z+    5z+    7z+

Taking a = -1, z = 1:

                             2      2      2
 -pi/2    i         2     1+1    1+2    1+3
e      = i  = 1 - -----  -----  -----  ----- ...
                   2+     3+      5+     7+

The next program in UBASIC shows that adding 100 terms to the continued
fraction, the precision is incremented in 75 decimal digits
approximately.

   10   word 400
   20   point 200
   30   input "Terms";T
   40   S#=0
   50   for K=T to 1 step -1
   60   S#=(K*K+1)/(K+K+1+S#)
   70   next K
   80   F#=1-2/(2+S#)
   90   G#=#i
  100   H#=G#^G#
  110   print H#-F#

[mfgoode]

Originally Posted by Bob Silverman

The question is not quite posed correctly. i^i does not have a unique value.
The question should either be: what is the value of i^i assuming the
principal branch of the logarithm function, OR "what is the smallest possible
absolute value of i^i" OR "classify all possible values of i^i" etc.

QUOTE=S80780]1) What is the value of i^i?
i^i = e^[(2k+0.5)*pi*i*i] = e^[-(2k+0.5)*pi], k in Z, so it is real.

2) What is the value of i^-i?
i^-i = (i^i)^-1 = e^[(2k+0.5)*pi], k in Z, so it is real.

I expect the main value is k=0.

A more interesting question here would be, if e^pi is in Q.

Benjamin[/QUOTE]

Dear Bob,
Compared to your qualifications and career (see Jinydu's post in this thread)
I am just an amateur mathem'cian with a Mechanical engg. background groping into the darker recesses of maths with the advantage of having traversed the globe for 35 yrs and then retired.

I give below the numerical values of the foll:

i^i =0.207 879 576 350 761 908 546 955 ------

i^-i = reciprocal of i^i =4.810 477 389 ---------

e^pi =23.140 692 632 779 269 005 729 086 -----

This is a transcendental no. Hence in the set R

pi^e =22.459 157 718 361 045 473 427 152 --------
Q.E.I.
It is not known whether this no. is rational or irrational so it could be in Q .
Hence this no. is open for enquiry and proof

Source: "The Penguin dictionary of curious and intresting numbers" by David Wells"

Mally