
Re: What's the geometry meaning of i^i ?
Posted:
Jul 27, 2012 1:44 PM


On Thursday, July 26, 2012 1:16:34 PM UTC+5:30, Narasimham wrote: > On Jul 25, 9:28 am, Hongyi Zhao <hongyi.z...@gmail.com> wrote: > > Hi all, > > > > What's the geometry meaning of i^i ? Here, i is the imaginary unit. > > > > Regards > >  > > .: Hongyi Zhao [ hongyi.zhao AT gmail.com ] Free as in Freedom :. > > i^i is not an imaginary number or numbers, but a set of equally spaced > real numbers. > > Just as product of two negative numbers returns > positive,exponentiation of pure imaginary numbers returns as pure real > number. > > From Euler's identity the set (its log actually, missed out before) is seen to be odd multiples of pi/2, > (4*k + 1)pi/2. Geometrically seen on the Argand diagram they are all their logarithms are > on the real axis, say formed by a point of a rolling circle 2 units > radius rolling on real axis. i.e., (.., 7,3,1,5,9, ..)* pi/2. > HTH > Narasimham
It appears your question in other words is:
What happens to a complex number z when it is raised to 'i'th power, and show geometrically what happens to z, i.e., how it maps.
Actually this is a very good question, not properly addressed in undergraduate books even, at least those that I had chance to read. It is true we cannot visualize it as readily as we can visualize z e^( i theta). It as good special case of z^z. We have
z^i = ( r e^ (i theta) ) ^i = r^i / e^ theta = ( cos(log(r))+ i sin(log(r)) ) / e^ theta
I could upload 3D plots of real and imaginary parts as functions of modulus and argument of z.
Narasimham

