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Topic: What's the geometry meaning of i^i ?
Replies: 16   Last Post: Jul 29, 2012 2:34 PM

 Messages: [ Previous | Next ]
 Narasimham Posts: 359 Registered: 9/16/06
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 &lt;hongyi.z...@gmail.com&gt; wrote:
> &gt; Hi all,
> &gt;
> &gt; What&#39;s the geometry meaning of i^i ?  Here, i is the imaginary unit.
> &gt;
> &gt; Regards
> &gt; --
> &gt; .: 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&#39;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

Date Subject Author
7/25/12 Hongyi Zhao
7/25/12 Pfsszxt@aol.com
7/25/12 Frederick Williams
7/25/12 Brian Q. Hutchings
7/25/12 Gottfried Helms
7/26/12 Hongyi Zhao
7/26/12 Gottfried Helms
7/26/12 Narasimham
7/27/12 Narasimham
7/26/12 Richard Tobin
7/27/12 Ken S. Tucker
7/27/12 Butch Malahide
7/27/12 Ken S. Tucker
7/27/12 Frederick Williams
7/27/12 Ki Song
7/29/12 Michael Press
7/29/12 Brian Q. Hutchings