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        riddles >> putnam exam (pure math) >> Complex powers
        
(Message started by: Sameer on Sep 18th, 2007, 10:14pm)
Title: Complex powers
Post by Sameer on Sep 18th, 2007, 10:14pm
If ii^i...http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif= A + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/imath.gifB, then find

1) tan(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifA/2)

2) A2 + B2


Note: How do you do multiple sup? The question above is i^i^i...http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif
Title: Re: Complex powers
Post by JP05 on Sep 18th, 2007, 10:35pm
Since your are asking does  i^i = e^(-1/2 pi) or i^i = (-1)^ (1/2 i)? That is, for starters anyway.
Title: Re: Complex powers
Post by Sameer on Sep 18th, 2007, 10:46pm

on 09/18/07 at 22:35:40, JP05 wrote:
Since your are asking does  i^i = e^(-1/2 pi) or i^i = (-1)^ (1/2 i)? That is, for starters anyway.


Former.. Latter is not valid!! That would what a "certain individual" would do and give bad answers!!  ::)
Title: Re: Complex powers
Post by TenaliRaman on Sep 18th, 2007, 10:51pm
[hide]Bluntly,
x = i^x
log(x) = x log i
(1/2)(A^2 + B^2) + i arctan(A/B) = iA(pi/2) - B(pi/2)
tan(A(pi/2)) = A/B
A^2 + B^2 = -Bpi
Ofcourse, we could go further with x = W(-log(i))/log(i) and probably separate A and B as two series. Havent tried it, maybe it simplifies further.
[/hide]
Title: Re: Complex powers
Post by JP05 on Sep 18th, 2007, 10:59pm

on 09/18/07 at 22:46:53, Sameer wrote:
Former.. Latter is not valid!! That would what a "certain individual" would do and give bad answers!!  ::)


Really?  That's not what I know about complex numbers.  That is, both are valid.
Title: Re: Complex powers
Post by JP05 on Sep 18th, 2007, 11:06pm
As a follow up, I am not putting forth my remarks in reference to "any particular person" but I am simply being as precise mathematically as I know how. In this end,  i^i  = (-1)^(1/2 i) is precise in one of a number of ways i^i can be wrote.

Title: Re: Complex powers
Post by Sameer on Sep 18th, 2007, 11:07pm

on 09/18/07 at 22:51:51, TenaliRaman wrote:
[hide]Bluntly,
x = i^x
log(x) = x log i
(1/2)(A^2 + B^2) + i arctan(A/B) = iA(pi/2) - B(pi/2)
tan(A(pi/2)) = A/B
A^2 + B^2 = -Bpi
Ofcourse, we could go further with x = W(-log(i))/log(i) and probably separate A and B as two series. Havent tried it, maybe it simplifies further.
[/hide]


For 1) I actually have the inverse!! And for 2) I have  your answer on an exponent!!

Note: Alright, these look like easy problems.. I will move to the next chapter in my book!!  ;)


on 09/18/07 at 23:06:59, JP05 wrote:
As a follow up, I am not putting forth my remarks in reference to "any particular person" but I am simply being as precise mathematically as I know how. In this end,  i^i  = (-1)^(1/2 i) is precise in one of a number of ways i^i can be wrote.


Ah don't worry about that. It was my feeble attempt at being funny which came across terribly. I am not sure this representation is legal unless someone here can correct me or support me!!

Edit: -1 = eihttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/suppi.gifimplies both are same things!! So you should be able to use either!! (So both are legal)
Title: Re: Complex powers
Post by JP05 on Sep 18th, 2007, 11:17pm
There is often a lot of nonsense when it comes to complex numbers and it simply has to do with the fact they they are, well, complex, and really don't play out like we expect real numbers to. In fact, over time you will discover what I just said the hard way.

So, someone give us a limit for that infinite complex exponentiation so we can put this thing to bed. I am looking for a limit -- that is, at least we can have  one-sided continuity.
Title: Re: Complex powers
Post by JP05 on Sep 18th, 2007, 11:31pm
Dont write -1 = e^pi because -1 has other representations besides e^pi. I have seen mathematicians write stuff like that but they were being funny at the time.

Say,  e^pi = -1.
Title: Re: Complex powers
Post by Sameer on Sep 18th, 2007, 11:33pm

on 09/18/07 at 23:31:54, JP05 wrote:
Dont write -1 = e^pi because -1 has other representations besides e^pi. I have seen mathematicians write stuff like that but they were being funny at the time.

Say,  e^pi = -1.


Of course, I am an engineer so I tend to overlook these things...
Title: Re: Complex powers
Post by JP05 on Sep 18th, 2007, 11:48pm
Cool. I am still looking for that limit though. I would be pleased to work with it here if you can give me one, as 1 and 2 in this problem would be trivial.

The thing to consider is really what happens as you continue to exponentiate  i, regardless if the exponents resolve to real or complex numbers ultimately. This is why I want to see that limit formula.

Well, there is one: how many shapes can it take: i or -1 ...huh?  I know the answer!
Title: Re: Complex powers
Post by ThudanBlunder on Sep 19th, 2007, 6:14am
When http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rmi.gif is exponentiated upwards n times (downwards is normal), it is alternately real or complex, depending on n. To say that it equals a + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rmi.gifb when exponentiated an infinite number of times is to say that infinity is either odd or even!
Title: Re: Complex powers
Post by towr on Sep 19th, 2007, 7:22am

on 09/19/07 at 06:14:21, ThudanBlunder wrote:
When http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rmi.gif is exponentiated n times, it is alternately real or complex, depending on n. To say that it equals a + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/rmi.gifb when exponentiated an infinite number of times is to say that infinity is either odd or even!
Unless it converged on 0.
(But it doesn't, does it?)
Title: Re: Complex powers
Post by ThudanBlunder on Sep 19th, 2007, 8:57am

on 09/19/07 at 07:22:25, towr wrote:
Unless it converged on 0.
(But it doesn't, does it?)

Doesn't the sequence repeat itself?
e-http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2 to the power of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/fraki.gif = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pm.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/fraki.gif, right?
Title: Re: Complex powers
Post by Sameer on Sep 19th, 2007, 6:20pm
You don't need to find A and B by themselves...
Title: Re: Complex powers
Post by srn347 on Sep 19th, 2007, 6:36pm
How do you apply complex powers? I'm only familiar with e being the base when there are complex powers(or 1 which is still 1 always). e^pi is not -1 though.
Title: Re: Complex powers
Post by Sameer on Sep 19th, 2007, 6:53pm

on 09/19/07 at 18:36:35, srn347 wrote:
How do you apply complex powers? I'm only familiar with e being the base when there are complex powers(or 1 which is still 1 always).


You can use the fact that ehttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/supiota.gif2khttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/suppi.gif= 1

This is a little beyond high school math. You would want to pick a book that has a chapter on complex numbers, Argand's diagram, De Moivre's theorem, hyperbolics, etc. That would help!


on 09/19/07 at 18:36:35, srn347 wrote:
e^pi is not -1 though.


Corrected!
Title: Re: Complex powers
Post by srn347 on Sep 19th, 2007, 10:29pm
that is something I already understand. How do I apply complex powers to something that isn't e(or 1)?
Title: Re: Complex powers
Post by Sameer on Sep 19th, 2007, 11:00pm

on 09/19/07 at 22:29:58, srn347 wrote:
that is something I already understand. How do I apply complex powers to something that isn't e(or 1)?


I am not sure I understand the source of your confusion. A complex number can be expressed as rehttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/supi.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/suptheta.gif

A complex number can be thought up as a point in a z plane, thus expressable in terms of distance from origin and angle from positive x axis.
Title: Re: Complex powers
Post by Grimbal on Sep 20th, 2007, 7:08am

on 09/19/07 at 22:29:58, srn347 wrote:
How do I apply complex powers to something that isn't e(or 1)?


Let's take xy

If y is an integer, you can define it as successive multiplications.  If x is e, you can define it as exp(x) where the exp() function is defined as
   exp(x) = sum(xn/n!)

In the general case
   xy = exp(y·ln(x)).
where the ln() function is the inverse of the exp() function.

The problem in the general case is that the solution to  
  exp(z) = x
is not unique for a given x.  If z is a solution,
  z' = z + i·k·2·pi
is also a solution.  If y is integer it doesn't matter, the different solutions fold back to a single value after applying exp().  If y is rational, y=p/q, the solutions fold back to q distinct values.  In the general case there can be an infinity of values, and you have to choose one.

Out of my head, I cannot think of an good use for complex exponents, except for the case ez which is just a way to write the exp() function.

[Sorry, wrong button -Eigenray]
Title: Re: Complex powers
Post by ThudanBlunder on Sep 20th, 2007, 8:16am

on 09/20/07 at 07:08:37, Grimbal wrote:
Out of my head, I cannot think of an good use for complex exponents...

Fourier Transforms?
Title: Re: Complex powers
Post by Grimbal on Sep 20th, 2007, 8:19am
I meant other than the special case ez.
Title: Re: Complex powers
Post by Eigenray on Sep 20th, 2007, 6:52pm

on 09/20/07 at 07:08:37, Grimbal wrote:
Out of my head, I cannot think of an good use for complex exponents, except for the case ez which is just a way to write the exp() function.

[link=http://en.wikipedia.org/wiki/Dirichlet_series]Dirichlet series[/link].

Yes, it is pretty rare to see wz (unless zhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbz.gif, or w=e, or whttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif), but this is probably because it is not well-defined.  The whole point of common notation, after all, is that you don't have to define it every time you use it.

[Argh!  I accidentally replaced Grimbal's entire post with this one.  Lucky I noticed before closing the tab!  And this happened right after I posted about this very problem!!]
Title: Re: Complex powers
Post by srn347 on Sep 26th, 2007, 7:27pm
How do you define ni when n is not 1, -1, or or e to some power.
Title: Re: Complex powers
Post by Grimbal on Sep 27th, 2007, 12:28am

on 09/26/07 at 19:27:37, srn347 wrote:
How do you define ni when n is not 1, -1, or or e to some power.

Just apply what I explained in post 19.
Title: Re: Complex powers
Post by srn347 on Sep 27th, 2007, 4:48pm
Your 19th post doesn't fully define it, since some numbers have multiple natural logs(all numbers have multiple logs actually).
Title: Re: Complex powers
Post by Grimbal on Sep 28th, 2007, 8:47am
You are right, complex logs are not fully defined.

So what are possible values for ln(n) in the complex plane?

What does ei·ln(n) give for each value?
Title: Re: Complex powers
Post by srn347 on Sep 28th, 2007, 4:47pm
For any log, 2pi(i)can be added or subtracted infinitely, since it is the same as multiplying or dividing by 1.


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