Modern Algebra

Date: 07/10/97 at 17:55:34
From: Cathy Fyle
Subject: Modern algebra

Our professor has asked us to show that the natural log of i 
to the 1/2 is equal to i times pi over 4. I have not had complex 
numbers nor do I know how to go about showing this is true.  
Can you help?

Date: 07/11/97 at 17:27:51
From: Doctor Anthony
Subject: Re: Modern algebra

If you are new to complex numbers, I think your professor is expecting 
rather a lot of you to work with natural logs and complex numbers.  
However, let us see what can be done.

   ln(i^(1/2)) = (1/2)ln(i)

Now let  ln(i) = x

             i = e^(x)

But we also have the identity:  

 i = cos(pi/2) + i.sin(pi/2)  =  e^(i.pi/2)   

and equating the two expressions for i

        e^(x) = e^(i.pi/2)   so the powers of e must be equal

            x = i.pi/2

       (1/2)x = i.pi/4

   (1/2)ln(i) = i.pi/4

  ln(i^(1/2)) = i.pi/4

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/