Modern AlgebraDate: 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/ |
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