Complex power raised over real number.

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Homework Help Overview

The discussion revolves around the operation of raising a complex number to a real power, specifically questioning the validity of such operations and exploring the implications of different approaches to the problem.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss whether raising a complex power over a real number is permissible and share their methods for solving the problem. There are mentions of numerical accuracy and the importance of retaining more digits in calculations. Some participants also raise questions about the existence of multiple solutions based on the choice of branch or argument.

Discussion Status

The discussion is active, with participants providing feedback on each other's methods and questioning assumptions about the periodic nature of trigonometric functions. There is no explicit consensus, but several participants have offered guidance on maintaining accuracy in numerical computations.

Contextual Notes

Some participants note that the original question is not commonly found in textbooks, indicating a potential gap in resources for learning about complex numbers at a pre-college level.

PrashntS
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1. I actually don't know if such kind of operation is even allowed.

A friend of mine raised this question, that can we raise a complex power over a real number. I solved it this way. Is this correct?

http://i45.tinypic.com/254vwux.jpg

Homework Statement


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The Attempt at a Solution

 
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Yes, that's fine.
 
Yes, you did it correctly. I am not quite sure approximating the transcandental functions with decimals, but I suppose that's fine.
 
PrashntS said:
1. I actually don't know if such kind of operation is even allowed.

A friend of mine raised this question, that can we raise a complex power over a real number. I solved it this way. Is this correct?

http://i45.tinypic.com/254vwux.jpg

Your method is OK, but if you plan to use the results in further numerical computations, you should keep more digits of accuracy; nowadays, in this computer age, there is no barrier to retaining more digits. For example, it might be better (depending on future uses) to write
[tex]3^{5i} = e^{i 5 \ln 3 } = \cos(5 \ln 3) + i \sin(5 \ln 3) \doteq <br /> 0.7037573 - 0.7104404 i\, .[/tex]

RGV
 
Ray Vickson said:
Your method is OK, but if you plan to use the results in further numerical computations, you should keep more digits of accuracy; nowadays, in this computer age, there is no barrier to retaining more digits. For example, it might be better (depending on future uses) to write
[tex]3^{5i} = e^{i 5 \ln 3 } = \cos(5 \ln 3) + i \sin(5 \ln 3) \doteq <br /> 0.7037573 - 0.7104404 i\, .[/tex]

RGV

Yeah this is obviously much better, I was just looking whether it is even possible or not as none of the books I use has such question.
Would be awesome if you could suggest some good book for complex number (pre collage).
 
Are you also taking into account that there are infinitely-many solutions depending

on your choice of branch/argument?
 
Bacle2 said:
Are you also taking into account that there are infinitely-many solutions depending

on your choice of branch/argument?

well that's obvious, isn't it? trigo fn is periodic, i just stayed in principle branch
 

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