Complex conjugate of absolute exponential

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Discussion Overview

The discussion revolves around finding the complex conjugate of the expression (-0.5)^abs(x), exploring the implications of complex conjugation and the nature of the expression based on the absolute value function. The scope includes conceptual clarification and mathematical reasoning.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant, Brian, seeks assistance in determining the complex conjugate of (-0.5)^abs(x).
  • Another participant questions the definition of complex conjugation and suggests applying it to the expression.
  • A participant notes that abs(x), denoted |x|, is a nonnegative real number, implying that the expression is real and thus its conjugate is straightforward to compute.
  • However, a different participant challenges this by asking what happens if x=1/4, indicating that the situation may not be as simple as it seems.
  • There is a suggestion to express the problem using complex exponential or trigonometric notation (cos + i sin) to facilitate the computation of the conjugate.
  • A participant expresses confusion and seeks clarification on the equations related to complex exponentials and their relationship to the original expression.
  • Another participant elaborates on the idea of representing the expression in the form r e^(i theta) for some theta and r, confirming the previous mention of cos + i sin notation.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are differing views on the nature of the expression and the implications for finding its complex conjugate. Some argue it is straightforward due to its real nature, while others raise questions about specific cases that complicate the matter.

Contextual Notes

There are limitations in the discussion regarding the assumptions about the values of x and the implications of using absolute values in the context of complex numbers. The discussion does not resolve these nuances.

vg8open
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Hello all,

I am trying to figure out how to solve for the complex conjugate of the following: (-0.5)^abs(x)

Thanks for your help.

-Brian
 
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What is the definition of complex conjugation?? Try to apply that first on your expression.
 
vg8open said:
Hello all,

I am trying to figure out how to solve for the complex conjugate of the following: (-0.5)^abs(x)

Thanks for your help.

-Brian

abs(x), usually denoted |x|, is a nonnegative real number whether x is real or complex.

So your expression is real. Which makes its conjugate very easy to compute!
 
SteveL27 said:
abs(x), usually denoted |x|, is a nonnegative real number whether x is real or complex.

So your expression is real. Which makes its conjugate very easy to compute!

Not necessarily. What if x=1/4?
 
Char. Limit said:
Not necessarily. What if x=1/4?

Oops missed the minus sign. Thanks.
 
vg8open said:
Hello all,

I am trying to figure out how to solve for the complex conjugate of the following: (-0.5)^abs(x)

Thanks for your help.

-Brian

It's a whole lot easier if you put it into complex exponential, or even better, cos + i sin notation.
 
Char. Limit said:
It's a whole lot easier if you put it into complex exponential, or even better, cos + i sin notation.

I think I missing something here... Are you talking about these equations?
a^b = e^{(\ln(r) + \phi i)b} and e^{ix} = \cos(x) +i\sin(x)
 
Well, by complex exponential, I just mean putting it into r e^(i theta) for some theta and r. But the cos + i sin notation I was talking about, yeah, you got it.
 

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