I'm having (another) thick moment (complex numbers)

In summary: The complex conjugate of \cos(x) is \cos(x), as it is a real number. Therefore, the switching of the sign does not go in front of the e. In summary, the equation \cos(x) = \frac{1}{2}(e^{ix}+e^{-ix}) is valid for all real and complex x, and its complex conjugate is also \cos(x). The switching of the sign does not go in front of the e.
  • #1
Brewer
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0
This is a question that's stumping both myself, and my friends who are on maths degrees!

So...

cos(x) can be written as [tex]\frac{1}{2}(e^{ix}+e^{-ix})[/tex] correct?

so does that make its conjugate [tex]\frac{1}{2}(e^{-ix}+e^{ix})[/tex], i.e. cos(x) again? or does the switching of the sign go in front of the e? Its been a long time since I used complex numbers, so I (and my friends) are a little rusty! Any help would be appreciated.

Thanks

Brewer
 
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  • #2
You are assuming that [tex]e^{iz}[/tex] and [tex]e^{-iz}[/tex] are conjugates of each other. This is only true if z is pure real.

If [tex]z=x+iy, (x,y)\in\mathbb R\times \mathbb R)[/tex], then [tex]e^{iz}=e^{-y}e^{ix}[/tex] and [tex]e^{-iz}=e^{y}e^{-ix}[/tex]. Taking the conjugates, [tex]e^{iz^\ast}=e^{-y}e^{-ix} \ne e^{-iz}[/tex] and [tex]e^{-iz^\ast}=e^{y}e^{ix} \ne e^{iz}[/tex].
 
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  • #3
Brewer said:
This is a question that's stumping both myself, and my friends who are on maths degrees!

So...

cos(x) can be written as [tex]\frac{1}{2}(e^{ix}+e^{-ix})[/tex] correct?

so does that make its conjugate [tex]\frac{1}{2}(e^{-ix}+e^{ix})[/tex], i.e. cos(x) again? or does the switching of the sign go in front of the e? Its been a long time since I used complex numbers, so I (and my friends) are a little rusty! Any help would be appreciated.

Thanks

Brewer
In order that your equation [tex]\frac{1}{2}(e^{ix}+e^{-ix})[/tex] be correct, x must be a real number and then cos(x) is a real number. Is the complex conjugate of cos(x) equal to cos(x)? Of course it is: the complex conjugate of any real number is itself!
 
  • #4
You are assuming that \(\displaystyle e^{ix}\) and \(\displaystyle e^{-ix}\) are conjugates of each other. This is only true if \(\displaystyle x\) is pure real.

Well in the example I'm doing this is the case.
 
  • #5
HallsofIvy said:
In order that your equation [tex]\frac{1}{2}(e^{ix}+e^{-ix})[/tex] be correct, x must be a real number and then cos(x) is a real number. Is the complex conjugate of cos(x) equal to cos(x)? Of course it is: the complex conjugate of any real number is itself!
Good that makes me feel better, as that's the reasoning I came up with, and the other thought was conceived by 2 maths students!
 
  • #6
HallsofIvy said:
In order that your equation [tex]\frac{1}{2}(e^{ix}+e^{-ix})[/tex] be correct, x must be a real number ...

Halls, you should know better!

The equation [tex]\cos(x) = \frac{1}{2}(e^{ix}+e^{-ix})[/tex] follows directly from Euler's formula, [tex]e^{ix} = \cos(x) + i\sin(x)[/tex], which is valid for all real and complex x. Thus the given expression for [tex]\cos(x)[/tex] is valid for all real and complex x.
 

Related to I'm having (another) thick moment (complex numbers)

1. What are complex numbers?

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the square root of -1.

2. What is a "thick moment" in the context of complex numbers?

A "thick moment" is a phrase used to describe a moment of confusion or difficulty when working with complex numbers. It could refer to a time when someone is having trouble understanding a concept or solving a problem involving complex numbers.

3. How are complex numbers used in science?

Complex numbers are used in many areas of science, including physics, engineering, and mathematics. They are particularly useful in fields that involve waves or oscillations, such as electromagnetism and quantum mechanics.

4. Can you give an example of a complex number in real life?

One example of a complex number in real life is in the field of electricity, where the impedance of a circuit is represented by a complex number. The real part of the number represents the resistance of the circuit, while the imaginary part represents the reactance.

5. How can I improve my understanding of complex numbers?

One way to improve your understanding of complex numbers is to practice solving problems involving them. You can also read textbooks or watch online tutorials to learn more about their properties and how they are used in different fields of science. Working with a tutor or joining a study group can also be helpful in gaining a better understanding of complex numbers.

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