Is this property valid for complex exponentials?

[biometrix]

Hello,

suppose we have the complex exponential [MATH]e^{j\cdot x}.[/MATH] and consider the following computation:

[MATH]e^{j\cdot x} = (e^{j\cdot 2\pi})^{\frac{x}{2\pi}} = 1^{\frac{x}{2\pi}} = 1.[/MATH]
Is this power property invalid for the complex exponentials ?

 

[Romsek]

[MATH]e^{j x} \neq \left(e^{j 2\pi }\right)^{\frac{x}{2\pi}}[/MATH]
letting \(\displaystyle x=\pi\) should convince you of that

[MATH]-1 = e^{j \pi} \neq \left(e^{j 2\pi}\right)^{1/2} = 1^{1/2} = 1[/MATH]

 

[biometrix]

[MATH]e^{j x} \neq \left(e^{j 2\pi }\right)^{\frac{x}{2\pi}}[/MATH]
letting \(\displaystyle x=\pi\) should convince you of that

[MATH]-1 = e^{j \pi} \neq \left(e^{j 2\pi}\right)^{1/2} = 1^{1/2} = 1[/MATH]

Can you provide me any source of theory about that topic ? Any book that specifies that? Thanks.

 

[Romsek]

Can you provide me any source of theory about that topic ? Any book that specifies that? Thanks.

you can search google as well as I can

 

[pka]

Can you provide me any source of theory about that topic ? Any book that specifies that?

Any standard textbook in complex variables will discuss this topic.
The fact is that if each of \(w~\&~z\) is a complex number then we define \(z^w=\exp(w\log(z))\).
Because that definition involves the complex logarithm the valid rules for exponents must be restricted.
This is from a standard text: Some of the rules for exponents carry over from real numbers but not all.
i) \(z^{-w}=\dfrac{1}{w}\)
ii) \(z^c z^d=z^{c+d}\)
iii) \(\dfrac{z^c}{z^d}=z^{c-d}\)
iv) BUT \(\bf\left(z^w\right)^C=z^{Cw}\) only if \(\bf C\) is an integer.
Doesn't part iv) answer your question?

 

[biometrix]

Any standard textbook in complex variables will discuss this topic.
The fact is that if each of \(w~\&~z\) is a complex number then we define \(z^w=\exp(w\log(z))\).
Because that definition involves the complex logarithm the valid rules for exponents must be restricted.
This is from a standard text: Some of the rules for exponents carry over from real numbers but not all.
i) \(z^{-w}=\dfrac{1}{w}\)
ii) \(z^c z^d=z^{c+d}\)
iii) \(\dfrac{z^c}{z^d}=z^{c-d}\)
iv) BUT \(\bf\left(z^w\right)^C=z^{Cw}\) only if \(\bf C\) is an integer.
Doesn't part iv) answer your question?

Yes indeed that answers it.Thanks!

 

[Steven G]

I used to think that you can replace equals with equals always. Then one day I realized that is not true for powers. For example (-1)^(1/2) = i. Note that 2/4 = 1/2. But (-1)^(2/4) = 1

 

[topsquark]

Using the parenthesis you wrote [math](-1)^{(1/2)} = i[/math] and [math](-1)^{(2/4)} = (-1)^{(1/2)} \neq 1[/math]!

Taking powers of negative numbers can be very problamatic.

-Dan

 

[Cubist]

FYI: In this post(and the one after) are the constraints for when the "power of power" rule can be applied with real inputs (possibly returning a complex result)