Equivalence of two complex expressions

[TheCanadian]

I found the above while going through my textbook, where the textbook was trying to explain that the principal value of the product of two complex numbers raised to an exponent is not necessarily equivalent to the product of the two complex number each raised to the same exponent first.

Based on the above, what exactly is the difference in the two expressions? Is not $e^{-i \frac {3\pi}{8}}$ equivalent to $e^{i \frac {5\pi}{8}}$?

 

[Mark44]

Is not $e^{-i \frac {3\pi}{8}}$ equivalent to $e^{i \frac {5\pi}{8}}$?

No, they're not equivalent. They refer to points on a unit circle that diametrically opposite one another.

 

[fresh_42]

Isn't $e^{-i \frac{3 \pi}{8}}= e^{2i\pi -i \frac{3 \pi}{8}}= e^{i \frac{13 \pi}{8}} \neq e^{i \frac{5 \pi}{8}}$?

You might want to read the Insight here on PF about it:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/

 

[TheCanadian]

No, they're not equivalent. They refer to points on a unit circle that diametrically opposite one another.

Yikes, it's been a long night. For some reason I mistook the 8 in the denominator of the fraction as a 4. Thank you.