I have seen the proof by induction for all integers, and I have also seen in a textbook that we can use Euler's formula to prove it true for all rational n, but nowhere in the book does it say its true for irrational n.

I have also looked over the internet and there seems to be some problem with non-integer values for n (as I understand, a problem of uniqueness, but I'm not sure).

I would appreciate it if someone could just clarify this for me.

Thanks in advance!

  • 5
    $\begingroup$ $\cos (z\varphi) + i \sin (z\varphi)$ is for all complex $z$ (why stop at real?) one possible value of $(\cos \varphi + i\sin\varphi)^z$, if that is your question. $\endgroup$ – Daniel Fischer Dec 3 '13 at 11:24
  • $\begingroup$ I see. Thankyou. $\endgroup$ – Joe S Dec 3 '13 at 11:27

The formula is actually true in a more general setting: if $z$ and $w$ are complex numbers, then $\left(\cos z + i\sin z\right)^w$ is a multi-valued function while $\cos (wz) + i \sin (wz)$ is not. However, it still holds that $\cos (wz) + i \sin (wz)$ is one value of $\left(\cos z + i\sin z\right)^w$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.