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For example, how do I find the value of:

$$ (\cos(e) + i\times\sin(e))^{\frac{\pi}{2}} $$

I've did a bit of searching and found this amazing question, and it gave me two ways "to raise a non-complex number to an irrational power":

  1. You can pick a sequence of rational numbers $x_n$ converging to $x$ (i.e., $\lim\limits_{n\to\infty} x_n = x$) and define $$a^x = \lim_{n\to\infty} a^{x_n}.$$
  2. You can use the exponential function $e^x$ (defined in many ways, say as $e^x = \lim_{n\to\infty} (1+\frac{x}n)^n$ or with a power series), and its inverse the logarithmic function that satifies $e^{\ln t} = t$ for all positive $t$, and since $a = e^{\ln a}$, define $$a^x = e^{x \ln a}.$$

The first way won't work with transcendental numbers such a $ \pi $, $ \frac{\pi}{2}$ and its family.

The second way will end up giving me $ \ln(i) $ in an exponent, which I'd rather not endure right now.

I tried De Moivre's, but I noticed that it only works for integer powers.

So how should I approach $(\cos(e) + i*\sin(e))^{\frac{\pi}{2}} $?

1 Answers1

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$$ (\cos e + i \sin e) ^ {\pi \over 2} $$ $$ = \left( e^ {i \times e} \right) ^ {\pi \over 2} $$ (by Euler's formula) $$ = e ^ {i \times \left({\pi \over 2} \times e\right)} $$ $$ = \cos \left({\pi \over 2} \times e\right) + i \sin \left({\pi \over 2} \times e\right) $$

Warning: As GEdgar wrote in the comment below, because of branch cut, $e^{x\times y}=\left(e^x\right)^y$ is not true for all complex $x$ and $y$. Wikipedia reference.

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