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Suppose we have the complex exponential $e^{jx}.$

I manipulate the exponential as follows:

$e^{jx} = (e^{j2\pi})^{\frac{x}{2\pi}} = 1^{\frac{x}{2\pi}} = 1.$

My question is if the power property in the first equation is valid.

Also if you can provide some additional information or theory about that it would be appreciated.

Toni Ivanov
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    Cf. this question; the general rule $(a^m)^n=a^{m\times n}$ does not always work when $m$ and $n$ are not integers – J. W. Tanner Mar 29 '20 at 17:44
  • Im studying about fourier series and a lot of computation of this kind needs to be done. Does that mean I will have to remember when the rule works? Because I havent seen any reference about that in the textbook im using. – Toni Ivanov Mar 29 '20 at 18:12

1 Answers1

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You can use the general rule $(a^m)^n=a^{m\times n}$ for real $a>0$ or when $m$ and $n$ are integers.

In other situations, it could lead to nonsense such as $-1=(-1)^1=((-1)^2)^{1/2}=1^{1/2}=1$.

J. W. Tanner
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