Imaginary Part of a Complex Function: How to Find It?

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SUMMARY

The discussion focuses on finding the imaginary part of the complex function $\displaystyle \frac{1}{i}xe^{-ix}+e^{ix}$. The first term is simplified using the property $\displaystyle i^{n}=i^{n+4k}$, while the second term utilizes Euler's formula, $\displaystyle e^{\theta i}=\cos(\theta)+i\sin(\theta)$. The user successfully derived the imaginary part with these techniques, confirming the effectiveness of the suggested methods.

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How do I find the imaginary part of $\displaystyle \frac{1}{i}xe^{-ix}+e^{ix}$?
 
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For the first term, I would rewrite i with a negative exponent, then apply:

$\displaystyle i^{n}=i^{n+4k}$ where $\displaystyle k\in\mathbb{Z}$

For the second term, apply Euler's formula:

$\displaystyle e^{\theta i}=\cos(\theta)+i\sin(\theta)$
 
Sorry there was an $i$ missing from the first part. But I got the answer using your suggestion. Thanks.
 

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