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

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To find the imaginary part of the function $\frac{1}{i}xe^{-ix}+e^{ix}$, the first term can be rewritten using a negative exponent for $i$, applying the property $i^{n}=i^{n+4k}$. The second term utilizes Euler's formula, which states that $e^{\theta i}=\cos(\theta)+i\sin(\theta)$. After correcting a missing $i$ in the calculations, the user successfully derived the answer. The discussion highlights the importance of understanding complex exponentials and their relationship to trigonometric functions. Overall, the method effectively illustrates how to extract the imaginary part of complex functions.
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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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