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Poly1
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How do I find the imaginary part of $\displaystyle \frac{1}{i}xe^{-ix}+e^{ix}$?
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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.
Obviously, there is something elementary I am missing here.
To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix
in the real plane; taking the transpose we get
which then corresponds to a-bi...
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