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I have a function $f: \mathbb{R}\to\mathbb{C}$. How can I proof/argue that $$\int\Re(f(x))\,\mathrm{d}x=\Re\left(\int f(x)\,\mathrm{d}x\right)$$ (and the same for the imaginary part)? I'm afraid I don't have any idea how to start…

The reason I ask is that I need to proof $\widehat{\overline{f}}(n)=\overline{\widehat{f}(-n)}$ and I'm coming to a point where I need the step from $\frac{1}{2\pi}\int_{-\pi}^\pi\overline{f(x)e^{inx}}\,\mathrm{d}x$ to $\frac{1}{2\pi}\overline{\int_{-\pi}^\pi f(x)e^{inx}\,\mathrm{d}x}$.

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How are you defining integrals of complex-valued functions? –  Chris Eagle Oct 16 '12 at 20:00
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Yes, it would be just the definition... –  Berci Oct 16 '12 at 20:06
    
@ChrisEagle Well, good question, I'm not sure. Are there multiple ways to define it? Honestly I never thought about that… Probably that solves the problem, as I just "asked to proof the definition" :-P Thanks! –  siegi Oct 16 '12 at 20:07
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2 Answers 2

Write $f(x)=u(x)+iv(x)\,,$ $u(x)$ and $v(x)$ are real, and $e^{inx}=\cos(nx)+i\sin(nx)$, then express it in terms of real and imaginary parts. Evaluate each integral and then compare the results.

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up vote 0 down vote accepted

Eventually I realized, it is not necessary to pull $\Re$ and $\Im$ out of the integral: I just want to proof that $\int \overline{f(x)}\,\mathrm{d}x=\overline{\int f(x)\,\mathrm{d}x}$ for some complex function $f$, so I only need to pull out the complex conjugation. I think this could be done like this:

$$ \int \overline{f(x)}\,\mathrm{d}x= \int \overline{\Re(f(x))+i\Im(f(x))}\,\mathrm{d}x=\\ \int \Re(f(x))-i\Im(f(x))\,\mathrm{d}x= \int \Re(f(x))\,\mathrm{d}x-\int i\Im(f(x))\,\mathrm{d}x=\\ \overline{\int \Re(f(x))\,\mathrm{d}x+i\int \Im(f(x))\,\mathrm{d}x}= \overline{\int \Re(f(x))+i\Im(f(x))\,\mathrm{d}x}=\\ \overline{\int f(x)\,\mathrm{d}x} $$

As this is valid for all complex functions (please comment if I'm wrong), it is also valid for $f(x)e^{inx}$.

Thanks to all who helped me getting in the right direction! :-)


Now that the question is somewhat misleading, should I delete it or edit the question/title?

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You ask if you should do something with the question/answer - but it seems ok as it stands. But if you think the other answer merits your thanks, I might recommend giving an upvote on it as well (or not - the choice is yours). –  mixedmath Oct 23 '12 at 21:06
    
@mixedmath I only asked because the questions title and the question itself do not match the answer I was searching for. If this is ok, I'm happy, but I don't want to "leave a mess behind"… Thanks for taking a look! –  siegi Oct 24 '12 at 7:16
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