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 $\frac{1}{2\pi}\overline{\int_{-\pi}^\pi f(x)e^{inx}\,\mathrm{d}x}$.