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Ok, so if $f$ is a real function, then I know how to calculate $\int f(x) dx$.

If $f(x)$ is a complex-valued function, i.e. an expression that involves i, then $\int f(x) dx$ is calculated in the same way as before: I just treat $i$ as a constant.

However, now I am reading a book which writes integrals as $\int Re(f(x)) dx$ where $Re()$ denotes the real part of $f$.

And this, I don't understand. I don't know what the real part of $f$ is, so this expression seems to me impossible to calculate? The book also doesn't tell me how I can calculate this.

So how do I calculate this integral? In other words, how do I figure out what the real part of a function is? Or can the integral be calculated in another way?

sugarpit
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  • I don't know what $f$ is here, but that doesn't make these integrals non-meangingful. – Angina Seng Oct 05 '19 at 01:43
  • I am not saying they aren't meaningful, I am saying how do you calculate them? If I give you a complicated expression for $f$, you can always calculate it. Worst case scenario, you use a trapezoid rule and use an approximation instead. But you can always do something because you know what $f$ is. But you don't know what $Re(f)$ is so calculation is impossible, isn't it? – sugarpit Oct 05 '19 at 01:47
  • I don't quite understand the difficulty. The symbol $\mathrm{Re}$ is just the name of another function from $\mathbb C$ to $\mathbb C$. Any complex number $z$ can be written $u + iv$ where $u$ and $v$ are real, and then $\mathrm{Re}(z) = u+i0.$ Now if you take $\mathrm{Re}(f(x))$ you simply have chained two functions together, nothing more complicated than $g(f(x))$ where $g$ is any other function. Perhaps if you would edit the question to show exactly what the book was asking you to do, the question would be clearer. – David K Oct 05 '19 at 02:24

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Assuming that $x$ is a real number the $Re(f(x))$ is the real part of $f(x)$ which is the part of $f(x)$ which does not involve $i$

For example if you have $$f(x) = 3x^2 + 2ix^3$$

Then the real part of $f(x)$ is $$ Re(f) = 3x^2$$ and the integral is $$\int Re(f(x))dx = \int 3x^2 dx = x^3+C $$

Mohammad Riazi-Kermani
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