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The $f_i(x)$ are complex functions and I know that $\int_{-\infty}^{\infty} f_i (x ) \, d x=c_i$.

How can solve this integral?

$$\int_{-\infty}^{\infty} \left| \sum_{i=0}^{N} f_i(x) \right|^2 d x $$

Abolfazl Abdollahi
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  • $f_i$ must be zero, because its integral is a constant. That is, the sum is 0. Integrating 0 squared is just another (complex) constant. – Coolwater Jan 13 '15 at 11:43
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    Which set do you integrate over? Are you integrating over the entire real line or are you perhaps finding an antiderivative? And what else do you assume about your functions? Try to give more details so that others can help. – Joonas Ilmavirta Jan 13 '15 at 11:46
  • By "solve" I think he means "evaluate". And he is integrating over a fixed domain (so they are definite integrals). – GEdgar Jan 13 '15 at 12:20
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    In general, even for the case $N=1$ we cannot do this. Even if we know $\int_0^1 f(x),dx = c$, this does not determine $v=\int_0^1 |f(x)|^2,dx$. You can find two functions $f$ with the same mean $c$ but not the same second moment $v$. – GEdgar Jan 13 '15 at 12:23

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