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I believe that since $|x|^2=x^2$ then we have the Fourier transforms

$$\int_{-\infty}^{\infty} \mathrm dx \frac{\exp{iux}}{a^2+|x|^2} =\int_{-\infty}^{\infty}\mathrm dx \frac{\exp{iux}}{a^2+x^2}$$

even if the function $|x|$ is not analytic at $x=0$. Also, if we are summing over complex numbers the two sums $\sum_n (c_n)^2$ and $\sum_n |c_n|^2$ are not equal if the $c_n$ are complex numbers.

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Yes - the integrals are equal, the sums are not equal. The reason the sums are not equal is far more elementary and relevant than $z$ not being analytic at $z=0.$ What exactly is your question? –  Ragib Zaman Oct 14 '11 at 10:25
    
Yes the integrals are the same since $x$ is real. Also, for complex $c_n$ the sums differ. –  AD. Oct 14 '11 at 10:26
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up vote 1 down vote accepted

The integrals are equal has $x$ is a real number.

The equality $\sum_n c_n^2=\sum_n|c_n|^2$ does not hold in general. The LHS may be a negative real number: take $c_n:=i2^{-n}$.

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