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\int_{-\tau}^{+\tau} \frac{e^{-itx} - e^{-ity}} {it}\, \phi_X(t)\, dt.</math>
In general this is an [[improper integral]]; the function being integrated may be only conditionally integrable rather than [[Lebesgue integral|Lebesgue-integrable]], i.e. the integral of its [[absolute value]] may be infinite.
Characteristic functions are used in the most frequently seen proof of the [[central limit theorem]].
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