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I am bit confused by Carr and Madan's paper. In it the authors write that the Fourier transform $ c_T(k)$ is defined by

\begin{align} \psi_T(v) = \int_{ - \infty}^{\infty} e^{ivk} c_T(k)dk \end{align}

Yet traditionally it is know that the Fourier Transform formula is \begin{align} X( \omega) = \int_{- \infty}^{\infty} x(t) e^{ - i \omega t} dt \end{align}

The negative being the difference between the two.

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    $\begingroup$ as has been said, it's just a convention that varies from field to field. I present various different approaches and show how they agree, in my book More Mathematical Finance. $\endgroup$ – Mark Joshi Mar 27 '17 at 5:18
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Different fields use different conventions to define the Fourier transform. The one in Carr and Madan is often referred to as the "probabilist's Fourier transform". It coincides with the definition of the characteristic function.

See https://www.johndcook.com/blog/fourier-theorems/ for a nice overview and the link between the different definitions.

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