# Mathematica's definition of Fourier Transform

I would to know how is defined this function in Mathematica:

FourierTransform[expr,t,ω]


That function give the Fourier Transform of a complex function of real variable with the angular frequency ω=2πv. Is correct?

## Update

Thanks for the comments, but why if I try to input the code below isn't the same thing?

1/Sqrt[2 \[Pi]]*Integrate[(Sin[t] + I)*E^(I \[Omega] t), {t, -Infinity, Infinity}, PrincipalValue -> True, Assumptions -> \[Omega] \[Element] Reals]

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Have you looked at the documentation? Have you tried clicking on "More Information"? – rm -rf Oct 3 '12 at 15:34
Take a look for yourself: reference.wolfram.com/… – David Carraher Oct 3 '12 at 15:35
Two reasons: 1. you're using curly braces ({ and }) as grouping constructs - change them to parentheses, and 2. E^I\[Omega]t is treating I\[Omega]t as a single variable - add spaces between the symbols. – rcollyer Oct 3 '12 at 15:52
You need to add PrincipalValue -> True and assume w is real in your integration. You can do the rest of the math – rm -rf Oct 3 '12 at 16:07
No, the FT and integral you show are not the same thing. The integral is not convergent. Fourier transforms are defined as integrals for a large class of functions (the ones in $L_1(\mathbb R)$). But they extend also to certain classes of functions that are not integrable over the entire real line. – Daniel Lichtblau Oct 3 '12 at 16:08