Basic Definitions of the Fourier Transform
\begin{align*}
H(f)&=\int_{-\infty}^{\infty}h(t)\,\ee^{-\jj2\pi ft}\,\dd t &&\text{(Fourier transform)},\\
h(t)&=\int_{-\infty}^{\infty}H(f)\,\ee^{\jj2\pi ft}\,\dd f &&\text{(inverse Fourier transform)}.
\end{align*}
In general \(H(f)=R(f)+\jj I(f)\), where
\(H(f)\) is a complex function;
\(R(f)\) is the real part of \(H(f)\);
\(I(f)\) is the imaginary part of \(H(f)\).
Then
\[
\abs{H(f)}=\sqrt{R^2+I^2},\qquad \theta(f)=\tan^{-1}\frac IR.
\]
Some other useful definitions In terms of \(\omega=2\pi f\),
\[
H(\omega)=a_1\int_{-\infty}^{\infty}h(t)\,\ee^{-\jj\omega t}\,\dd t\qquad\&\qquad
h(t)=a_2\int_{-\infty}^{\infty}H(\omega)\,\ee^{\jj\omega t}\,\dd\omega,
\]
with \(a_1a_2=\dfrac1{2\pi}\). Common choices are
\[
a_1=\frac1{2\pi},\ a_2=1;\qquad a_1=1,\ a_2=\frac1{2\pi};\qquad a_1=a_2=\frac1{\sqrt{2\pi}}.
\]
(The choice \(a_1=a_2=1\) is possible only with \(f\) as the variable, as above.)
Fourier transformation in position (\(x\)) and wave-vector (\(k\)) space
\[
F(k)=\int_{-\infty}^{\infty}f(x)\,\ee^{\ii kx}\,\dd x,\qquad
f(x)=\frac1{2\pi}\int_{-\infty}^{\infty}F(k)\,\ee^{-\ii kx}\,\dd k.
\]
