Small explanation for people not introduced to signals theory:
Fourier transform is the decomposition of an arbitrary signal into many sinusoids, such that their sum gives a signal identical to the starting one. Each of these sinusoids can be represented using Euler's formula as e^(i * omega * t). You can see that the exponent is purely imaginary.
There is a generalization of this transform (i.e. this transform is a particular case of a more general one): what if instead of a purely imaginary exponent we use a generic complex value a+ib?
Then e^((a+ib) * t) = e^(a * t)e^(ib * t): the new term that appears is a real-valued exponential, so exponential curves can also be used describe the starting signal!
However, making a discrete-time, sample-based algorithm of this transform is tricky, and the corresponding inverse transform (the "sum" of the components to get the starting signal back) didn't exist before
There is a generalization of this transform (i.e. this transform is a particular case of a more general one): what if instead of a purely imaginary exponent we use a generic complex value a+ib? Then e^((a+ib) * t) = e^(a * t)e^(ib * t): the new term that appears is a real-valued exponential, so exponential curves can also be used describe the starting signal!
However, making a discrete-time, sample-based algorithm of this transform is tricky, and the corresponding inverse transform (the "sum" of the components to get the starting signal back) didn't exist before