What are the different forms of Fourier notation and how are they connected?

[Jhenrique]

The Fourier integrals and series can be written of 3 forms (possibly of 4):

the "real cartesian":
a(ω)cos(ωt) + b(ω)sin(ωt)

the "real polar":
A(ω)cos(ωt - φ(ω))

where:
A² = a² + b²
sin(φ) = b/A
cos(φ) = a/A
tan(φ) = b/a

the "complex polar"
A(ω)exp(iφ(ω))exp(iωt)

And my doubts are: 1) exist a "complex cartesian" correspondent? 2) is possible to connect the real forms with the complex forms?

 

[mathman]

The Euler identity is the connection.
exp(ix) = cos(x) + isin(x)