Representations of the Fourier's integral

[Jhenrique]

If exist 3 representations for Fourier series (sine/cosine, exponential and amplite/phase) and at least two Fourier integral that I know
$$f(t)=\int_{0}^{\infty }A(\omega)cos(\omega t) + B(\omega)sin(\omega t)d\omega$$
$$f(t)=\int_{-\infty }^{+\infty }\frac{e^{+i\omega t}}{\sqrt{2\pi}} F(\omega)d\omega$$
So, exist too a representation for Fourier integral in amplitude/phase notation? How is it?
Other question: $A(\omega)$ and $B(\omega)$ are the sine and cosine transforms?
And another ask: I can relates F(ω) with A(ω) and B(ω), like we do in series Fourier $\left (|c_{n}|=\frac{1}{2}\sqrt{a_{n}^{2} + b_{n}^{2}} \right )$?

 

[jvicens]

Hello, thank you for your questions. Yes, there is also a representation for Fourier integral in amplitude/phase notation. It is given by:

f(t)=\int_{-\infty }^{+\infty }A(\omega)\cos(\omega t + \phi(\omega))d\omega

where A(ω) is the amplitude and φ(ω) is the phase shift of the signal at frequency ω.

To answer your second question, A(ω) and B(ω) are not necessarily the sine and cosine transforms. They are the coefficients of the Fourier series expansion in the sine/cosine representation. In the exponential representation, they are the real and imaginary parts of the complex Fourier coefficients.

Finally, to relate F(ω) with A(ω) and B(ω), you can use the following relationship:

F(\omega) = \frac{1}{\sqrt{2\pi}}\left(A(\omega) - iB(\omega)\right)

This is known as the complex Fourier transform. So, you can find the amplitude and phase information from the complex Fourier transform. I hope this helps clarify things for you. Let me know if you have any further questions.