# Learning Fourier Integral

I am learning Fourier Integral for real numbers. I downloaded a presentation of some university. I am summing up what I know and hopefully someone will correct me where I am wrong.

• Fourier integral can be calculated only for functions that are decaying (what ever that means) and non-periodic
• The formula for Fourier Integral : $$f(x) = \int_{0}^{\infty}(A(\lambda)Cos(\lambda x)+B(\lambda)Sin(\lambda x)))\ d\lambda$$ where $\lambda$ is real numbers greater than or equal to zero.

Also my question is:
Fourier Series allows you to express a piece-wise continuous, periodic function as a sum of infinite terms. What does Fourier integral do ?

• You can think of the Fourier transform as being a generalization of the coefficients you see in Fourier series. – oldrinb Jul 24 '13 at 3:49

Basically, yes. "Decaying" can be interpreted as "integrable" or "square integrable" (in terms of function spaces, $L^1$ or $L^2$). "Non-periodic" is redundant, since a function integrable in any sense cannot be periodic unless it's identically zero.