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To properly understand introductory quantum mechanics, I want to understand what the Fourier transform actually gives me mathematically. What book do you recommend? I found one book, but it doesn't get to Fourier transformations until after seven long chapters. Is that what I have to expect?

In case you are wondering what I already know. Here it is:

Single- and multivariable calculus

Linear algebra

Complex analysis (with some topology of complex spaces)

Real analysis (metric spaces, measure theory, Lebesgue integration, some functional analysis)

Abstract algebra (group, ring and field theory)

I just need to understand what the solution of the Schrodingers equation for the free particle actually tells me (for which I have to use the Fourier integral). The "intuitive" explanations are not satisfactory for me.