Consider the simple first-order low pass filter, described by
$\tau \frac{d \mathcal{O}}{dt} + \mathcal{O}(t)=\mathcal{I}(t)$
Considered as a linear system (ie $\mathcal{L} \mathcal{O} = \mathcal{I}$), $\mathcal{L}$ has transfer function $\frac{1}{1 + i \omega \tau}$ (up to maybe some constants, minus signs etc etc depending on convention). By transfer function, I mean $\tilde{R}(k)$ such that if $\tilde{\mathcal{I}}$ denotes the Fourier/Laplace transform of the input, then $\tilde{\mathcal{O}}(t) = \tilde{R}(t)\tilde{\mathcal{I}}(t)$. Questions:
1) Is the transfer function the 'same thing' as the frequency response? The reference textbook I have (Signals and Systems by Oppenheim & Willsky) talks of the frequency response of the 1st order LPF $H(i\omega) = \frac{1}{1 +i \omega \tau}$ which as far as I can tell amounts to something very similar.
2) If we solve the LPF ODE for a sinusoidal input $sin(\omega t)$, then we get an expression for the output amplitude $A(\omega) = \frac{1}{\sqrt{1+\omega^2 \tau^2}}$. How is this related to the transfer function/frequency response?
Supplementary information/question: I'm actually a mathematics student in the situation of having to learn a lot of signals processing very quickly this summer. Are there any textbooks written out there which explain signals and systems in the perspective of a maths student? As far as I can tell, most books seem to pitched towards engineers (and also use $j$ instead of $i$), and are a little different to what I'm used to - I don't want to learn lots of things twice from two different perspectives.
Thank you very much for you time.
