Amplitude vs Frequency Response 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 &amp; 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.
The answer to your suplemental question: watch Openheim's free youtube digital signal processing MIT lectures, especially the lectures number 2,3,4, and you will get answers on the first question as well. The lectures are well grounded in math, and I am sure that this is the most efficient way to learn DSP on your own, from the point of view of a mathematician.