Is this what you would expect to see? This is magnitude of change in gain with frequency and phase of change in gain with frequency with the above equation used.
I chose a gain of R2/R1 as 0.5
R2 varies by 1% at a frequency of 1kHz.
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As I increase the R2 percent variation, the peak gets wider and takes more spectrum. If it is a much smaller variation, the peak is much more local around the R2 frequency. No matter what, they always approach infinity at the R2 frequency and the DC gain is always 0.5 (R2(0)/R1). I am interpreting the infinite peak at the R2 frequency as change in R2 at this frequency divided by change in R1, which would be infinity. At any other frequency, it is the phase and frequency of the input signal interacting with this R2 frequency. Also, if I lower the frequency of R2 change close to DC, the gain is always -6dB (0.5) across the entire spectrum such that the variation has no frequency affect.
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You are right. I just am not sure if a rapid varying resistor can be defined in the laplace domain as I did and have it still work out in circuit analysis. But, R != R in your case, so it has to be defined by something else . .