P: 18 Ok I put a capacitor parallel to the resistor and derive the new impedance which is indeed $$Z = \frac{R}{\left(1+(2\pi fRC)^2\right)}$$ So $$\phi= 4ZkT = \frac{4RkT}{\left(1+(2\pi fRC)^2\right)}$$ Integrating over df gives: $$\frac{2kT}{ \pi C} \arctan{2 \pi C f R}$$ filling in f with limits from 0 to infinity gives: $$\frac{2kT}{ \pi C} \frac{\pi}{2} - 0 = \frac{kT}{C}$$ which is of course finite with constant temperature. The idea that phi is a good approximation up to a certain frequency is because the capacitance of the parallel capacitor is very small so the term $$(2 \pi f R C)^2$$ is negligible and phi is the "original" phi for the ideal resistor. That is what I can think of and seems correct, Thanks