Square-wave response of an RC low-pass filter ($v_p=A\tanh [T/(4RC)]$)

For an equal-duty square wave alternating between $\pm A$ with period $T$, the steady periodic capacitor voltage has extrema $\pm v_p$. A positive half-period maps $-v_p$ to $A-(A+v_p)e^{-T/(2RC)}=v_p$, proving the title formula. The negative half-period is its sign-reversed copy. Short periods give a small nearly triangular output; long periods give input-like plateaux with exponential transitions.