Complex, Simply Yours

For the post "Rectified Waveform To The Rescue",

The response using a full normal wave is just the superposition of a rectified wave and an inverted rectified wave, but frequency halved.  The results does not contradict.

But the expression,

$\cfrac{d^2T}{d\,t^2}=i\omega_dV_oI_oe^{i2\omega_d t}cos(\theta)$

is a complex valued frequency  $i\omega_d$,  as needed in the post "Complex Frequency And SuperConductivity".  Which is dealt with,

$\cfrac{d^2T}{d\,t^2}=e^{i\pi/2}\omega_dV_oI_oe^{i2\omega_d t}cos(\theta)$

$\cfrac{d^2T}{d\,t^2}=\omega_dV_oI_oe^{i(2\omega_d t+\pi/2)}cos(\theta)$

A phase delay in time.

That is to say, a complex number, $e^{iA}$ multiplied to a time function  $e^{iwt}$,  travels back or forth in time by a phase  $A$.