Actually, the fourier analysis might be very beneficial, if I understood it better. I was trying to think of an easy way to derive the terms, without resorting to tedious math, and it hit me.

I was thinking about the fourier series of the along the imaginary axis, which is cyclic with period 2*pi*i. So we would need sines and cosines with of the form , with a_k and b_k complex. These could be expressed as . Well, at 0, the slog is real-valued, so the c_k would have to be real as well.

This is where it hit me. The c_k would look like a power series in terms of e^z, which isn't terribly fascinating, except for the minor detail that , and we have a way of calculating the power series for . Therefore, we have the coefficients c_k already!

This in and of itself essentially validates a few things. First, it validates that the slog will be periodic with period 2*pi*i, though I've already figure out other arguments to demonstrate this. But it's nice to have an independent verification.

Second, it validates the very notion of creating these singularities with the corrugations caused cyclic functions. These corrugations first create ripples, viewed as the smooth sections of various branches, but in some places the ripples add up to singularities.

I was thinking about the fourier series of the along the imaginary axis, which is cyclic with period 2*pi*i. So we would need sines and cosines with of the form , with a_k and b_k complex. These could be expressed as . Well, at 0, the slog is real-valued, so the c_k would have to be real as well.

This is where it hit me. The c_k would look like a power series in terms of e^z, which isn't terribly fascinating, except for the minor detail that , and we have a way of calculating the power series for . Therefore, we have the coefficients c_k already!

This in and of itself essentially validates a few things. First, it validates that the slog will be periodic with period 2*pi*i, though I've already figure out other arguments to demonstrate this. But it's nice to have an independent verification.

Second, it validates the very notion of creating these singularities with the corrugations caused cyclic functions. These corrugations first create ripples, viewed as the smooth sections of various branches, but in some places the ripples add up to singularities.

~ Jay Daniel Fox