Zooming out another factor of 10, the non-circularity of the contours becomes obvious.
Finally, we're zoomed out enough here to see the origin. Notice that we're almost to the interesting part, where our ovals are no longer connected loops. (Technically, they're not connected now, because the imaginary parts increase counter-clockwise, due to the logarithmic singularity. However, the branches are all the same, at least when we calculate the regular slog at 2.)
This is where I can describe the peculiar spacing of the contours. When I take the logarithm of points near the origin, they will towards negative infinity. To get any kind of decent picture of what happens in the left half plane, I had to make my contours bunch up around the origin. And because of how I calculate each region as the logarithm of a region closer to the singularity, I had to effect this strategy from the outset. Therefore, I created a cyclic mapping of the real parts, \( c\left(\Re(z)\right) = \Re(z) - \frac{\sin\left(2\pi \Re(z)\right)}{2\pi} \)
This gives me a slope of 0 at the integers, so that I can get values very close to 0 when I get back to the origin. In the next posts, you'll see the effect I was going for.
Finally, we're zoomed out enough here to see the origin. Notice that we're almost to the interesting part, where our ovals are no longer connected loops. (Technically, they're not connected now, because the imaginary parts increase counter-clockwise, due to the logarithmic singularity. However, the branches are all the same, at least when we calculate the regular slog at 2.)
This is where I can describe the peculiar spacing of the contours. When I take the logarithm of points near the origin, they will towards negative infinity. To get any kind of decent picture of what happens in the left half plane, I had to make my contours bunch up around the origin. And because of how I calculate each region as the logarithm of a region closer to the singularity, I had to effect this strategy from the outset. Therefore, I created a cyclic mapping of the real parts, \( c\left(\Re(z)\right) = \Re(z) - \frac{\sin\left(2\pi \Re(z)\right)}{2\pi} \)
This gives me a slope of 0 at the integers, so that I can get values very close to 0 when I get back to the origin. In the next posts, you'll see the effect I was going for.
~ Jay Daniel Fox
