08/31/2007, 05:05 PM
(This post was last modified: 08/31/2007, 05:07 PM by hyper4geek.)
bo198214 Wrote:How did you verify them? Numerically computing the inverse of Andrew's slog and then numerically computing the derivatives?That's exactly how I did it.
The formulas are even valid for complex values of x. I used a series expansion derived from Andrew's formulas and computed them also in the complex plane. Despite of the radius of convergence, using some known recurrence relations, you can make the function convergent for almost any point in the complex plane. I just have still difficulties in the region very roughly around +/- I (imaginary i), where it doesn't converge regardless of what relations I use.
It seems that at these points we have branch points and actually I found two distinct ways of placing branch cuts around these points. See the three images about TetraLog for this. The first two are centered at the origin and toward the left/right approach -Infinity/Infinity and toward the top/bottom I*Infinity / -I*Infinity. You can see the branch cuts along roughly I to Infinity and -I to -Infinity. An alternative placement of branch cuts is seen in the third image, which repeats its branch cut every 2 I Pi k (not seen in the image though). This image runs from -4 to 4 on the real axis (x) and from -4*I to 4*I on the imaginary axis (y).
That TetraLog (slog) function is then easily (well numerically) invertible to give (what I call) TetraExp, however, because of the non-convergent TetraLog (slog) for the two regions I mentioned, I can only generalize the TetraExp function for complex values up to (real)x +/- 1 I. See the two images about TetraExp for this. They run from -4 to 4 on the real axis (x) and -1 I to I on the imaginary axis (y).
Of course the images don't explain why of if the formulas are valid, but you can see that I had a lot of values to try out in the complex plane, and for every single point, the formulas were true even regardless of what branch cut definition I used for TetraLog. It's still verified numerically only.
I have no way of proving the formulas. Maybe they are only numerically valid because of a flaw in Andrew's paper, but if they are in fact true, then this should be at least something one can work with.
