I recently compared the complex values of the matrix power super-exponential (msexp) and the intuitive super-expoential (isexp), i.e. the inverse of the intuitive slog.
The graphic method is ideal for powerseries depictions, because powerseries dont converge beyond their convergence radius.
I call it conformal polar plots: they show how a mesh of circles and rays is mapped under the powerseries/function. Images of circles are green, images of rays are red.
The following both pictures show the images of circles and lines inside a radius of 1.5 around the devlopment point 0 under the corresponding sexp to base e. Both are made from 50x50 Carleman matrices.
msexp
isexp
Its clear that they "feel unpleasant" near the fixed point.
The isexp seems to start crimping near the fixed point, does that mean it is not injective there?
Unfortunately everything can be explained with the limited precision.
Indeed both methods seem to converge very slowly, which would make it perhaps impossible to decide the question numerically.
We can continue the sexp to the left and right via exponentiating, logarithmating; but we can not do in direction of the imaginary axis. So the powerseries development theirby limited. In the above case of development point 0 the convergence radius should be 2 (because there is a singularity at -2 and hopefully nowhere else). However only the msexp can be driven to near 2, while the isexp already before starts to wildly oscillate.
Here is how the msexp extends to radius 1.9:
PS: If you add the conjugate complex half, the pictures look like strange pears, thatswhy I called it funny!
The graphic method is ideal for powerseries depictions, because powerseries dont converge beyond their convergence radius.
I call it conformal polar plots: they show how a mesh of circles and rays is mapped under the powerseries/function. Images of circles are green, images of rays are red.
The following both pictures show the images of circles and lines inside a radius of 1.5 around the devlopment point 0 under the corresponding sexp to base e. Both are made from 50x50 Carleman matrices.
msexp
isexp
Its clear that they "feel unpleasant" near the fixed point.
The isexp seems to start crimping near the fixed point, does that mean it is not injective there?
Unfortunately everything can be explained with the limited precision.
Indeed both methods seem to converge very slowly, which would make it perhaps impossible to decide the question numerically.
We can continue the sexp to the left and right via exponentiating, logarithmating; but we can not do in direction of the imaginary axis. So the powerseries development theirby limited. In the above case of development point 0 the convergence radius should be 2 (because there is a singularity at -2 and hopefully nowhere else). However only the msexp can be driven to near 2, while the isexp already before starts to wildly oscillate.
Here is how the msexp extends to radius 1.9:
PS: If you add the conjugate complex half, the pictures look like strange pears, thatswhy I called it funny!