Outline of a General Construction Method for Riemann Surfaces of Compositions of Elementary Functions

  • Avoid discontinuities due to branch cuts
  • Branch cuts are "defined", only branch points matter

ArcCosh[z] // TrigToExp
[Graphics:../Images/TalkGD99_gr_44.gif]
[Graphics:../Images/TalkGD99_gr_45.gif]
[Graphics:../Images/TalkGD99_gr_46.gif]

[Graphics:../Images/TalkGD99_gr_47.gif]

[Graphics:../Images/TalkGD99_gr_48.gif]

Class of functions: [Graphics:../Images/TalkGD99_gr_49.gif]

1.) All branch points come from either arctrig functions, logarithms or radicals.
     Determining the branch points will be done using Solve.
     (Numerical root--finding techniques inside rectangles are under work.)

2.) Divide the complex [Graphics:../Images/TalkGD99_gr_50.gif]--plane radially and azimuthally into sectors generated by the
     outer product of the radial and azimuthal coordinates of the branch points.
     No sector has interior branch points.

3.) Derive a set of coupled nonlinear ordinary differential equations describing [Graphics:../Images/TalkGD99_gr_51.gif]
      as a rational function in [Graphics:../Images/TalkGD99_gr_52.gif] and [Graphics:../Images/TalkGD99_gr_53.gif] for all radical, root, and log expressions.

4.) Generate some possible sheets of the function.
     For radicals and Root--objects generate all possible sheets, and for logarithmic terms use [Graphics:../Images/TalkGD99_gr_54.gif](=3)
     out of the infinitely many sheets.

5.) Solve the differential equation radially outwards for all sheets of every sector.
     Start quite near to the branch points.
     
6.) Generate polygons and display the Riemann surface.



Introduction | Construction Method | Implementation | Outlook |