The Weierstrass Continuation Method for Log[1+z]Here we will construct a part of the Riemann surface of by Weierstrass's method. It is not applicable for more complicated examples. But it is instructive to see this method carried out in detail for a real example, as it is taught in every complex analysis course. In addition, this method fits well with the subject of this article. On the other hand, all elementary transcendental functions are built only from log functions, so knowledge of the analytic continuation of log is sufficient to deal with all functions under consideration here. We start with an analytic function having a Taylor expansion around . The superscript 0 on the refers to the expansion point . Inside the disk of convergence of the Taylor expansion we can re-expand around a regular point . There has again a Taylor expansion (with coefficients ). Using elementary algebra, it can be shown [1] that the coefficients can be expressed through the coefficients in the following way. We use this formula repeatedly by encircling the branch point of . The th coefficient () of the Taylor series of around is given by .
The function
Here is the first step of the analytic continuation, .
Be aware that such terms as in the previous expression do not "simplify" to . They contain the branch cut information relevant for us. We now carry out the analytic continuation repeatedly along a path formed by, say, 10 points , . We do not specify explicit numerical values for the yet so that the formulas will be easier to read.
Here are the Taylor coefficients of the first few steps. The zeroth term contains explicit logarithms.
We can now use
We see mainly the
Now let us plug in explicit expansion points. We will encircle the point (the branch point of
The picture below shows the expansion points together with their disks of convergence.
As the picture already suggests, the new expansion points lie inside the disks of convergence of the old ones.
Let us piecewise define a function
We look at the resulting Riemann surface by showing the values of the various
After one round about the point the function has changed (as expected) by .
This means that the function Converted by Mathematica |