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The Weierstrass Continuation Method for Log[1+z]

Here we will construct a part of the Riemann surface of [Graphics:../Images/index_gr_1.gif] 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 [Graphics:../Images/index_gr_2.gif] having a Taylor expansion around [Graphics:../Images/index_gr_3.gif]. The superscript 0 on the [Graphics:../Images/index_gr_4.gif] refers to the expansion point [Graphics:../Images/index_gr_5.gif].

[Graphics:../Images/index_gr_6.gif]

Inside the disk of convergence of the Taylor expansion we can re-expand [Graphics:../Images/index_gr_7.gif] around a regular point [Graphics:../Images/index_gr_8.gif]. There [Graphics:../Images/index_gr_9.gif] has again a Taylor expansion (with coefficients [Graphics:../Images/index_gr_10.gif]).

[Graphics:../Images/index_gr_11.gif]

Using elementary algebra, it can be shown [1] that the coefficients [Graphics:../Images/index_gr_12.gif]can be expressed through the coefficients [Graphics:../Images/index_gr_13.gif] in the following way.

[Graphics:../Images/index_gr_14.gif]

We use this formula repeatedly by encircling the branch point [Graphics:../Images/index_gr_15.gif] of [Graphics:../Images/index_gr_16.gif]. The [Graphics:../Images/index_gr_17.gif]th coefficient ([Graphics:../Images/index_gr_18.gif]) of the Taylor series of [Graphics:../Images/index_gr_19.gif] around [Graphics:../Images/index_gr_20.gif] is given by [Graphics:../Images/index_gr_21.gif].

[Graphics:../Images/index_gr_22.gif]
[Graphics:../Images/index_gr_23.gif]
[Graphics:../Images/index_gr_24.gif]
[Graphics:../Images/index_gr_25.gif]

The function analyticContinuation implements this formula for calculating new coefficients of the Taylor series from the old ones. (This function makes heavy use of Mathematica's symbolic summation capabilities.) Starting with the series term of the form SeriesTerms[term0, termi, i] (in the index variable i) it calculates the series terms around the next expansion point. Because of the different form, we keep the zeroth order term as an extra element in SeriesTerms.

[Graphics:../Images/index_gr_26.gif]

Here is the first step of the analytic continuation, [Graphics:../Images/index_gr_27.gif].

[Graphics:../Images/index_gr_28.gif]
[Graphics:../Images/index_gr_29.gif]

Be aware that such terms as [Graphics:../Images/index_gr_30.gif] in the previous expression do not "simplify" to [Graphics:../Images/index_gr_31.gif]. 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 [Graphics:../Images/index_gr_32.gif], [Graphics:../Images/index_gr_33.gif]. We do not specify explicit numerical values for the [Graphics:../Images/index_gr_34.gif] yet so that the formulas will be easier to read.

[Graphics:../Images/index_gr_35.gif]

Here are the Taylor coefficients of the first few steps. The zeroth term contains explicit logarithms.

[Graphics:../Images/index_gr_36.gif]
[Graphics:../Images/index_gr_37.gif]

We can now use Sum to get closed-form expressions for all the series.

[Graphics:../Images/index_gr_38.gif]
[Graphics:../Images/index_gr_39.gif]
[Graphics:../Images/index_gr_40.gif]
[Graphics:../Images/index_gr_41.gif]

We see mainly the Log terms. If we would neglect branch cuts (the whole point here is, of course, not to do this), then all terms just reduce to [Graphics:../Images/index_gr_42.gif].

[Graphics:../Images/index_gr_43.gif]
[Graphics:../Images/index_gr_44.gif]

Now let us plug in explicit expansion points. We will encircle the point [Graphics:../Images/index_gr_45.gif] (the branch point of Log[1+z]). Because the radius of convergence of the starting power series is 1, we have to choose the next expansion point within a distance 1 from 0. Here is a possible choice for the [Graphics:../Images/index_gr_46.gif].

[Graphics:../Images/index_gr_47.gif]

The picture below shows the expansion points together with their disks of convergence.

[Graphics:../Images/index_gr_48.gif]

[Graphics:../Images/index_gr_49.gif]

As the picture already suggests, the new expansion points lie inside the disks of convergence of the old ones.

[Graphics:../Images/index_gr_50.gif]
[Graphics:../Images/index_gr_51.gif]

Let us piecewise define a function log which is represented by the summed forms of the various series.

[Graphics:../Images/index_gr_52.gif]

We look at the resulting Riemann surface by showing the values of the various log inside their disks of convergence.

[Graphics:../Images/index_gr_53.gif]
[Graphics:../Images/index_gr_54.gif]

[Graphics:../Images/index_gr_55.gif]

After one round about the point [Graphics:../Images/index_gr_56.gif] the function has changed (as expected) by [Graphics:../Images/index_gr_57.gif].

[Graphics:../Images/index_gr_58.gif]
[Graphics:../Images/index_gr_59.gif]

This means that the function log is the analytic continuation of the built-in function Log.


Converted by Mathematica     

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