# Question:Is there a function to conjugate algcurves.puiseux series?

## Question:Is there a function to conjugate algcurves.puiseux series?

Maple

*** Edited to make more clear ***

The function algcurves.puiseux creates generator series for an algebraic function at a point.  The code below generates a few terms of each of the five generator series for the 1,2,3,4 and 5-cycle branch of  w(z) at the origin written implicitly as f in the code below.  Consider the 3-cycle branch:  I extract the 3-cycle generator and plot the imaginary sheet.  Note that's only one surface of a 3-valued function..  In order to plot the other two surfaces the generator series has to be conjugated.   The documentation doesn't appear to have a option to do this.

If there's no procedure for doing this already and some are interested in this, we need to conjugate each individual monomial separately and  multiply each power of z expressed in terms of p/3 by the root of unity raised to the power p.   Not hard to do manually  for a few terms (see updated notebook below)  but I would like to generates several hundred terms.  Note in my updated notebook below how each monomial is multiplied by (exp(2k Pi i/3))^p (for k=1,2) where p is the numerator of the exponent in this case over 3 for the associated power of z.  We can graphically check if this is correct by stitching the sheets together as a Riemann surface via plots:-display(convert({sheet1Plot, sheet2Plot, sheet3Plot}, list)) which shows a 3-valued analytically-continuous surface wraping  around the z-plane three times.   I'll try to write a procedure but I'm brand new to Maple so might take a while.  Maybe call it as

conjugatedSeries=conjugatePuiseux(generator,k)

for sheet k.

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