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This question is from a math-essentials booklet for physicists. The function is not analytic from the C-R conditions but that is all I know.

$ \rlap{\textbf{-------------------------------------------------------------------------------------------------}}{\mbox{Its derivative should be}}$ $2z + 2\bar{z} =4Re[z]$ So are its critical points all over the real axis?

The question also asks for a roiugh sketch of the graph. How would I sketch $(x,y)\mapsto (x+x^2-y^2,2xy-y)$ ?

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What do you mean by its derivative? $\bar{z}$ is not complex differentiable. –  Alex Becker Mar 17 '11 at 1:30
    
@Alex I thought critical values were where the derivative of the function goes to zero i.e extremal and inflection points. I would correct the mistake, though I am even more lost now. –  Please Delete Account Mar 17 '11 at 1:34
    
@Approximist: That's the definition for real functions and I believe for holomorphic complex functions as well. In this context, however, I have no idea what the definition would be. –  Alex Becker Mar 17 '11 at 1:37
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2 Answers

Hint: This problem has nothing to do with complex analysis. We just have a map $$f:\ {\mathbb R}^2\to{\mathbb R}^2, \quad (x,y)\mapsto (u,v):=(x+x^2-y^2, 2xy-y).$$ The graph of this map lives in ${\mathbb R}^4$ and so is of no help in visualizing $f$. I suggest drawing images of some lines $x=$const. (horizontal parabolas) and $y=$const. (again horizontal parabolas). Along the circle $x^2+y^2={1\over4}$ the Jacobian of $f$ vanishes; this will lead to special effects in the picture.

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About the second question: here are some nice ways to plot complex functions: http://www.maa.org/pubs/amm_complements/complex.html , http://www.mai.liu.se/~halun/complex/ , http://www1.math.american.edu/People/lcrone/ComplexPlot.html

Or http://www.wolframalpha.com/input/?i=+z%5E2+%2B+Conj%5Bz%5D

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thanks for the links. In addition, Terry Tao has some java pplets on his coursepage for complex mappings. This question however is from a math-essentials booklet for physicists from the 70s, and I think its intention is that the student should be able to draw a rough sketch without referring to computers. –  Please Delete Account Mar 17 '11 at 2:10
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