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Cauchy-Riemann Equations - Complex Analysis

  1. Aug 13, 2011 #1
    Hello everyone,

    The question:
    My attempt:
    I'll try my hand at the analytic part if I could get some clarification on this part first. :)
     
  2. jcsd
  3. Aug 13, 2011 #2

    HallsofIvy

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    Science Advisor

    Everything is correct except your statement
    If f is differentiable only at (0, 1), that makes no sense except for z= i.

    As for analytic- a function is analytic at a point if and only if it is differentiable in some neighborhood of that point.
     
  4. Aug 13, 2011 #3
    HallsofIvy was a bit too generous in saying everything is correct.

    You cannot evaluate the function at x = 0 and then compute the second set of Cauchy-Riemann equations as you did. This amounts to evaluating a real function f(x,y) at x = 0, computing the partial derivative with respect to y, and then claiming that the result is actually [itex]\frac{\partial f}{\partial y}[/itex].

    You must compute the Cauchy-Riemann equations first, then look at the set of (x,y) that satisfy the equations. Then you can determine where the function is analytic (by HallsofIvy's given definition), if anywhere.
     
  5. Aug 13, 2011 #4
    So basically you're saying the part about figuring that [tex]x = 0[/tex] should just shift down a bit?
     
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