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MathGroup Archive 1999

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Re: Some problems with complex functions like Sqrt[z]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18217] Re: [mg18168] Some problems with complex functions like Sqrt[z]
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Tue, 22 Jun 1999 20:41:12 -0400
  • References: <199906200354.XAA27497@smc.vnet.net.>
  • Sender: owner-wri-mathgroup at wolfram.com

By the way, an approach for verifying analyticity that is often cleaner
is as follows. Expand in terms of the function and its conjugate
(telling ComplexExpand to assume the variable is complex-valued), then
show the derivative with respect to the conjugate is zero.

In[81]:= InputForm[f = ComplexExpand[z^(1/3), z,
        TargetFunctions->Conjugate] /. Conjugate[z]->zbar]
         
Out[81]//InputForm= 
(z*zbar)^(1/6)*Cos[(-I*Log[z] + I/2*Log[z*zbar])/3] + 
 I*(z*zbar)^(1/6)*Sin[(-I*Log[z] + I/2*Log[z*zbar])/3]

In[82]:= Simplify[D[ff, zbar]]
Out[82]= 0


Daniel Lichtblau
Wolfram Research


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