Re: Some problems with complex functions like Sqrt[z]
- To: mathgroup at smc.vnet.net
- Subject: [mg18206] Re: [mg18168] Some problems with complex functions like Sqrt[z]
- From: "Richard Finley" <rfinley at medicine.umsmed.edu>
- Date: Mon, 21 Jun 1999 22:50:45 -0400
- Sender: owner-wri-mathgroup at wolfram.com
Robert,
Try the following:
In[1]:= f[z_]:=Sqrt[z]
In[2]:= re = ComplexExpand[Re[f[a + I b]], TargetFunctions->{Re,Im}];
In[3]:= im = ComplexExpand[Im[f[a + I b]], TargetFunctions->{Re, Im}];
In[4]:= D[{re,im},a].D[{re,im},b]//FullSimplify
Out[4]:= 0
Hope that helps..... RF
>>> Robert Prus <robert at fuw.edu.pl> 06/19/99 09:54PM >>>
Hi,
For any complex (holomorphic) function f[z] I should obtain 0 as the
result of the following calculations:
Mathematica 3.0 for Silicon Graphics
Copyright 1988-97 Wolfram Research, Inc.
-- Motif graphics initialized --
In[1]:= f[z_]:=z^3
In[2]:= re=Re[f[a+I b]]//ComplexExpand
3 2
Out[2]= a - 3 a b
In[3]:= im=Im[f[a+I b]]//ComplexExpand
2 3
Out[3]= 3 a b - b
In[4]:= D[{re,im},a].D[{re,im},b]//Simplify
Out[4]= 0
But if I choose more complicated function, like Sqrt[z], I obtain:
In[5]:= f[z_]:=Sqrt[z]
In[6]:= re=Re[f[a+I b]]//ComplexExpand
Arg[a + I b]
Out[6]= Sqrt[Abs[a + I b]] Cos[------------]
2
In[7]:= im=Im[f[a+I b]]//ComplexExpand
Arg[a + I b]
Out[7]= Sqrt[Abs[a + I b]] Sin[------------]
2
In[8]:= D[{re,im},a].D[{re,im},b]//Simplify
I 2 2 2
- (Abs'[a + I b] + Abs[a + I b] Arg'[a + I b] )
4
Out[8]= -------------------------------------------------
Abs[a + I b]
In the following one can use the substitutions:
Abs'[x_+I y_] -> x/Sqrt[x^2+y^2]
Arg'[x_+I y_] -> -y/(x^2+y^2)
(one can check them using:
z=Abs'[x+I y]-x/Sqrt[x^2+y^2] (* or z=Arg'[x+I y]+y/(x^2+y^2) *)
Plot[Chop[z],{x,-10,10},{y,-10,10}]
Plot[Chop[z],{x,Random[],Random[]},{y,Random[],Random[]}]
)
As the final result I obtain:
In[10]:= %8/.{Abs'[x_+I y_] -> x/Sqrt[x^2+y^2], Arg'[x_+I y_] ->
-y/(x^2+y^2)}/.Abs[x_+I y_] -> Sqrt[x^2+y^2]//Together
I
-
4
Out[10]= -------------
2 2
Sqrt[a + b ]
Under Mathematica 2.0 as the final result I have:
In[9]:= %8/.{Abs'[x_+I y_] -> x/Sqrt[x^2+y^2], Arg'[x_+I y_] ->
-y/(x^2+y^2)}/.Abs[x_+I y_] -> Sqrt[x^2+y^2]//Together
b
Out[9]= -------------------------
2 2
4 (a - I b) Sqrt[a + b ]
But it should be equal to zero !!!
Any comments?
Robert Prus, robert at fuw.edu.pl
Institute of Theoretical Physics, Warsaw University
Hoza 69, 00-681 Warsaw, Poland