Some problems with complex functions like Sqrt[z]
- To: mathgroup at smc.vnet.net
- Subject: [mg18168] Some problems with complex functions like Sqrt[z]
- From: Robert Prus <robert at fuw.edu.pl>
- Date: Sat, 19 Jun 1999 23:54:29 -0400
- Organization: Warsaw University, Physics Department, Institute of Theoretical Physics
- Sender: owner-wri-mathgroup at wolfram.com
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) *)
Plot3D[Chop[z],{x,-10,10},{y,-10,10}]
Plot3D[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
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- Re: Some problems with complex functions like Sqrt[z]