       Re: Complex Analysis using Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg52628] Re: [mg52621] Complex Analysis using Mathematica
• From: "David Park" <djmp at earthlink.net>
• Date: Sun, 5 Dec 2004 02:08:04 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Pratik,

I may not understand your question, but why not just use ComplexExpand. I
don't think you need all those definitions. ComplexExpand assumes that
symbols are Real, unless you tell it otherwise.

Sinh[p(a + b I)]
ComplexExpand[%]
Sinh[(a + I*b)*p]
I*Cosh[a*p]*Sin[b*p] + Cos[b*p]*Sinh[a*p]

Or if you want to define real and imaginary part functions

u[p_][x_, y_] = ComplexExpand[Re[Sinh[p(a + b I)]]]
Cos[b p] Sinh[a p]

v[p_][x_, y_] = ComplexExpand[Im[Sinh[p(a + b I)]]]
Cosh[a p] Sin[b p]

ComplexExpand is the real workhorse in doing complex analysis with
Mathematica.

David Park

From: Pratik Desai [mailto:pdesai1 at umbc.edu]
To: mathgroup at smc.vnet.net

Here we go again,

I have to define a complex function
So I go through this procedure to define that the variables are  really
"real"

TagSet[p, Im[p], 0];
TagSet[a, Im[a], 0];
TagSet[b, Im[b], 0];
TagSet[p, Re[p], p];
TagSet[a, Re[a], a];
TagSet[b, Re[b], b];
lamda = a + I*b
z = ComplexExpand[lamda*p]
x=Re[z]
y=Im[z]
TagSet[u, Im[u[x, y]], 0];
TagSet[v, Im[v[x, y]], 0];
TagSet[x, Re[x], x];
TagSet[y, Re[y], y];
TagSet[u, Re[u[x, y]], u[x, y]];
TagSet[v, Re[v[x, y]], v[x, y]];

Then I define my actual function

u1 = TrigToExp[Sinh[z]] (*By this time I have realized that
Mathematica or for that matter most of the CAS work better with
exponentials when it comes to complex analysis*)

u[x, y] = Re[u1]
v[x, y] = Im[u1]

The problem I face is that the software is not able to identify x and y
as I have defined above. May be I am making a trivial mistake. Please