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Re: condition and pattern matching

• To: mathgroup at smc.vnet.net
• Subject: [mg44892] Re: [mg44885] condition and pattern matching
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sat, 6 Dec 2003 04:45:30 -0500 (EST)
• References: <200312051031.FAA09180@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

ram wrote:
> 
> I would like to integrate the function from {0,h}.
> 
> \!\(\((E\^\(\(-I\)\ k\ Abs[\(-h\) + z]\)\/Abs[\(-h\) + z] +
>         E\^\(\(-I\)\ k\ Abs[h + z]\)\/Abs[h + z] -
>         \(2\ E\^\(\(-I\)\ k\ Abs[z]\)\ Cos[h\ k]\)\/Abs[z])\)\ Csc[h\ k]\^2\
>     Sin[k\ \((h - Abs[z])\)]\)
> 
> Is there any way that I can replace Abs function by its values?  I know that
> the following cases for condition matching work
> 
> fac[n_ /; n > 0] := n!
> m /:Greater[m,n]=True;
> fac[m]
> gg[m_,n_]:=m-n /; m>n
> u/:Greater[u,v]=True;
> gg[u,v]
> 
> But I can not find a way to apply  above ideas when expressions are
> involved.  Thanks for any help.
> 
> Ram Kochhar

It diverges, due to a singularity at the origin.

In[2]:= InputForm[Integrate[
  (1/(E^(I*k*Abs[-h + z])*Abs[-h + z]) + 1/(E^(I*k*Abs[h + z])*Abs[h +
z]) - 
  (2*Cos[h*k])/(E^(I*k*Abs[z])*Abs[z]))*Csc[h*k]^2*Sin[k*(h - Abs[z])],
  {z,0,h}, Assumptions->{h>0,k>0}]]
                        
Integrate::idiv: 
                             1                             2 Cos[h k]
   Integral of (---------------------------- + <<1>> -
------------------) 
                 I k Abs[-h + z]                        I k Abs[z]
                E                Abs[-h + z]           E          
Abs[z]
     <<1>> does not converge on {0, h}.

Integrate::idiv: 
                             1                             2 Cos[h k]
   Integral of (---------------------------- + <<1>> -
------------------) 
                 I k Abs[-h + z]                        I k Abs[z]
                E                Abs[-h + z]           E          
Abs[z]
     <<1>> does not converge on {0, h}.

Out[2]//InputForm= 
Csc[h*k]^2*Integrate[(1/(E^(I*k*Abs[-h + z])*Abs[-h + z]) + 
    1/(E^(I*k*Abs[h + z])*Abs[h + z]) -
(2*Cos[h*k])/(E^(I*k*Abs[z])*Abs[z]))*
   Sin[k*(h - Abs[z])], {z, 0, h}, Assumptions -> {h > 0, k > 0}]

The numeric example below may help to show what is happening. You have a
term of the form 1/z so the integral diverges logarithmically as you
approach the origin from the right.

In[14]:= InputForm[NIntegrate[
  (1/(E^(I*k*Abs[-h + z])*Abs[-h + z]) + 1/(E^(I*k*Abs[h + z])*Abs[h +
z]) - 
  (2*Cos[h*k])/(E^(I*k*Abs[z])*Abs[z]))*Csc[h*k]^2*Sin[k*(h - Abs[z])]
/.
  {k->2.2,h->4.6}, {z,.01,4.6}]]
                           
Out[14]//InputForm= -5.610184273698055 - 7.452163944484216*I

In[15]:= InputForm[NIntegrate[
  (1/(E^(I*k*Abs[-h + z])*Abs[-h + z]) + 1/(E^(I*k*Abs[h + z])*Abs[h +
z]) - 
  (2*Cos[h*k])/(E^(I*k*Abs[z])*Abs[z]))*Csc[h*k]^2*Sin[k*(h - Abs[z])]
/.
  {k->2.2,h->4.6}, {z,.001,4.6}]]
                           
Out[15]//InputForm= -11.068854032540559 - 7.409240422031397*I

In[16]:= InputForm[NIntegrate[
  (1/(E^(I*k*Abs[-h + z])*Abs[-h + z]) + 1/(E^(I*k*Abs[h + z])*Abs[h +
z]) - 
  (2*Cos[h*k])/(E^(I*k*Abs[z])*Abs[z]))*Csc[h*k]^2*Sin[k*(h - Abs[z])]
/.
  {k->2.2,h->4.6}, {z,.0001,4.6}]]
                           
Out[16]//InputForm= -16.583460196486023 - 7.4048907215947875*I

In[17]:= InputForm[NIntegrate[
  (1/(E^(I*k*Abs[-h + z])*Abs[-h + z]) + 1/(E^(I*k*Abs[h + z])*Abs[h +
z]) - 
  (2*Cos[h*k])/(E^(I*k*Abs[z])*Abs[z]))*Csc[h*k]^2*Sin[k*(h - Abs[z])]
/.
  {k->2.2,h->4.6}, {z,.00001,4.6}]]
                           
Out[17]//InputForm= -22.103615633575806 - 7.4044551823817875*I


Daniel Lichtblau
Wolfram Research

• References:
  • condition and pattern matching
    • From: "ram" <amli@comcast.net>