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MathGroup Archive 1999

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RE: Can someone explain this?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg19165] RE: [mg19106] Can someone explain this?
  • From: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
  • Date: Thu, 5 Aug 1999 23:59:08 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Benjamin Lotto  wrote,
----------------------
No matter what variation I try, I can't get Mathematica to replace
Tan[x] with Sin[x]/Cos[x].


In[1]:= Tan[x]/.Tan[z_]->Sin[z]/Cos[z]

Out[1]= Tan[x]
 
-----------------------

It actually does the replacement, but it after the replacement is made the
kernel simplifies Sin[x]/Cos[x] to guess what?  Tan[x]

Before I give a solution I have to bring to your attention a problem with 
your use of (lhs->rhs).  Consider the following:

In[2]:= Tan[x]/.Tan[z_]->f2[z]

Out[2]= f2[z]


OK that works fine, so you continue doing more stuff.
-----------------------

In[3]:= z=2.234 + 4.565 I;

....
....
....
(*  You do a lot more stuff  *)
(*  and the value of z hasn't changed.  *)


In[134]:= Tan[x]/.Tan[z_]->f2[z]

Out[134]= f2[2.234 + 4.565 I]


Opps! That isn't right.

Whenever you use (lhs->rhs) and (lhs) uses a symbol to name a pattern the
global value of the symbol will be used in (rhs).  In that case this you
should use (lhs:>rhs) as below.

I have to inform people of this problem about twice a month.  I plan to put
it on a web page before I develop carpal tunnel syndrome from typing it over
and over.

-------------------------

In[135]:= Tan[x]/.Tan[z_]:>f2[z]

Out[135]= f2[x]


In[136]:= z

Out[136]:= 2.234 + 4.565 I


The replacement worked above even though z has a global value.
---------------------------

Now back to your problem.  You can use HoldForm to 
ensure the kernel doesn't evaluate Sin[x]/Cos[x].

In[137]:= expr=Tan[x]/.Tan[z_]:>HoldForm[Sin[z]/Cos[z]]

Out[137]= Sin[x]/Cos[x]


But now look what happens when you try to use the result.
---------------------------

In[138]:= D[expr,x]

Out[138]= (1+Tan[x]^2) HoldForm'[Tan[x]]

------------------

You can use ReleaseHold to remove use of HoldForm, Hold, or HoldComplete.
For example the line below succeeds in doing the derivative.

In[139]:= D[ReleaseHold[expr],x]

Out[139]= Sec[x]^2


One of the features I am dying to see in a future version is a way to
prevent further evaluation without requiring the use of ReleaseHold when the
result is used later.
-------------------

Regards,
Ted Ersek


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