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Re: Piecewise vs. /; ?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103901] Re: Piecewise vs. /; ?
  • From: Szabolcs Horvát <szhorvat at gmail.com>
  • Date: Mon, 12 Oct 2009 06:35:38 -0400 (EDT)
  • References: <hashpc$qu8$1@smc.vnet.net>

On 2009.10.11. 14:07, Erik Max Francis wrote:
> I'm defining a piecewise function:
>
> mmin = 0.08;
> mmax = 100;
>
> \[Xi][m_] := Piecewise[
>      {{0.035 m^-1.3, 0.08<= m<= 0.50},
>       {0.019 m^-2.2, 0.50<  m<= 1.00},
>       {0.019 m^-2.7, 1.00<  m<= 100}}]; (* stars per pc^3 *)
>
> This works fine for plotting (which I won't show) and integrating:
>
> In[5]:= Integrate[\[Xi][m], {m, mmin, mmax}]
>
> Out[5]= 0.136978
>
> If I try to define \[Xi][m] with /;, it doesn't work.  This version:
>
> \[Xi][m_ /; 0.08<= m<= 0.50] = 0.035 m^-1.3 ;
> \[Xi][m_ /; 0.50<  m<= 1.00] = 0.019 m^-2.2;
> \[Xi][m_ /; 1.00<  m<= 100] = 0.019 m^-2.7;
> (* stars per pc^3 *)
>
> plots correctly but integrate refuses to do anything:
>
> In[26]:= Integrate[\[Xi]2[m], {m, mmin, mmax}]
>
> Out[26]= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(0.08`\), \(100\)]\(\[Xi]2[
>      m] \[DifferentialD]m\)\)
>
> This version (putting the /; in a different place):
>
> \[Xi]3[m_] = 0.035 m^-1.3  /; 0.08<= m<= 0.50;
> \[Xi]3[m_] = 0.019 m^-2.2 /; 0.50<  m<= 1.00;
> \[Xi]3[m_] = 0.019 m^-2.7 /; 1.00<  m<= 100;
> (* stars per pc^3 *)
>
> neither plots properly (the plot is blank) nor integrates properly:
>
> In[27]:= Integrate[\[Xi]3[m], {m, mmin, mmax}]
>
> Out[27]= \!\(
> \*SubsuperscriptBox[\(\[Integral]\), \(0.08`\), \(100\)]\(\((
> \*FractionBox[\(0.035`\),
> SuperscriptBox[\(m\), \(1.3`\)]] /;
>       0.08`<= m<= 0.5`)\) \[DifferentialD]m\)\)
>
> I was under the impression that these were equivalent; what am I missing?
>

In short, /; is a programming construct that affects evaluation. 
Piecewise[] is used to represent a mathematical concept: piecewise 
functions.

If you want to do mathematical operations like integration, then use 
Piecewise[].  Condition[] (/;) is a lot more general than Piecewise[], 
for example one could write a "function" that returns a different value 
in the evening than in the morning, which is obviously not a function in 
the mathematical sense, and thus can't be integrated.

The third version of your example doesn't work (can't be plotted) 
because the syntax is incorrect: when /; is placed at the end, := must 
be used, not = .  This can be easily discovered if one tries to evaluate 
that function with a specific numerical value.


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