       Re: Conditions on patterns in Flat functions

• To: mathgroup at smc.vnet.net
• Subject: [mg13401] Re: [mg13332] Conditions on patterns in Flat functions
• From: David Withoff <withoff>
• Date: Thu, 23 Jul 1998 03:32:44 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```> Hi all, I have a problem with conditions on patterns in flat functins,
> here is an example:
>
>
> In:= CosPlusISin[expr_]:= expr //. {
>                 ((a_. Cos[th_] + b_. Sin[th_] /; b === I a ) :> a E^(I th)),
>                 ((a_. Cos[th_] + b_. Sin[th_] /; b === - I a ) :> a E^(-I th))
>         }
>
> In:= 4 Cos[x]+4 I Sin[x]
>
> Out= 4 Cos[x] + 4 I Sin[x]
>
> In:= CosPlusISin[%]
>
>            I x
> Out= 4 E
>
> In:= test=4 Cos[x]+4 I Sin[x] + something
>
> Out= something + 4 Cos[x] + 4 I Sin[x]
>
> In:= CosPlusISin[test]
>
> Out= something + 4 Cos[x] + 4 I Sin[x]
>
> The solutions I found:
>
> In:= CosPlusISin1[expr_]:= expr //. {
>                 ((a_. Cos[th_] + b_. Sin[th_] +c___ /; b === I a ) :>
> a E^(I th)+c),
>                 ((a_. Cos[th_] + b_. Sin[th_] +c___ /; b === - I a ) :>
> a E^(-I th)+c)
>          }
>
> In:= CosPlusISin2[expr_]:= expr //. {
>                 ((a_. Cos[th_] + b_. Sin[th_] +c_. /; b === I a ) :> a
> E^(I th)+c),
>                 ((a_. Cos[th_] + b_. Sin[th_] +c_. /; b === - I a ) :> a
> E^(-I th)+c)
>          }
>
> In:= CosPlusISin1[test]
>
>             I x
> Out= 4 E    + something
>
> In:= CosPlusISin2[test]
>
>             I x
> Out= 4 E    + something
>
> The questions:
>
> Which solution of the two is better, and why does the original idea not
> work since Plus is Flat ?
>
> Tobias

The original idea didn't work because the left-hand side of the rule

CosPlusISin[expr_]:= expr //. {
((a_. Cos[th_] + b_. Sin[th_] /; b === I a ) :> a E^(I
th)),
((a_. Cos[th_] + b_. Sin[th_] /; b === - I a ) :> a
E^(-I th))
}

is a Condition pattern rather than an expression with a head of Plus.
One way to address that concern is to put the condition on the
right-hand side of the rule.

In:= CosPlusISin[expr_]:= expr //. {
(a_. Cos[th_] + b_. Sin[th_]) :> a E^(I th) /; b === I
a,
(a_. Cos[th_] + b_. Sin[th_]) :> a E^(-I th) /; b === -I
a
}

In:= CosPlusISin[something + 4 Cos[x] + 4 I Sin[x]]

I x
Out= 4 E    + something

There is also a built-in function that does this:

In:= TrigToExp[something + 4 Cos[x] + 4 I Sin[x]]

I x
Out= 4 E    + something

Dave Withoff
Wolfram Research

```

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