Optional (x_^n_.), (x_+y_.), (x_*y_.)


Optional (_.)  is used to indicate that a default value should be used of a
value isn't provided.  The feature can only be used with heads that have
assigned values for either Default[h], Default[h, n], or Default[h, n, i].  
The only heads that have built-in default values are Plus, Times, Power, and
they are each demonstrated below.

Example using (x_^n_.)


In the first example (f) tells us the base and exponent of (s^n).

Clear[f, s]  f[x_^n_.] := StringJoin["base =", ToString[x], ",  exponent =", ToString[n]]  f[s^2]

base =s,  exponent =2


The next cell (s) matches the pattern (x_^n_.) and the pattern matcher uses
the default exponent of (1).  The use of (n_.) instead of (n_) indicates that
the default value should be used if one is not provided explicitly.

f[s]

base =s,  exponent =1

Example using (x_+y_.)


In the next example (s) matches the pattern (x_+y_.) and the pattern matcher
uses the default value of (0) for (y_.) when only a single term is provided.  
Because Plus has the Flat attribute the pattern matcher recognizez that
(s1+s2+s3+s4)  is a case of  Plus[s1, Plus[s2,s3,s4] ].

Clear["Global`*"] ;  f[x_ + y_.] := Add[x, y]  {f[s], f[s1 + s2], f[s1 + s2 + s3 + s4]}

{Add[s, 0], Add[s2, s1], Add[s2 + s3 + s4, s1]}


Next we see that the rule is stored as a definition for  f[ y_. + x_ ]  
instead of a definition for  f[ x_ + y_. ]  as entered.  I suppose that
explains why (y) is matched with (s1) in (s1+s2+s3+s4).

??f

Global`f

f[y_. + x_] := Add[x, y]

Example using (x_*y_.)


In the next example (s) matches the pattern (x_*y_.) and the pattern matcher
uses the default value of (1) for (y_.) when only a single factor is
provided.  Because Times has the Flat attribute the pattern matcher
recognizez that (s1  s2  s3  s4)  is a case of  Times[s1, Times[s2, s3, s4]
].

Clear["Global`*"] ;  f[x_ * y_.] := Multiply[x, y]  {f[s], f[s1 s2], f[s1 s2 s3 s4]}

{Multiply[s, 1], Multiply[s2, s1], Multiply[s2 s3 s4, s1]}


Next we see that the rule is stored as a definition for  f[ (y_.)  x_ ]  
instead of a definition for  f[ x_ (y_.) ]  as entered.  I suppose that
explains why (y) is matched with (s1) in

(s1  s2  s3  s4).

??f

Global`f

f[y_. x_] := Multiply[x, y]

Other applications (rarely used)

You can also use arguments like (y_.) as arguments for functions other  than Power, Plus, Times, but only if you first assign values via Default[f],  Default[f,i] or Default[f,i,n].  To see examples of this goto the section on  Default.

Clear[f] ;  Default[f] = z ;  f[x_, y_.] := {x, y}


In the next cell the default value (z) is used when (f) is give a single
argument.

f[a, b]  f[5]

{a, b}

{5, z}

Created by Mathematica  (May 16, 2004)

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