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You have certainly noticed the use of the underscore ( _ ) character when defining functions in Mathematica. The use of the underscore is an important concept in Mathematica called pattern matching. A pattern is an expression such as x_ that contains an underscore character. The pattern can stand for any expression. Thus, f[x_] specifies how the function f should be applied to any argument. When you define a function such as f[x_]= x2, you are telling Mathematica to automatically apply the transformation rule f[x_] x2 whenever possible. In contrast, a transformation rule for f[x] without an underscore specifies only how the literal expression f[x] should be transformed, and does not say anything about the transformation of f[y], f[z], etc.
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Clear[f] f[x_]= x ; f[x] x
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Clear[f] f[x]= x2; f[x] x2 f[y] f[y] f[a + b] f[a + b]
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x is matched only by x.
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f[y] y2 f[a + b] (a + b)2
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x_ is matched by any expression.
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1 + xp + xq /. xq _ Log[q] 1 + Log[p] + Log[q] 1 + xp + xq /. xq Log[q] 1 + xp + Log[q]
Only xq is transformed. All exponentials are transformed to Log.
Patterns can specify the type of an expression as well as its format. For example, _Integer stands for an integer pattern. Similarly, _Rational, _Real, and _Complex are acceptable patterns representing other types of numbers.
EXAMPLE 6 The Mathematica function Factorial[n] computes n! if n is a positive integer and [1 + n] if n is a positive real number. For certain applications, it might be useful to leave the factorial of a non-integer undefined.
fact[n_Integer]= Factorial[n]; fact[n_Real] = "unde ned"; fact[5] 120 fact[5.5] unde ned
EXAMPLE 7 This example defines the function
Factorial[5] 120 Factorial[5.5] 287.885
x y f ( x , y) = x + y x y
if both x and y are integers if both x and y are real if x or y is an integer and the other is real
APPENDIX
f[a_Integer, b_Integer]= a b; f[a_Real, b_Real]= a + b; f[a_Real, b_Integer]= f[a_Integer, b_Real]= a b; f[2, 3] 6 f[2., 3.] 5. f[2., 3] 1. f[2, 3.] 1. f[I, 1] f[ ,1]
f is unde ned for complex arguments.
A.3 Contexts
It is common practice to define symbols using names that are reminiscent of the symbol s purpose. Sometimes, however, the names get unwieldy and cumbersome to work with. Contexts are used as a tool to help organize the symbols used in a Mathematica session. The complete name of a symbol is divided into two parts, a context and a shorter name, separated by a backquote ( ` ) character. Used for this purpose the backquote is called a context mark.
EXAMPLE 8 atomicnumber`au and atomicweight`au are two distinct symbols with a common short name,
au. (Au is the chemical symbol for gold.) atomicnumber`au = 79; atomicweight`au = 196.967; atomicnumber`au 79 atomicweight`au 196.967
When you begin a Mathematica session, the default context is Global`. Thus, for example, the symbol object is equivalent to Global`object. The default can be changed by redefining the symbol $Context.
$Context is the current default context. Context[symbol] returns the context of symbol.
EXAMPLE 9
atomicnumber`au = 79; atomicweight`au = 196.967; $Context = "atomicweight`" ; au 196.967 $Context = "atomicnumber`" ; au 79
Context names are strings; quotes are important.
Built-in Mathematica symbols have context System`.
APPENDIX
EXAMPLE 10
Context[Pi] System`
It is common for symbols in different contexts to have the same short name. If only the short name is referenced, Mathematica decides which is called by its position in a list called $ContextPath.
$ContextPath is the current search path.
EXAMPLE 11
au = "gold"; atomicnumber`au = 79; atomicweight`au = 196.967; $ContextPath
{PacletManager`,
WebServices`, System`, Global`}
Default context path.
$ContextPath = Join[$ContextPath, {"atomicweight`"}, {"atomicnumber`"}]
{PacletManager`, WebServices`, System`, Global`, atomicweight`, atomicnumber`}
au gold Remove[Global`au] au 196.967 Remove[atomicweight`au] au 79
atomicnumber is now the rst element of $ContextPath in which au appears. Global` comes before atomicweight` and atomicnumber` in $ContextPath. atomicweight is now the rst element of $ContextPath in which au appears.
A.4 Modules
Mathematica, by default, assumes that all objects are global. This means, for example, that if you define x to have a value of 3, x will remain 3 until its value is changed. In contrast, a local object has a limited scope valid only within a certain group of instructions. Modules allow you to define local variables whose values are defined only within the module. Outside of the module, the object may either be undefined or have a completely different value.
Module[{var1, var2, . . .}, body] defines a module with local variables var1, var2, . . . Module[{var1 = v1, var2 = v2, . . .}, body] defines a module with local variables var1, var2, . . . initialized to v1, v2, . . . , respectively.
Global variable x is set to 3. Module is de ned with local variable x initialized to 8. x is incremented. Global x is called. Original value of x is returned.
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