Transformation Rules and Definitions
The notation "/." tells Mathematica TE to use the rule x 
1 + a on the algebraic expression 
.
In[1]:= 1 + x^2 + 3 x^3 /. x -> 1 + a
Out[1]= 
You can give transformation rules for any expression. This uses a rule for f[2].
In[2]:= {f[1], f[2], f[3]} /. f[2] -> b
Out[2]= 
This replaces f[anything], where anything is named n, by n^2.
In[3]:= {f[1], f[2], f[3]} /. f[n_] -> n^2
Out[3]= 
Here is a Mathematica TE function definition. It specifies that f[n] is always to be transformed to n^2.
In[4]:= f[n_] := n^2
The definition for f is automatically used whenever it applies.
In[5]:= f[3] + f[a + b]
Out[5]= 
Here is the recursive rule for the factorial function.
In[6]:= fac[n_] := n fac[n-1]
This gives a rule for the end condition of the factorial function.
In[7]:= fac[1] := 1
Here are the two rules you have defined for fac.
In[8]:= ?fac
Mathematica TE can now apply these rules to find values for factorials.
In[9]:= fac[20]
Out[9]= 
Mathematica TE lets you give rules for transforming any expression. This defines ln of a product to be a sum of ln functions.
In[10]:= ln[x_ y_] := ln[x] + ln[y]
Mathematica TE uses the definition you have given to expand out this expression.
In[11]:= ln[a b c d]
Out[11]= 
You can add a rule for each property of the function.
In[12]:= ln[x_ ^ a_] := a ln[x]
Now the rules for ln apply to a greater variety of expressions.
In[13]:= ln[a^2 b^3]
Out[13]= 
