Calculses of Finite Differences

E f(x)  =  f ( x+n )

Operator of E and ∆

Meaning of E

                        if y = f (x) is any function then operation of E an f (x) implies that gives an increment .

Given increment to the value of x in the function .

If this increment is of quantity H, then the operation for E means that put (x + h) in the function whenever there is x

i.e.,  |E f(x) = f(x+n)|

E f(x) does not mean that E is multiply to f(x) but this means that E is operator on f(x) and as such it is only a symbol and not an algebric number.

 E²    f(x) means  that operator E is applied twice only function f(x)

i.e.,         E²     f(x)  = E    E f(x)

                            = E    f(x+n)

                            = E     (x + n + n)     [x=x+n]

                            = E     (x + 2n)

Relatives   or    ∆= E - 1

        ∆ f(x) = f(x+n) - f(x) 

                      = E f(x)  - f(x)

                      = E f(x)  - [ E - 1]

  

           ∆ f(x)  =  f(x) [E -1]

              ∆      = E - 1 

               |E = 1+∆|