View source for Finite-difference calculus

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A branch of mathematics in which functions are studied under a discrete change of the argument, as opposed to differential and integral calculus, where the argument changes continuously. Let  $  f $
be a function defined at the points  $  x _ {k} = x _ {0} + kh $(
where  $  h $
is a constant and  $  k $
is an integer). Then

$$ 
\Delta y _ {k} \  = \ 
\Delta f _ {k} \  = \ 
f (x _ {k + 1 }  ) -
f (x _ {k} )
$$

are the (finite) first-order differences,

$$ 
\Delta  ^ {2} y _ {k} \  = \ 
\Delta  ^ {2} f _ {k} \  = \ 
\Delta f _ {k + 1 }  -
\Delta f _ {k} \  = \ 
f (x _ {k + 2 }  ) -
2f (x _ {k + 1 }  ) +
f (x _ {k} )
$$

are the second-order differences $  \dots $

$$ 
\Delta  ^ {n} y _ {k} \  = \ 
\Delta  ^ {n} f _ {k} \  = \ 
\Delta ^ {n - 1 }
f _ {k + 1 }  -
\Delta ^ {n - 1 } f _ {k}  $$

are the  $  n $-
th order differences. The differences are conveniently arranged in a table:

<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  x $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  y $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  \Delta y $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  \Delta  ^ {2} y $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \Delta  ^ {3} y $
</td> <td colname="6" style="background-color:white;" colspan="1"> $  \dots $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  x _ {0} $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  y _ {0} $
</td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"> $  \Delta y _ {0} $
</td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  x _ {1} $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  y _ {1} $
</td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  \Delta  ^ {2} y _ {0} $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"> $  \Delta y _ {1} $
</td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"> $  \Delta  ^ {3} y _ {0} $
</td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  x _ {2} $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  y _ {2} $
</td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  \Delta  ^ {2} y _ {1} $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"> $  \dots $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"> $  \Delta y _ {2} $
</td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"> $  \Delta  ^ {3} y _ {1} $
</td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  x _ {3} $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  y _ {3} $
</td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  \Delta  ^ {2} y _ {2} $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"> $  \dots $
</td> <td colname="4" style="background-color:white;" colspan="1"></td> <td colname="5" style="background-color:white;" colspan="1"> $  \dots $
</td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  x _ {4} $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  y _ {4} $
</td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  \dots $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  \dots $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  \dots $
</td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  \dots $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> <td colname="6" style="background-color:white;" colspan="1"></td> </tr> </tbody> </table>

</td></tr> </table>

An  $  n $-
th order difference can be expressed in terms of the quantities  $  y _ {0} ,\  y _ {1} \dots $
by the formula

$$ 
\Delta  ^ {n} y _ {k} \  = \ 
y _ {k + n }  -
\left ( \begin{array}{c}
n \\
 1 
\end{array}
 \right )
y _ {k + n - 1 }  +
\left ( \begin{array}{c}
n \\
 2 
\end{array}
 \right )
y _ {k + n - 2 }  - \dots +
(-1)  ^ {n} y _ {k} .
$$

As well as the forward differences  $  \Delta y _ {k} $, 
one uses the backward differences:

$$ 
\nabla y _ {k} \  = \ 
f (x _ {k} ) -
f (x _ {k - 1 }  ).
$$

In a number of problems (in particular in the construction of interpolation formulas) central differences are used:

$$ 
\delta ^ {2m - 1 } y _ {i + 1/2 }  ,\ \ 
\delta  ^ {2m} y _ {i} ,
$$

which are defined as follows:

$$ 
\delta y _ {i + 1/2 }  \  = \ 
y _ {i + 1 }  - y _ {i} ,
$$

$$ 
\delta  ^ {2} y _ {i} \  = \  \delta y _ {i + 1/2 }  - \delta y _ {i - 1/2 }  ,
$$

$$ 
{\dots \dots \dots \dots \dots }
$$

$$ 
\delta ^ {2m - 1 } y _ {i +1/2 }  \  = \  \delta ^ {2m - 2 }
y _ {i + 1 }  - \delta ^ {2m - 2 } y _ {i} ,
$$

$$ 
\delta  ^ {2m} y _ {i} \  = \  \delta ^ {2m - 1 } y _ {i +1/2 }  - \delta ^ {2m - 1 } y _ {i - 1/2 }  .
$$

The central differences  $  \delta  ^ {n} y _ {l} $
and the ordinary differences  $  \Delta  ^ {n} y _ {k} $
are related by the formula

$$ 
\delta  ^ {2m} y _ {i} \  = \ 
\Delta  ^ {2m} y _ {i - m }  ,\ \ 
\delta ^ {2m + 1 }
y _ {i + 1/2 }  \  = \ 
\Delta ^ {2m + 1 }
y _ {i - m }  .
$$

In the case when the intervals  $  ( x _ {k + 1 }  ,\  x _ {k} ) $
are not of constant size, one also considers divided differences:

$$ 
[ x _ {0} ] \  = \  y _ {0} ,\ \ 
[ x _ {1} ] \  = \  y _ {1} ,
$$

$$ 
[x _ {0} ; \  x _ {1} ] \  = \  
\frac{y _ {0} - y _ {1} }{x _ {0} - x _ {1} }
 ,
$$

$$ 
[x _ {0} ; \  x _ {1} ; \  x _ {2} ] \  = \  
\frac{[x _ {0} ; \  x _ {1} ] - [x _ {1} ; \  x _ {2} ] }{x _ {0} - x _ {2} }
 ,
$$

$$ 
{\dots \dots \dots \dots \dots \dots }
$$

$$ 
[x _ {0} ; \dots ; \  x _ {n} ] \  = \  
\frac{[x _ {0} ; \dots ; \ 
x _ {n - 1 }  ] - [x _ {1} ; \dots ; \  x _ {n} ] }{x _ {0} - x _ {n} }
 .
$$

The following formula holds:

$$ 
[x _ {0} ; \dots ; \  x _ {n} ] \  = \ 
\sum _ {j = 0 } ^ { n } 

\frac{y _ {j} }{\prod _ {i \neq j } (x _ {j} - x _ {i} ) }
 .
$$

Sometimes, instead of  $  [x _ {0} ; \dots ; \  x _ {n} ] $
the notation  $  f (x _ {0} ; \dots ; \  x _ {n} ) $
is used. If  $  x _ {n} = x _ {0} + nh $, 
$  n = 0,\  1 \dots $
then

$$ 
[x _ {0} ; \dots ; \  x _ {n} ] \  = \ 

\frac{\Delta  ^ {n} y _ {0} }{n! h  ^ {n} }
 .
$$

If  $  f $
has an  $  n $-
th derivative  $  f ^ {\  (n) } $
throughout the interval  $  ( x _ {k} ,\  x _ {k + n }  ) $, 
then

$$ 
\Delta  ^ {n} y _ {k} \  = \ 
h  ^ {n} f ^ {\  (n) } ( \xi ),\ \ 
x _ {k} < \xi < x _ {k + n }  .
$$

The calculus of finite differences is closely related to the general theory of approximation of functions, and is used in approximate differentiation and integration and in the approximate solution of differential equations, as well as in other questions. Suppose that the problem is posed (an interpolation problem) on recovering a function  $  f $
from the known values of  $  f $
at points  $  x _ {0} \dots x _ {n} $. 
One constructs the interpolation polynomial  $  P $
of degree  $  n $
having the same values as  $  f $
at the indicated points. It can be expressed in various forms: in the form of Lagrange, Newton, etc. In Newton's form it is:

$$ 
P _ {n} (x) \  = \ 
f (x _ {0} ) +
[x _ {0} ; \  x _ {1} ]
(x - x _ {0} ) +
$$

$$ 
+ 
\dots + [x _ {0} ; \dots ; \  x _ {n} ] (x - x _ {0} ) \dots (x - x _ {n - 1 }  ),
$$

and in the case when the values of the independent variable are equally spaced:

$$ 
P _ {n} (x) \  = \ 
f (x _ {0} ) +
\sum _ {k = 1 } ^ { n } 

\frac{\Delta  ^ {k} f (x _ {0} ) }{k! h  ^ {k} }

(x - x _ {0} ) \dots
(x - x _ {k - 1 }  ).
$$

The function  $  f $
is set approximately equal to  $  P _ {n} $. 
If  $  f $
has a derivative of order  $  n + 1 $, 
then the error in replacing  $  f $
by  $  P _ {n} $
can be estimated by the formula

$$ 
f (x) \  = \ 
P _ {n} (x) +

\frac{f ^ {\  (n + 1) } ( \xi ) }{(n + 1)! }

(x - x _ {0} ) \dots
(x - x _ {n} ),
$$

where  $  \xi $
lies in the interval in which the points  $  x,\  x _ {0} \dots x _ {n} $
ly. If  $  f $
is a polynomial of degree  $  \leq  n $, 
then  $  f = P _ {n} $.

As the number of nodes increases without bound, the interpolation polynomial  $  P _ {n} $
becomes in the limit a polynomial  $  P $
of  "infinite"  degree, and the following question naturally arises: When does  $  f = P $? 
In other words, when does the following equation hold:

$$ \tag{1 }
f (x) \  = \ 
f (x _ {0} ) +
\sum _ {k = 1 } ^  \infty  

\frac{\Delta  ^ {k} f (x _ {0} ) }{k! h  ^ {k} }

(x - x _ {0} ) \dots
(x - x _ {k - 1 }  )
$$

(for simplicity, the case of equally-spaced intervals is considered)? Let  $  x _ {0} = 0 $, 
$  h = 1 $, 
so that  $  x _ {n} = n $(
$  n \geq  0 $). 
If the series (1) converges at a point  $  \alpha $
that is not a node (the series (1) always converges at the nodes  $  x _ {0} ,\  x _ {1} ,\dots $), 
then it converges in the half-plane  $   \mathop{\rm Re} \  x > \alpha $
and is an analytic function in this half-plane which satisfies the following condition in any half-plane  $   \mathop{\rm Re} \  x \geq  \beta > \alpha $(
where  $  \epsilon > 0 $):

$$ 
| f (re ^ {i \phi } ) | \  < \ 
c e ^ {r h ( \phi ) }
r ^ {\alpha + \epsilon + 1/2 } ,
$$

$$ 
h ( \phi ) \  = \  \cos \  \phi \   \mathop{\rm ln} (2 \  \cos \  \phi ) + \phi \  \sin \  \phi .
$$

Conversely, if  $  f $
is analytic in some half-plane and has similar order of growth (or somewhat better than it), then it can be represented by the series (1). Thus, functions of a very narrow class (only the analytic functions of the indicated growth) can be expanded in a series (1) (a so-called Newton series). Newton series are studied when the nodes are general complex numbers. These series have found great application in transcendental number theory. Suppose next that the interpolation nodes form a triangular array

$$ 

\begin{array}{llll}
x _ {0,0}  &{}  &{}  &{}  \\
x _ {1,0}  &x _ {1,1}  &{}  &{}  \\
\dots  &\dots  &\dots  &\dots  \\
x _ {n,0}  &x _ {n,1}  &\dots  &x _ {n,n}  \\
\dots  &\dots  &\dots  &\dots ,  \\
\end{array}

$$

and that the interpolation polynomial  $  P _ {n} $
is constructed from the nodes of the  $  (n + 1) $-
th row. The class of functions  $  f $
for which  $  P _ {n} $
converges to  $  f $
as  $  n \rightarrow \infty $
depends on the array of nodes. E.g., if

$$ 
x _ {n,k} \  = \  \cos \ 

\frac{2k + 1 }{2n }
 \pi ,\ \ 
k = 0 \dots n
$$

( $  x _ {n,k} $
are the roots of the [[Chebyshev polynomials|Chebyshev polynomials]]), then in order that the interpolation process converges on the interval  $  [-1 ,\  1] $
it suffices that the following condition holds:

$$ 
\lim\limits _ {n \rightarrow \infty } \ 
\omega \left (
{
\frac{1}{n}
 } \right ) \ 
 \mathop{\rm ln} \  n \  = \  0,
$$

where  $  \omega ( \delta ) $
is the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of  $  f $
on  $  [-1,\  1] $.

Another important problem in the calculus of finite differences is the problem of the summation of functions. Let  $  f $
be a given function. It is required to find in finite form, exact or approximate, the sum

$$ 
S _ {n} \  = \ 
f (x _ {0} ) +
f (x _ {0} + h) + \dots +
f (x _ {0} + nh),
$$

for fixed  $  x _ {0} $, 
$  h $
and large  $  n $, 
when certain analytic properties of  $  f $
are known. In other words, the asymptotic behaviour of  $  S _ {n} $
as  $  n \rightarrow \infty $
is studied. Let  $  x _ {0} = 0 $, 
$  h = 1 $(
for simplicity) and suppose that a function  $  F $
can be found such that

$$ \tag{2 }
\Delta F (x) \  = \ 
F (x + 1) -
F (x) \  = \  f (x).
$$

Then

$$ \tag{3 }
S _ {n} \  = \ 
F (n + 1) - F (0).
$$

E.g., let  $  f (x) = x  ^ {2} $. 
The solution of (2) is sought in the form of a polynomial of degree three,

$$ 
Q (x) \  = \ 
ax  ^ {3} + bx  ^ {2} + cx ,
$$

with undetermined coefficients. On substituting into equation (2) and equating coefficients of corresponding powers of  $  x $
on the left-hand and right-hand sides, the polynomial has the form:

$$ 
Q (x) \  = \ 
{
\frac{x  ^ {3} }{3}
 } -
{
\frac{x  ^ {2} }{2}
 } +
{
\frac{x}{6}
 } \  = \ 
{
\frac{x (x - 1) (2x - 1) }{6}
 }
$$

and

$$ 
{
\frac{n (n + 1) (2n + 1) }{6}
 }
\  = \  1  ^ {2} + 2  ^ {2} + \dots + n  ^ {2} .
$$

The solution to equation (2) cannot always be obtained in finite form. It is therefore useful to have approximate formulas for the  $  S _ {n} $. 
Such a formula is the [[Euler–MacLaurin formula|Euler–MacLaurin formula]]. If  $  f $
has  $  k $
derivatives and  $  k $
is even, the Euler–MacLaurin formula can be written as

$$ 
\sum _ {x = 0 } ^ { {n }  - 1 }
f (x) \  = \ 
\int\limits _ { 0 } ^ { n } 
f (x) \  dx +
$$

$$ 
+ 
\sum _ {\nu = 1 } ^ { {k }  - 1 } 
\frac{B _  \nu  }{\nu ! }
 \{ f ^ {\  ( \nu - 1) } (n) - f ^ {\  ( \nu - 1) } (0) \} + 
\frac{B _ {k} }{k! }
 \sum
_ {x = 0 } ^ { {n }  - 1 } f ^ {\  (k) } (x + \theta ),
$$

where  $  0 < \theta < 1 $(
in general  $  \theta $
depends on  $  n $), 
and the  $  B _  \nu  $
are the [[Bernoulli numbers|Bernoulli numbers]]. If  $  f $
is a polynomial of degree less than  $  k $, 
the remainder term vanishes.

There is a similarity between the problems of the calculus of finite differences and those of differential and integral calculus. The operation of finding the difference corresponds to that of finding the derivative; the solution of equation (2), which, as an operation, is the inverse of finding the finite difference, corresponds to finding a primitive, that is, an indefinite integral. Formula (3) is a direct analogue of the [[Newton–Leibniz formula|Newton–Leibniz formula]]. This similarity emerges in considering finite-difference equations. By a finite-difference equation is meant a relation

$$ 
F (x,\  f (x),\ 
\Delta f (x) \dots
\Delta  ^ {n} f (x)) \  = \  0,
$$

where  $  F $
is a given function and  $  f $
is the required function. If all the  $  \Delta  ^ {n} f (x) $
are expressed in terms of  $  f (x),\  f (x + 1) \dots f (x + n) $, 
then the finite-difference equation is written in the form

$$ \tag{4 }
\Phi (x,\  f (x),\ 
f (x + 1) \dots
f (x + n)) \  = \  0.
$$

Its solution with respect to  $  f (x + n) $
is:

$$ \tag{5 }
f (x + n) \  = \ 
\psi (x,\  f (x) \dots
f (x + n - 1)).
$$

For given initial values  $  f (x _ {0} ) \dots f (x _ {0} + n - 1) $
one can successively find  $  f (x _ {0} + n),\  f (x _ {0} + n + 1) $, 
etc. After solving (4) with respect to  $  f (x) $:

$$ 
f (x) \  = \ 
\phi (x,\  f (x + 1) \dots
f (x + n)),
$$

one can, by setting  $  x = x _ {0} - 1 $, 
find  $  f (x _ {0} - 1) $, 
then  $  f (x _ {0} - 2) $, 
etc. Thus, from the equation in terms of the initial data one can find the values of  $  f $
at all the points  $  x _ {0} + k $, 
where  $  k $
is an integer. Consider, for  $  x = 0,\  1 \dots $
the linear equation

$$ \tag{6 }
f (x + k) +
P _ {1} (x)
f (x + k - 1) + \dots +
P _ {k} (x) f (x) \  = \ 
Q (x),
$$

where  $  P _ {1} (x) \dots P _ {k} (x) $
and  $  Q (x) $
are given functions on the set  $  x = 0,\  1 ,\dots $. 
The general solution of the inhomogeneous equation (6) is the sum of a particular solution of the inhomogeneous equation and the general solution of the homogeneous equation

$$ \tag{7 }
f (x + k) +
P _ {1} (x)
f (x + k - 1) + \dots +
P _ {k} (x) f (x) \  = \  0.
$$

If  $  f _ {1} \dots f _ {k} $
are linearly independent solutions of (7), then the general solution of (7) is given by the formula

$$ 
f (x) \  = \ 
c _ {1} f _ {1} (x) + \dots +
c _ {k} f _ {k} (x),
$$

where  $  c _ {1} \dots c _ {k} $
are arbitrary constants. The constants  $  c _ {1} \dots c _ {k} $
can be found by prescribing the initial conditions, that is, the values of  $  f (0) \dots f (k - 1) $. 
Linearly independent solutions  $  f _ {1} \dots f _ {k} $(
a fundamental system) are easily found in the case of an equation with constant coefficients:

$$ \tag{8 }
f (x + k) +
a _ {1} f (x + k - 1) + \dots +
a _ {k} f (x) \  = \  0.
$$

The solution of (8) is sought in the form  $  f (x) = \lambda  ^ {x} $. 
The characteristic equation for  $  \lambda $
is:

$$ 
\lambda  ^ {k} +
a _ {1} \lambda ^ {k - 1 } + \dots +
a _ {k} \  = \  0.
$$

Suppose that its roots  $  \lambda _ {1} \dots \lambda _ {k} $
are all distinct. Then the system  $  \lambda _ {1}  ^ {x} \dots \lambda _ {k}  ^ {x} $
is a fundamental system of solutions of (8) and the general solution of (8) can be represented by the formula:

$$ 
f (x) \  = \ 
c _ {1} \lambda _ {1}  ^ {x} + \dots +
c _ {k} \lambda _ {k}  ^ {x} .
$$

If  $  \lambda _ {1} $
is a root of the characteristic equation of multiplicity  $  s $, 
then the particular solutions corresponding to it are  $  \lambda _ {1} ,\  x \lambda _ {1} \dots x ^ {s - 1 } \lambda _ {1} $.

Suppose, for example, that one considers the sequence of numbers starting with 0 and 1 in which every successive number is equal to the sum of its two immediate predecessors: 0, 1, 1, 2, 3, 5, 8, 13 $  ,\dots $(
the [[Fibonacci numbers|Fibonacci numbers]]). One seeks an expression for the general term of the sequence. Let  $  f (x) $, 
$  x = 0,\  1 \dots $
be the general term of the sequence; the conditions

$$ 
f (x + 2) \  = \ 
f (x + 1) + f (x),\ \ 
f (0) \  = \  0,\ \ 
f (1) \  = \  1,
$$

form a difference equation with given initial conditions. The characteristic equation has the form  $  \lambda  ^ {2} - \lambda - 1 = 0 $, 
its roots being  $  \lambda _ {1} = (1 + \sqrt 5 )/2 $, 
$  \lambda _ {2} = (1 - \sqrt 5 )/2 $. 
Therefore

$$ 
f (x) \  = \  c _ {1} \left (
{
\frac{1 + \sqrt 5 }{2}
 }
\right )  ^ {x} +
c _ {2} \left (
{
\frac{1 - \sqrt 5 }{2}
 }
\right )  ^ {x} ;
$$

$  c _ {1} = 1/ \sqrt 5 $
and  $  c _ {2} = - 1/ \sqrt 5 $
being found from the initial conditions.

Equation (4) can be found not only when  $  x $
varies discretely, taking values  $  0,\  1 \dots $
but also when  $  x $
varies continuously. Let  $  f $
be arbitrarily defined in the half-open interval  $  [0,\  n) $. 
From (5) one obtains  $  f (n) $
by setting  $  x = 0 $. 
If  $  f $
is continuously defined in  $  [0,\  n) $, 
then  $  f $
may prove to be discontinuous in the closed interval  $  [0,\  n] $. 
If one wishes to deal with continuous solutions,  $  f $
needs to be defined on  $  [0,\  n) $
in such a way that by virtue of (5) it proves to be continuous in  $  [0,\  n] $. 
Knowledge of  $  f $
in  $  [0,\  n] $
enables one to find from (5)  $  f (x) $
for  $  x \in (n,\  n + 1] $, 
then for  $  x \in (n + 1,\  n + 2] $, 
etc.

More general than (8) is the equation

$$ \tag{9 }
f (x + h _ {k} ) +
a _ {1} f (x + h _ {k - 1 }  ) + \dots +
a _ {k} f (x) \  = \  0.
$$

Here  $  0 < h _ {1} < \dots < h _ {k} $
are not necessarily integers and are not necessarily commensurable relative to one another. Equation (9) has the particular solutions

$$ 
f (x) \  = \  e ^ {\lambda x } ,
$$

where  $  \lambda $
is a root of the equation

$$ 
L ( \lambda ) \  \equiv \ 
e ^ {h _ {k} \lambda } +
a _ {1} e ^ {h _ {k - 1 }  \lambda } + \dots +
a _ {k} \  = \  0.
$$

This equation has an infinite number of roots  $  \lambda _ {1} ,\  \lambda _ {2} ,\dots $. 
Consequently, (9) has an infinite number of particular solutions  $  e ^ {\lambda _ {m} x } $, 
$  m = 1,\  2 ,\dots $. 
Suppose that all the roots are simple. To express the solutions of (9) in terms of these elementary particular solutions, it is convenient to write the equation in the form:

$$ \tag{10 }
\int\limits _ { 0 } ^  \alpha  
f (x + t) \  d \sigma (t) \  = \  0,\ \ 
\alpha = h _ {k} ,
$$

where  $  \sigma (t) $
is a step function having jumps at the points  $  0,\  h _ {1} \dots h _ {k} $, 
equal to  $  a _ {k} ,\  a _ {k-1} \dots 1 $, 
respectively. Let

$$ 
\psi _  \nu  (t) \  = \ 

\frac{- e ^ {\lambda _  \nu  t } }{L  ^  \prime  ( \lambda _  \nu  ) }

\int\limits _ { 0 } ^ { t } 
e ^ {\lambda _  \nu  \xi } \ 
d \sigma ( \xi ),\ \ 
\nu \geq  1.
$$

The functions  $  \psi _  \nu  (t) $
have the property:

$$ 
\int\limits _ { 0 } ^  \alpha  
e ^ {\lambda _ {m} t }
\psi _  \nu  (t) \ 
dt \  = \  \delta _ {m \nu } 
$$

( $  \delta _ {m \nu }  = 1 $
if  $  m = \nu $, 
and  $  \delta _ {m \nu }  = 0 $
if  $  m \neq \nu $), 
that is, they form a system that is bi-orthogonal to the system  $  \{ e ^ {\lambda _ {m} t } \} $. 
On this basis, the solution  $  f $
of (10) corresponds to the series

$$ \tag{11 }
f (x) \  \sim \ 
\sum _ {\nu = 1 } ^  \infty  
c _  \nu  e ^ {\lambda _  \nu  x } ,
$$

$$ 
c _  \nu  \  = \  \int\limits _ { 0 } ^  \alpha   f (t) \psi _  \nu  (t) \  dt.
$$

If (9) has the form

$$ \tag{12 }
f (x + 2 \pi ) - f (x) \  = \  0
$$

(that is,  $  f $
is a periodic function with period  $  2 \pi $); 
$  L (x) = e ^ {2 \pi x } - 1 $; 
the roots of the equation  $  L ( \lambda ) = 0 $
are  $  mi $(
$  m = 0,\  \pm 1 ,\dots $); 
and (11) is the Fourier series of  $  f $
in complex form. The series (11) can be regarded as a generalization to the case of the difference equation (9) of the ordinary Fourier series corresponding to the simplest difference equation (12). Under certain conditions the series (11) converges to the solution  $  f $. 
If  $  f $
is an analytic function, then (9) is expressible in the form of an [[Equation of infinite order|equation of infinite order]]

$$ 
\sum _ {n = 0 } ^  \infty  
\alpha _ {n} f ^ {\  (n) } (x) \  = \  0.
$$

Differences of functions of several variables are introduced by analogy with differences of functions of one variable. For example, suppose that it is required to solve the problem of finding numerically the solution of the Laplace equation

$$ 

\frac{\partial   ^ {2} u }{\partial  x  ^ {2} }
 +

\frac{\partial   ^ {2} u }{\partial  y  ^ {2} }
 \  = \  0
$$

in the rectangle  $  0 \leq  x \leq  a $, 
$  0 < y < b $, 
for given values of  $  u $
on the boundary of the rectangle. The rectangle is divided into small rectangular cells with sides  $  \Delta x = a/N $, 
$  \Delta y = b/M $. 
The values of the solution are sought at the vertices of these cells. At the vertices that lie on the boundary of the original rectangle, the values of  $  u $
are known. By applying the approximation (where second-order differences are in the numerators)

$$ 

\frac{\partial   ^ {2} u }{\partial  x  ^ {2} }
 \  \approx \ 

\frac{u (x + \Delta x,\  y) - 2u (x,\  y) + u (x - \Delta x,\  y) }{\Delta x  ^ {2} }
 ,
$$

$$ 

\frac{\partial   ^ {2} u }{\partial  y  ^ {2} }
 \  \approx \  
\frac{u (x,\  y
+ \Delta y) - 2u (x,\  y) + u (x,\  y - \Delta y) }{\Delta y  ^ {2} }
 ,
$$

one obtains instead of the Laplace equation the following system of equations:

$$ 

\frac{u (x + \Delta x,\  y) - 2u (x,\  y) + u (x - \Delta x,\  y) }{\Delta x  ^ {2} }
 +
$$

$$ 
+ 

\frac{u (x,\  y + \Delta y) - 2u (x,\  y) + u
(x,\  y - \Delta y) }{\Delta y  ^ {2} }
 \  = \  0.
$$

The point  $  (x,\  y) $
runs through those vertices of cells that are situated in the interior of the original rectangle. Thus, a system of  $  (N - 1) (M - 1) $
equations is constructed containing the same number of unknowns. By solving this algebraic system of equations, the values of  $  u $
at the vertices of the rectangles are obtained. When  $  \Delta x $
and  $  \Delta y $
are small and the solution of the problem has a specified smoothness, the values so obtained are close to the exact values.

The calculus of finite differences was developed in parallel with that of the main branches of mathematical analysis. The calculus of finite differences first began to appear in works of P. Fermat, I. Barrow and G. Leibniz. In the 18th century it acquired the status of an independent mathematical discipline. The first systematic account of the calculus of finite differences was given by B. Taylor in 1715. The researches of mathematicians of the 19th century prepared the ground for the modern branches of the calculus of finite differences. The ideas and methods of the calculus of finite differences have been considerably developed in their application to analytic functions of a complex variable and to problems of numerical mathematics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Markov,   "The calculus of finite differences" , Odessa  (1910)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.O. [A.O. Gel'fond] Gelfond,   "Differenzenrechnung" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR></table>

====Comments====
See also [[Interpolation|Interpolation]]; [[Interpolation formula|Interpolation formula]]; [[Lagrange interpolation formula|Lagrange interpolation formula]]; [[Newton interpolation formula|Newton interpolation formula]]; [[Approximation of functions, linear methods|Approximation of functions, linear methods]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.M. Milne-Thomson,   "The calculus of finite differences" , Macmillan  (1933) {{ZBL|0008.01801}}; reprinted Dover (1981) {{ZBL|0477.39001}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Samarskii,   "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest & Portig K.-D.  (1984)  (Translated from Russian) {{ZBL|0543.65067}}</TD></TR>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Maurice V. Wilkes, "A short introduction to numerical analysis", Cambridge University Press (1966) ISBN 0-521-09412-7 {{ZBL|0149.10902}}

</TD></TR>
</table>