|
|
|
|
indefinite sum
|
(Definition)
|
|
|
Recall that the finite difference operator $\Delta$ defined on the set of functions $\mathbb{R}\to \mathbb{R}$ is given by $$\Delta f(x):= f(x+1)-f(x).$$ The difference operator can be thought of as the discrete version of the derivative operator sending a function to its derivative (if it exists). With the derivative operation, there corresponds an inverse operation called the antiderivative, which, given a function $f$ , finds its antiderivative $F$ so that the derivative of $F$ gives $f$ . There is also a discrete analog of this inverse operation, and it is called the indefinite sum.
The indefinite sum of a function $f:\mathbb{R}\to \mathbb{R}$ is the set of functions $$\lbrace F:\mathbb{R}\to \mathbb{R}\mid \Delta F = f \rbrace.$$ This set is often denoted by $\Delta^{-1}f$ or $\Sigma f$ , and any element in $\Delta^{-1}f$ is called an indefinite sum of $f$ .
Remark. Like the indefinite integral, the indefinite sum $\Delta^{-1}$ is shift invariant. This means that for any $F\in \Delta^{-1} f$ , then $F+c \in \Delta^{-1} f$ for any $c\in \mathbb{R}$ . But, unlike the indefinite integral, the indefinite sum is also invariant by a shift of a periodic real function of period $1$ . Conversely, the
difference of two indefinite sums of a function $f$ is a periodic real function of period $1$ .
In the following discussion, we consider the indefinite sum of a function as a function.
Basic Properties
- $\Delta \Delta^{-1}f =f$ , and $\Delta^{-1} \Delta f=f$ modulo a real function of period $1$ .
- Modulo a real number, and treating $\Delta^{-1}$ as an operator taking a function into a function, we see that $\Delta^{-1}$ is linear, that is,
- $\Delta^{-1} (rf) = r\Delta^{-1} f$ for any $r\in \mathbb{R}$ , and
- $\Delta^{-1}(f+g)=\Delta^{-1}f +\Delta^{-1}g$ .
- If $F(x)=\Delta^{-1}f(x)$ , then $F(x+a)=\Delta^{-1}f(x+a)$ .
- If $F=\Delta^{-1}f$ , then we see that \begin{eqnarray*} F(a+1)-F(a) &=& f(a), \\ F(a+2)-F(a+1)&=& f(a+1), \\ &\vdots& \\ F(x)-F(x-1)&=& f(x-1). \end{eqnarray*}where $x-a$ is a positive integer. Summing these expressions, we get $$F(x)-F(a)=\sum_{i=1}^{x-a} f(a+i-1).$$ This is the discrete version of the fundamental theorem of calculus.
Below is a table of some basic functions and their indefinite sums ($C$ is a real constant):
| $f(x)$ |
$\Delta^{-1}f(x)$ |
Comment |
| $r\in \mathbb{R}$ |
$rx+C$ |
|
| $x$ |
$\displaystyle{\frac{x(x-1)}{2}+C}$ |
|
| $x^2$ |
$\displaystyle{\frac{x(x-1)(2x-1)}{6}+C}$ |
|
| $x^3$ |
$\displaystyle{\frac{x^2(x-1)^2}{4}+C}$ |
|
| $x^n$ |
$T_n(x)+C$ |
See this link for detail |
| $a^x$ |
$\displaystyle{\frac{a^x}{a-1}+C}$ |
$a\ne 1$ |
| $(x)_n$ |
$\displaystyle{\frac{(x)_n}{n+1}+C}$ |
$(x)_n$ is the falling factorial of degree $n$ |
| $\displaystyle{\binom{x}{n}}$ |
$\displaystyle{\binom{x}{n+1}+C}$ |
$\displaystyle{\binom{x}{n}:=\frac{(x)_n}{n!}}$ |
| $\displaystyle{\frac{1}{x}}$ |
$\psi(x)+C$ |
$\psi(x)$ is the digamma function |
| $\ln{x}$ |
$\ln{\Gamma(x)}+C$ |
$\Gamma(x)$ is the gamma function |
| $\sin{x}$ |
$\displaystyle{-\frac{\cos(x-1/2)}{2\sin(1/2)}+C}$ |
|
| $\cos{x}$ |
$\displaystyle{\frac{\sin(x-1/2)}{2\sin(1/2)}+C}$ |
|
- 1
- C. Jordan. Calculus of Finite Differences, third edition. Chelsea, New York (1965)
|
"indefinite sum" is owned by CWoo. [ full author list (2) ]
|
|
(view preamble | get metadata)
Cross-references: gamma function, digamma function, degree, falling factorial, fundamental theorem of calculus, expressions, summing, integer, positive, real number, properties, conversely, period, real function, periodic, invariant, antiderivative, inverse, operation, derivative, discrete, difference, functions, operator, finite difference
There is 1 reference to this entry.
This is version 16 of indefinite sum, born on 2007-10-15, modified 2009-04-23.
Object id is 9999, canonical name is IndefiniteSum.
Accessed 1297 times total.
Classification:
| AMS MSC: | 39A99 (Difference and functional equations :: Difference equations :: Miscellaneous) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
