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2014-01-17T00:50:58Z

181.130.156.77: Undid revision 589532054 by Fraulein451 (talk)

In [[mathematics]] the '''indefinite sum''' operator (also known as the '''antidifference''' operator), denoted by <math>\sum _x \,</math> or <math>\Delta^{-1} \,</math>,<ref>{{PlanetMath|urlname=IndefiniteSum|title=Indefinite Sum}}</ref><ref>[http://hostel6.ru/books/_Papers/Computer_algebra/Summation/Man.%20Closed%20forms%20for%20symbolic%20summation.%20JSC%201993%20(22s).pdf On Computing Closed Forms for Indefinite Summations. Yiu-Kwong Man. J. Symbolic Computation (1993), 16, 355-376]</ref><ref>"If ''Y'' is a function whose first difference is the function ''y'', then ''Y'' is called an indefinite sum of ''y'' and denoted Δ<sup>-1</sup>''y''" [http://books.google.co.uk/books?id=5rFOeE0zvY4C&pg=PA41&dq=%22indefinite+sum%22 ''Introduction to Difference Equations''], Samuel Goldberg</ref> is the [[linear operator]], inverse of the [[difference operator|forward difference operator]] <math>\Delta \,</math>. It relates to the [[difference operator|forward difference operator]] as the [[indefinite integral]] relates to the [[derivative]]. Thus

:<math>\Delta \sum_x f(x) = f(x) \, .</math>

More explicitly, if <math>\sum_x f(x) = F(x) \,</math>, then

:<math>F(x+1) - F(x) = f(x) \, .</math>

If ''F''(''x'') is a solution of this functional equation for a given ''f''(''x''), then so is ''F''(''x'')+''C'' for any constant ''C''. Therefore each indefinite sum actually represents a family of functions, differing by an additive constant.

==Fundamental theorem of discrete calculus==

Indefinite sums can be used to calculate definite sums with the formula:<ref>"Handbook of discrete and combinatorial mathematics", Kenneth H. Rosen, John G. Michaels, CRC Press, 1999, ISBN 0-8493-0149-1</ref>

:<math>\sum_{k=a}^b f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)</math>

==Definitions==

===Laplace summation formula===

:<math>\sum _x f(x)=\int_0^x f(t) dt +\sum_{k=1}^\infty \frac{c_k\Delta^{k-1}f(x)}{k!} + C </math>

:where <math>c_k=\int_0^1 \frac{\Gamma(x+1)}{\Gamma(x-k+1)}dx</math> are the [[Bernoulli numbers of the second kind]].<ref>[http://mathworld.wolfram.com/BernoulliNumberoftheSecondKind.html Bernoulli numbers of the second kind on Mathworld]</ref>

===Newton's formula===
:<math>\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C</math>

:where <math>(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)}</math> is the [[falling factorial]].

===Faulhaber's formula===

:<math>\sum _x f(x)= \sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n(x) + C \, ,</math>

provided that the right-hand side of the equation converges.

===Mueller's formula===
If <math>\lim_{x\to{+\infty}}f(x)=0,</math> then

:<math>\sum _x f(x)=\sum_{n=0}^\infty\left(f(n)-f(n+x)\right)+ C.</math> <ref>[http://www.math.tu-berlin.de/~mueller/HowToAdd.pdf Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations] (note that he uses a slightly alternative definition of fractional sum in his work, i.e. inverse to backwards difference, hence 1 as the lower limit in his formula)</ref>

===Ramanujan's formula===

:<math>\sum _x f(x)= \int_0^x f(t) dt - \frac12 f(x)+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x) + C</math>

==Connection to the Ramanujan summation==
Often the constant C in indefinite sum is fixed from the following equation:

: <math>\int_1^2 \sum _x f(x) dx=0 </math>

or

: <math>\int_0^1 \sum _x f(x) dx=0 </math>

In this case, where

: <math>F(x)=\sum _x f(x) \,</math>

then Ramanjuan's sum is defined as

: <math>\sum_{x \ge 1}^{\Re}f(x)=F(0)\,</math>

or

: <math>\sum_{x \ge 1}^{\Re}f(x)=F(1)\,</math><ref>Bruce C. Berndt, [http://www.comms.scitech.susx.ac.uk/fft/math/RamanujanNotebooks1.pdf Ramanujan's Notebooks], ''Ramanujan's Theory of Divergent Series'', Chapter 6, Springer-Verlag (ed.), (1939), pp. 133–149.</ref><ref>Éric Delabaere, [http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf Ramanujan's Summation], ''Algorithms Seminar 2001–2002'', F. Chyzak (ed.), INRIA, (2003), pp. 83–88.</ref>

==Summation by parts==

{{main|Summation by parts}}

Indefinite summation by parts:
:<math>\sum_x f(x)\Delta g(x)=f(x)g(x)-\sum_x (g(x)+\Delta g(x)) \Delta f(x) \,</math>

:<math>\sum_x f(x)\Delta g(x)+\sum_x g(x)\Delta f(x)=f(x)g(x)-\sum_x \Delta f(x)\Delta g(x) \,</math>

Definite summation by parts:
:<math>\sum_{i=a}^b f(i)\Delta g(i)=f(b+1)g(b+1)-f(a)g(a)-\sum_{i=a}^b g(i+1)\Delta f(i)</math>

==Period rule==

If <math>T \,</math> is a period of function <math>f(x)\,</math> then

:<math>\sum _x f(Tx)=x f(Tx) + C\,</math>

==Alternative usage==

Some authors use the phrase "indefinite sum" to describe a sum in which the numerical value of the upper limit is not given. e.g.

:<math>\sum_{k=1}^n f(k)</math>

In this case a closed form expression F(k) for the sum is a solution of

:<math>F(x+1) - F(x) = f(x+1) \,</math> which is called the telescoping equation.<ref>[http://www.risc.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf Algorithms for Nonlinear Higher Order Difference Equations], Manuel Kauers</ref> It is inverse to [[backward difference]] <math>\nabla</math> operator.
It is related to the forward antidifference operator using the fundamental theorem of discrete calculus described earlier.

==List of indefinite sums==

This is a list of indefinite sums of various functions. Not every function has an indefinite sum that can be expressed in terms of elementary functions.

===Antidifferences of rational functions===

:<math>\sum _x a = ax + C \,</math>

:<math>\sum _x x = \frac{x^2}{2}-\frac{x}{2} + C</math>

:<math>\sum _x x^a = \frac{B_{a+1}(x)}{a+1} + C,\,a\notin \mathbb{Z}^-</math>
:where <math>B_a(x)=-a\zeta(-a+1,x)\,</math>, the generalized to real order [[Bernoulli polynomials]].

:<math>\sum _x x^a = \frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)}+ C,\,a\in\mathbb{Z}^-</math>
:where <math>\psi^{(n)}(x)</math> is the [[polygamma function]].

:<math>\sum _x \frac1x = \psi(1-x) + C </math>
:where <math>\psi(x)</math> is the [[digamma function]].

===Antidifferences of exponential functions===

:<math>\sum _x a^x = \frac{a^x}{a-1} + C \,</math>

===Antidifferences of logarithmic functions===

:<math>\sum _x \log_b x = \log_b \Gamma (x) + C \,</math>

:<math>\sum _x \log_b ax = \log_b (a^{x-1}\Gamma (x)) + C \,</math>

===Antidifferences of hyperbolic functions===

:<math>\sum _x \sinh ax = \frac{1}{2} \operatorname{csch} \left(\frac{a}{2}\right) \cosh \left(\frac{a}{2} - a x\right) + C \,</math>

:<math>\sum _x \cosh ax = \frac{1}{2} \coth \left(\frac{a}{2}\right) \sinh ax -\frac{1}{2} \cosh ax + C \,</math>

:<math>\sum _x \tanh ax = \frac1a \psi _{e^a}\left(x-\frac{i \pi }{2 a}\right)+\frac1a \psi _{e^a}\left(x+\frac{i \pi }{2 a}\right)-x + C</math>

:where <math>\psi_q(x)</math> is the [[q-analog|q-digamma]] function.

===Antidifferences of trigonometric functions===

:<math>\sum _x \sin ax = -\frac{1}{2} \csc \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}- a x \right) + C \,,\,\,a\ne n \pi </math>

:<math>\sum _x \cos ax = \frac{1}{2} \cot \left(\frac{a}{2}\right) \sin ax -\frac{1}{2} \cos ax + C \,,\,\,a\ne n \pi</math>

:<math>\sum _x \sin^2 ax = \frac{x}{2} + \frac{1}{4} \csc (a) \sin (a-2 a x) + C \, \,,\,\,a\ne \frac{n\pi}2</math>

:<math>\sum _x \cos^2 ax = \frac{x}{2}-\frac{1}{4} \csc (a) \sin (a-2 a x) + C \,\,,\,\,a\ne \frac{n\pi}2</math>

:<math>\sum_x \tan ax = i x-\frac1a \psi _{e^{2 i a}}\left(x-\frac{\pi }{2 a}\right) + C \,,\,\,a\ne \frac{n\pi}2</math>

:where <math>\psi_q(x)</math> is the [[q-analog|q-digamma]] function.

:<math>\sum_x \tan x=ix-\psi _{e^{2 i}}\left(x+\frac{\pi }{2}\right) + C = -\sum _{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1-z\right)+\psi \left(k \pi -\frac{\pi }{2}+z\right)-\psi \left(k \pi -\frac{\pi }{2}+1\right)-\psi \left(k \pi -\frac{\pi }{2}\right)\right) + C\,</math>

:<math>\sum_x \cot ax =-i x-\frac{i \psi _{e^{2 i a}}(x)}{a} + C \,,\,\,a\ne \frac{n\pi}2</math>

===Antidifferences of inverse hyperbolic functions===

:<math>\sum_x \operatorname{artanh}\, a x =\frac{1}{2} \ln \left(\frac{(-1)^x \Gamma \left(-\frac{1}{a}\right) \Gamma \left(x+\frac{1}{a}\right)}{\Gamma \left(\frac{1}{a}\right) \Gamma \left(x-\frac{1}{a}\right)}\right) + C</math>

===Antidifferences of inverse trigonometric functions===

:<math>\sum_x \arctan a x = \frac{i}{2} \ln \left(\frac{(-1)^x \Gamma (\frac{-i}a) \Gamma (x+\frac ia)}{\Gamma (\frac ia) \Gamma (x-\frac ia)}\right)+C</math>

===Antidifferences of special functions===

:<math>\sum _x \psi(x)=(x-1) \psi(x)-x+C \,</math>

:<math>\sum _x \Gamma(x)=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}e+C</math>

:where <math>\Gamma(s,x)</math> is the [[incomplete gamma function]].

:<math>\sum _x (x)_a = \frac{(x)_{a+1}}{a+1}+C</math>

:where <math>(x)_a</math> is the [[falling factorial]].

:<math>\sum _x \operatorname{sexp}_a (x) = \ln_a \frac{(\operatorname{sexp}_a (x))'}{(\ln a)^x} + C \,</math>
:(see [[super-exponential function]])

==See also==
*[[Indefinite product]]
*[[Time scale calculus]]
*[[List of derivatives and integrals in alternative calculi]]

==References==
{{reflist}}

==Further reading==
* "Difference Equations: An Introduction with Applications", Walter G. Kelley, Allan C. Peterson, Academic Press, 2001, ISBN 0-12-403330-X
* [http://www.math.tu-berlin.de/~mueller/HowToAdd.pdf Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations]
* [http://arxiv.org/abs/math/0502109 Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities]
* [http://www.springerlink.com/content/kj0jx24240756457/ S. P. Polyakov. Indefinite summation of rational functions with additional minimization of the summable part. Programmirovanie, 2008, Vol. 34, No. 2.]
* "Finite-Difference Equations And Simulations", Francis B. Hildebrand, Prenctice-Hall, 1968

{{DEFAULTSORT:Indefinite Sum}}
[[Category:Mathematical analysis]]
[[Category:Mathematical tables|Indefinite sums]]
[[Category:Finite differences]]
[[Category:Linear operators in calculus]]

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