Editing Finite difference (section)

===Rules for calculus of finite difference operators===
Analogous to [[Differentiation rules|rules for finding the derivative]], we have:
* '''Constant rule''': If {{mvar|c}} is a [[Constant (mathematics)|constant]], then <math display="block">\ \Delta c = 0\ </math>
* '''[[Linearity of differentiation|Linearity]]''': If {{mvar|a}} and {{mvar|b}} are [[Constant (mathematics)|constants]], <math display="block">\ \Delta (a\ f + b\ g) = a \ \Delta f + b \ \Delta g\ </math>

All of the above rules apply equally well to any difference operator as to {{math|Δ}}, including {{math|δ}} and {{nobr|{{math|∇}}.}}

* '''[[Product rule]]''': <math display="block">\begin{align}
\ \Delta (f g) &= f \,\Delta g + g \,\Delta f + \Delta f \,\Delta g \\[4pt]
\nabla (f g) &= f \,\nabla g + g \,\nabla f - \nabla f \,\nabla g
\ \end{align}</math>
* '''[[Quotient rule]]''': <math display="block">\ \nabla \left( \frac{f}{g} \right) = \left. \left( \det \begin{bmatrix} \nabla f & \nabla g \\ f & g \end{bmatrix} \right) \right/ \left( g \cdot \det {\begin{bmatrix} g & \nabla g \\ 1 & 1 \end{bmatrix}}\right) </math> or <math display="block"> \nabla\left( \frac{f}{g} \right)= \frac {g \,\nabla f - f \,\nabla g}{g \cdot (g - \nabla g)}\ </math>
* '''[[Fundamental theorem of calculus|Summation rules]]''': <math display="block">\begin{align}
\ \sum_{n=a}^b \Delta f(n) &= f(b+1)-f(a) \\
\sum_{n=a}^{b} \nabla f(n) &= f(b)-f(a-1)
\ \end{align}</math>

See references.<ref>
{{cite book
|last1=Levy|first1=H.
|last2=Lessman|first2=F.
|year=1992
|title=Finite Difference Equations
|publisher=Dover
|isbn=0-486-67260-3
}}
</ref><ref>
{{cite book
|last=Ames|first=W.F.
|year=1977
|title=Numerical Methods for Partial Differential Equations
|at=Section&nbsp;1.6
|publisher=Academic Press
|place=New York, NY
|isbn=0-12-056760-1
}}
</ref><ref>
{{cite book
|last=Hildebrand|first=F.B.|author-link=Francis B. Hildebrand
|year=1968
|title=Finite-Difference Equations and Simulations
|at=Section&nbsp;2.2
|publisher=Prentice-Hall
|place=Englewood Cliffs, NJ
}}
</ref><ref>
{{cite journal
| first1 = Philippe| last1 = Flajolet
| first2 = Robert| last2 = Sedgewick| author-link2 = Robert Sedgewick (computer scientist)
| year = 1995
| title = Mellin transforms and asymptotics: Finite differences and Rice's integrals
| journal = Theoretical Computer Science
| volume = 144| issue = 1–2| pages = 101–124
| doi = 10.1016/0304-3975(94)00281-M
| url = http://algo.inria.fr/flajolet/Publications/FlSe95.pdf
}}
</ref>