I have updated the Reference for Functional Equations. I have added several entries to the tables, updated the graphs, added some new graphs, added some explanations and additional notes on notation, corrected a few typos, and re-organised the document slightly. Get the new version here.
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In this new version of Reference for Functional Equations I added several more z-transform pairs. I also started to add binomial transform pairs. The definition for the binomial is not consistent among different authors. I arbitrarily chose one, and later I changed it. I will probably change it again. Several typos were fixed. I am working on a system to include proofs so that the tables can be checked more easily.
I have not posted in a while; one reason is that I got sucked into some interesting mathematics; the work-in-progress Reference for Functional Equations is the result. If you are interested in such things – have a look.
Original image by openDemocracy.
The document below contains tables and formulas useful for working with functional equations, especially di fference equations, and to a lesser extent, quotient equations (where differences are replaced by quotients).
The reference contains tables for forward differences, (indefinite) sums, quotients, and products. There is also a table of z-transforms, binomial transfroms, formulas for converting certain kinds of functional equations to difference equations and some discrete Taylor series. There are more than 400 formulas in its 48 pages.
This is a work in progress, so be sure to read the preface (which highlights some of the issues with this document). If you find any errors, please comment below.
Changes
Version 1.1
- Added Exponential Sums to differences and sums.
- Additions to the z-transform table.
- Added Binomial Transform pairs.
Version 1.2
- Expanded the section on the discrete power functions.
- Expanded the section that explain the sue of constants in the table.
- Added forms involving the following expressions to the sum (x + h) tables:
- Updated all the graphs, and added some new ones.
- Reorganized slightly, and fixed some typos.
- Added a few examples, explanations, and additional notations in the sum (
) tables.
Download
DescreteCalculusTables.pdf (685 KB).
Tags: binomial transform, difference equation, discrete calculus, functional equation, functional equations, Mathematics, z-transform
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