A Treatise on the Calculus of Finite Differences

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Macmillian and Company, 1880 - Difference equations - 336 pages
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Even though it was written in 1860, this book probably even more relevant today than it was then. Why? Because it provides the mathematical basis for solving higher and higher resolution partial differential equations with faster and faster computers for the complex domains of modern engineering problems. The faster, more accurate and more efficient solution to these types of problems provides the foundation for improving quality of life for everyone.
I like his definition of the calculus of finite differences, it says so much, succinctly:
"The Calculus of Finite Differences may be strictly defined as the science which is occupied about the ratios of the simultaneous increments of quantities mutually dependant."
This book has to be understood as a second volume to accompany Boole's first book on Differential Equations. He stresses the importance of the relationship between the two in his preface to TCFD. The discussion of the distinction between a calculus of limits and the calculus of differences highlights this connection.
Like most old math or science books, it is the discussion of important applications that feels dated. The ideas are timeless, but the important applications change with the problems and the technology of the time.
Another timeless idea that bears repeating around all practitioners of the numerical art (from Chapter 8):
"... we shall very often need to use the method of Finite Differences for the purpose of shortening numerical calculation, and here the mere knowledge that the series obtained are convergent will not suffice; we must also know the degree of approximation.
To render our results trustworthy and useful we must find the limits of the error produced by taking a given number of terms of the expansion instead of calculating the exact value of the function that gave rise thereto."
More about finite differences on my blog:

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Page 121 - A SERIES is said to be convergent or divergent according as the sum of its first n terms approaches or does not approach to a finite limit when n is indefinitely increased.
Page 7 - Jacobi polynomials will be found in the exercises at the end of this chapter.
Page 82 - The successive orders of figurate numbers are defined by this ; — that the **" term of any order is equal to the sum of the first x terms of the order next preceding, while the terms of the first order are each equal to unity.
Page 187 - ... (1) and also by n — 1 , other differential equations, of the second order, to which the calculus of variations conducts, as supplementary to the given equation (1), and which may be thus denoted : /(*.)- */'(**,)_ _/'(*.)-*/'(**.).
Page 317 - OS of the angle which the normal makes with the axis of x...
Page 317 - Hence integrating we find ...... ..... (3), Ex. 3. Required a curve such that a ray of light proceeding from a given point in its plane shall after two reflections by the curve return to the given point. The above problem has been discussed by Biot, whose solution as given by Lacroix (Diff. and Int.
Page 231 - ... 0. The solution of these equations, which are linear, can be made to depend upon that of a linear equation of the second order having t = 0 for a singularity : it appears that the integrals are normal in the vicinity of t = 0. Their full expression is...
Page 103 - I numerically. aa; (1 — e ) x=0 [Schlomilch, Grunert x. 342.] 10. Shew that the sum of all the negative powers of all whole numbers (unity being in both cases excluded) is unity ; 3 if odd powers are excluded it is 7 . 11.
Page 47 - The two radii which form a diameter of a circle are bisected, and perpendicular ordinates are raised at the points of bisection.

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