Concrete Mathematics by R. Graham, D. Knuth, O. Patashnik

By R. Graham, D. Knuth, O. Patashnik

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The symbols D and A are called operators because they operate on functions to give new functions; they are functions of functions that produce functions. If f is a suitably smooth function of real numbers to real numbers, then Df is also a function from reals to reals. And if f is any real-to-real function, so is Af. The values of the functions Df and Af at a point x are given by the definitions above. Early on in calculus we learn how D operates on the powers f(x) = x"'. In such cases Df(x) = mxmP’.

Sometimes we might even be so bold as to write just 1 f.. + n. We’ll be working with sums of the general form al + a2 + ... + a,, A term is how long this course lasts. 1) where each ok is a number that has been defined somehow. This notation has the advantage that we can “see” the whole sum, almost as if it were written out in full, if we have a good enough imagination. Each element ok of a sum is called a term. The terms are often specified implicitly as formulas that follow a readily perceived pattern, and in such cases we must sometimes write them in an expanded form so that the meaning is clear.

Joseph Fourier introduced this delimited t-notation in 1820, and it soon took the mathematical world by storm. Incidentally, the quantity after x (here ok) is called the summa&. 2), because the k in ok is unrelated to appearances of k outside the Sigma-notation. 2). The letter i is often used (perhaps because it stands for “index”), but we’ll generally sum on k since it’s wise to keep i for &i. , to specify the set of indices over which summation should take place. 2) can also be written as ak .

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