By Walter E. Thirring
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Additional info for A Course in mathematical physics / 4, Quantum mechanics of large systems
It is quicker, however, to use methods (iii) and (ii). Method (iii) gives D2 as (−24 + 36 − 44 + 52)/(4 + 6) = 2. 5 BINOMIAL EXPANSION x = −1 gives respectively 52 C D1 2 = − + , 24 6 2 4 36 B+C = − D1 + 2, 7 7 86 C−B D1 2 = − + . 63 7 3 9 These equations reduce to 4C − 12D1 = 40, B + C − 7D1 = 22, −9B + 9C − 21D1 = 72, with solution B = 0, C = 1, D1 = −3. Thus, ﬁnally, we may rewrite the original expression F(x) in partial fractions as F(x) = x + 2 + 1 3 2 . g. (x − α)m . Later in this book we will ﬁnd numerous occasions on which we wish to write such a product of repeated factors as a polynomial in x or, more generally, as a sum of terms each of which contains powers of x and α separately, as opposed to a power of their sum or diﬀerence.
43) The ﬁrst question that arises is that of how many terms there should be on the right-hand side (RHS). Although some complications occur when h(x) has repeated roots (these are considered below) it is clear that f(x) only becomes inﬁnite at the two values of x, α1 and α2 , that make h(x) = 0. Consequently the RHS can only become inﬁnite at the same two values of x and therefore contains only two partial fractions – these are the ones shown explicitly. This argument can be trivially extended (again temporarily ignoring the possibility of repeated roots of h(x)) to show that if h(x) is a polynomial of degree n then there should be n terms on the RHS, each containing a diﬀerent root αi of the equation h(αi ) = 0.
44) We now list several methods available for determining the coeﬃcients A1 and A2 . We also remind the reader that, as with all the explicit examples and techniques described, these methods are to be considered as models for the handling of any ratio of polynomials, with or without characteristics that make it a special case. (i) The RHS can be put over a common denominator, in this case (x+1)(x+2), and then the coeﬃcients of the various powers of x can be equated in the 19 PRELIMINARY ALGEBRA numerators on both sides of the equation.