By G. Stephenson

This textbook offers an excellent starting place to a couple of very important issues in arithmetic of curiosity to technological know-how and engineering scholars. integrated are tensor algebra, traditional differential equations, contour integration, Laplace and Fourier transforms, partial differential equations and the calculus of adaptations. The authors' technique is easy and direct with an emphasis at the analytical realizing of the cloth. The textual content is almost selfcontained, assuming purely that the scholar has a superb realizing of ancillary arithmetic. every one bankruptcy features a huge variety of labored examples, and concludes with difficulties for resolution, with solutions behind the book

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**Example text**

The series where a0, ax, a2, . . are constants, is said to be an asymptotic series of the function/(x), written /(x) ~ ^(x), provided that, for any ny the error involved in truncating the series at the term ajxn tends to zero faster than ajxn as x—>°°. In other words we require limx"[/(x)-S(x)] = 0. 51) Asymptotic series may be added, multiplied and integrated to obtain asymptotic series for the sum, product and integral of the corresponding functions. It is instructive to point out the difference between a convergent series and an asymptotic series: a convergent series tends to the corresponding function/(x) as n^oo for a given x, whereas an asymptotic series tends to f(x) as JC->» for a given n (that is, truncating the series at the term an/xn).

73) Hence ax = 0 unless v — \y but as mentioned earlier, this case will have values of m which differ by an integer and so may need to be considered separately. 71), the coefficient of the general term in jc m+r+2 must vanish giving ar+2 = - % 22 5 2 2) -v for r = 0, 1, 2, We now consider the two possible values of the constant m separately. 74) becomes (v + r + 2)2 - v2 so that (r + 2)(2v + r + 2)' 34 Special functions and so on for all values of r. 62), ~2 _4 •2) 2 . ]. 2 . 4 . 77) To define the Bessel function of order v uniquely it is conventional to choose a specific value of a0.

58). 2)) * 2 ^ + * £ + (*2-v2)y = 0. 61) We attempt a series solution of this equation by using the Frobenius series oc y =xm(a0 + a1x + a2x2 + . . 62) where, without loss of generality, we shall assume a o ^ 0 . ) 32 Special functions The conditions under which a series solution of this kind is valid are discussed in Chapter 4. 61). 61). The solutions may not be independent if the values of m are identical or differ by an integer (see Chapter 4) but the method will always generate one solution.