By G. Stephenson

This textbook offers a high-quality starting place to a few vital subject matters in arithmetic of curiosity to technological know-how and engineering scholars. integrated are tensor algebra, usual differential equations, contour integration, Laplace and Fourier transforms, partial differential equations and the calculus of diversifications. The authors' process is easy and direct with an emphasis at the analytical figuring out of the fabric. The textual content is nearly selfcontained, assuming in simple terms that the coed has a high-quality realizing of ancillary arithmetic. every one bankruptcy features a huge variety of labored examples, and concludes with difficulties for resolution, with solutions at the back of the book

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**Example text**

In fact, all the Yn(x) are singular at x = 0. 5. Alternative forms exist for the function Yn(x). 4- -•* 38 Special functions When v=£n, this is just a linear combination of Jv(x) and J-V(x) and so is a solution of Bessel's equation (independent of Jv(x)). When v = n> the numerator and denominator are both zero, so the limit as v->n must be taken using L'Hopital's rule. 105) J (2107) To see why this is a solution of Bessel's equation consider the solutions Jv and /_ v . These satisfy x2 ^ 4- x ^ + (x2 - v 2 )/_ v = 0.

We first give a definition of such a series. 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.

Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964). 2 The gamma-function 23 special functions and their properties, we may regard a problem or equation as being solved if its solution can be expressed in terms of these functions. Besides arising as solutions of differential equations, other special functions are defined in terms of integrals. These occur in some of the analysis which follows in later chapters. The next three sections deal briefly with these functions.