By G. Stephenson

ISBN-10: 0521363128

ISBN-13: 9780521363129

ISBN-10: 052136860X

ISBN-13: 9780521368605

This textbook presents an excellent starting place to a couple of very important themes in arithmetic of curiosity to technology and engineering scholars. integrated are tensor algebra, traditional differential equations, contour integration, Laplace and Fourier transforms, partial differential equations and the calculus of diversifications. The authors' method is straightforward and direct with an emphasis at the analytical figuring out of the cloth. The textual content is almost selfcontained, assuming merely that the scholar has an outstanding knowing of ancillary arithmetic. every one bankruptcy features a huge variety of labored examples, and concludes with difficulties for answer, with solutions behind 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.

162) = ds2. Hence, the quantity ds is a scalar since it is invariant under a general coordinate transformation. Tensor algebra and its associated calculus are important tools in the study of continuum mechanics and in the general theory of relativity. Problems 1 1. Write out aikXiXk in expanded form, assuming aik — akiy and i, k = 1, 2, 3. 2. Over which indices (if any) in the following expressions is summation implied? (i) dijbj, (ii) dijbjj, (iii) atibny (iv) au = bu. 3. Find the values of 6iy6iy, <5,y(5yVAm<5/m, ejklAkAh and dikeikmy all indices ranging from 1 to 3.

Then < 215S > <2156> which is a real function of x. 152). Iv(x) and Kv(x) are called modified Bessel functions and their properties can be obtained in a similar way to those of Jv(x) and Yv(x). The main properties of these functions are given in the next section. 157) can be solved in terms of the modified Bessel functions. 158) /0(JC) and K0(x). y(x) = Axko(x) + Bx±K0(x). 160) is y(x) =Ax%(x/2) + Bx*Kv(x/2). 163) Summary of the main properties of special functions 1. Bessel functions Bessel's equation of order v is x2 —^ + x j - + (x2 - v2)y = 0.

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