MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps

... useful in finding solutions of Laplace equation in two dimensions. 2. The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some ... complex techniques are very helpful in certain problems of physics and engineering, which are essentially problems defined in the real domain, complex numbers in quantum mech...

Ngày tải lên: 13/08/2014, 09:21

... [Eqs. (9. 141) and (9. 143)] for X to obtain (9. 144) mmin = -1. (9. 146) Since m changes by integer amounts, we could write hin = mmax - integer -1 = 1 - integer 21 = integer. ... TECHNIQUE AND THE CATEGORIES OF FACTORIZATION 135 1 k1=-, a=O bd ko = -z+ 22 z (9. 98) (9. 99) After writing c for d and adding b/2 to p(m) we obtain...

Ngày tải lên: 13/08/2014, 09:21

... 1 0 - sin $ cosCp sin4cos$ cosdsin+ -sin4 cos~cos$ (10 .93 ) Reversing the order we get cos $ (10 .94 ) sin $ sin Cp - sin $ cos Cp R2R1 = 0 cos Cp sin 4 [ sin $ - cos ... COORDINATES In threedimensional Euclidean space a Cartesian coordinate system can be constructed by choosing three mutually orthogonal straight lines. A point is defined by giving its...

Ngày tải lên: 13/08/2014, 09:21

... in Quantum Mechanics + L=?;’xT, (11.188) by replacing position and momentum with their operator counterparts, that is, ?+?, as L = 4i-P x a‘. (11.1 89) (1 1, 190 ) Writing L in ... properties that remain invariant under similarity transformations. One such 254 CONTINUOUS GROUPS AND REPRESENTATIONS Y’ Fig. 11.6 (z,y,z) and the (z‘,y’,z‘) coordinates we obta...

Ngày tải lên: 13/08/2014, 09:21

... INTEGRALS Many of the definite integrals encountered in physics and engineering can be evaluated by using the complex integral theorems and analytic continuation: I. Integrals of the form ... 0 and a-lrnl # 0, then is called a singular point of order m. Definition I11 Essential singular point: If m is infinity, then a is called an essential singular point....

Ngày tải lên: 13/08/2014, 09:21

... definition, 4 29 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING ... 177 1 78 1 78 1 79 180 181 183 164 186 188 188 188 1 89 1 89 1 89 1 89 190 193 195 196 197 197 197 199 801 201 x CONTENTS 10.10.5 Four- Tenso...

Ngày tải lên: 13/08/2014, 09:21

... write x = fl in the generating function Equation (2.65) we find (2.85) Expanding the left-hand side by using the binomial formula and comparing equal powers oft, we obtain 9 (1) = 1 ... part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics....

Ngày tải lên: 13/08/2014, 09:21

... that the value of the integral in Equation (3.38) can be obtained by expanding in powers of t and s and then by comparing the equal powers of tnsm with the left-hand side of Equation ... (4.52) 'I A This Page Intentionally Left Blank ORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating with re- spe...

Ngày tải lên: 13/08/2014, 09:21

... equations in physics and engineering can be solved by the method of separation of variables. This method helps us to reduce a second-order partial differential equation into a set of ordinary ... second kind. Note that N,(z) and Jm(z) are linearly independent even for the integer values of m. Hence it is common practice to take N,(z) and Jm(z) as the two lin...

Ngày tải lên: 13/08/2014, 09:21

... any g E G, 99 0 = 90 9 = 9. iv) For every g E G, there exists an inverse element, 9- l E G, such that 99 -1 = 9- 19 = go. (1 1.4) (11.5) 11.1.1 Terminology In n dimensions ... (r2sinog) +ae d (TaB) r2sinOaW +a (WE)] After simplifying, the Laplacian is obtained as r2sin6 dr 34 r2sin28 a4 ' (10.2 39) Here we have obtained a well-known f...

Ngày tải lên: 13/08/2014, 09:21