MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx

... + . d sin+ + p) ’ (9.89) where p and d are integration constants. With these Ico and Icl functions in Equation (9. 75) and the p(m) given in Equation (9. 85) we obtain r(z,m) ... exists a minimum value, mmin, thus determining X as X = mmin (9.143) To find mmin we equate the two expressions [Eqs. (9.141) and (9.143)] for X to obtain (9.144)...

Ngày tải lên: 13/08/2014, 09:21

... 56 2 56 4 56 5 56 7 56 7 56 7 56 9 56 9 57 2 57 2 57 3 57 5 57 9 58 2 58 3 58 4 59 3 59 4 59 6 59 7 603 606 606 609 19.2.3 Compounding Propagators 609 672 INDEX Divergent series, 4 65 ... Variables 44s 15. 8 Power Series 449 15. 8.1 Convergence of Power Series 450 15. 8.2 Continuity 450 Series 450 15. 8.4 Uniqueness Theorem 451 15. 8 .5 Inversio...

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

... POLYNOMIALS Writing this as (4 .53 ) and defining a new parameter (4 .54 ) 50 - = t, 2 and using the generating function [Eq. (4. 25) ], we obtain (4 .55 ) We now use the orthogonality ... 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...

Ngày tải lên: 13/08/2014, 09:21

... obtain the following recursion relations and First by adding and then by subtracting these equations we also obtain the relations m Jm-l(X) = -Jm(IL.) + J:,(X) (6 .52 ) X and (6 .53 ) ... taken as (6. 25) PROBLEMS 81 5. 7 Show the following special values: and 5. 8 Show the relations 5. 9 Using the generating function show that 5. 10 Show that Tn(z) and U...

Ngày tải lên: 13/08/2014, 09:21

... axn and Adding the first two equations and subtracting the last one from the result and after some rearrangement of indices we obtain (10.197) I0 COORDINATES and TENSORS Starting with ... choosing three mutually orthogonal straight lines. A point is defined by giving its coordinates, (q,z2,q), or by using the position vector 163 188 COORDINATES AND TENSORS Hence a...

Ngày tải lên: 13/08/2014, 09:21

... 1 - v2/c2 If the axis in K and R remain parallel but the velocity ?i' of (10. 256 ) (10. 257 ) (10. 258 ) (10. 259 ) ame ?T in frame K is arbitrary in direction, then the Lorentz ... respectively as and a; = (10.2 75) . (10.276) For the general linear transformation [Eq. (10. 250 )] matrix elements a$ can be obtained by using dZa dxp a; =...

Ngày tải lên: 13/08/2014, 09:21

... (1 1.348) (11.349) (1 1. 350 ) (11. 351 ) (1 1. 352 ) (1 1. 353 ) ( 1 1. 354 ) (1 1. 355 ) (11. 356 ) ( 1 1. 357 ) In terms of the three Euler angles the four independent parameters of u ... Equation (11. 153 ) and the relations g2 = I, A = eL, and A-l = ecL we can also write - - gAg = egLg = ePL; (11. 154 ) thus - gLg = - L. (11. 155 ) Sin...

Ngày tải lên: 13/08/2014, 09:21

... 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

... 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