MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf

... be expressed in terms of the poten- tials -A' and 4 as In the Lorentz gauge 2 and 4 satisfy and (10. 373 ) (10. 374 ) (10. 375 ) (10. 376 ) (10. 377 ) respectively. Defining a four-potential ... respectively as and a; = (10. 275 ) . (10. 276 ) For the general linear transformation [Eq. (10.250)] matrix elements a$ can be obtained by using dZa dx...

Ngày tải lên: 13/08/2014, 09:21

... 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. In fact, the number ... 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, w...

Ngày tải lên: 13/08/2014, 09:21

... start by integrating the above integral by parts twice to get and then (15. 175 ) (15. 176 ) We keep on integrating by parts to obtain the series 436 INFINITE SERIES Example 15 .7. Cuwhy ... (15.134) Integrating this by parts we obtain where we have used Bl(1) = a and Bl(0) = -4. In the above integral we now use (15.1 37) 1 Blk) = and integrate by parts again...

Ngày tải lên: 13/08/2014, 09:21

... 2 97 Chebyshev equation, 75 second kind, 76 Chebyshev polynomials first kind, 75 , 76 Gegenbauer polynomials, 76 generating function, 78 orthogonality and completeness, second kind, 76 ... 163 1 63 166 166 1 70 1 70 1 72 1 74 175 177 1 78 1 78 179 180 181 183 164 186 188 188 188 189 189 189 189 190 193 195 196 1 97 1 97 1 97 199...

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- spect...

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 ... can 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....

Ngày tải lên: 13/08/2014, 09:21

... Cancelling m on both sides and noting that (9.166) (9.1 67) and using Equation (9.164) we finally write Similarly We now define the ladder operators L+ and L- for the m index of ... 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. (9.1 4...

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 (10.1 97) I0 COORDINATES and TENSORS Starting with ... Xr(t), (10 .73 ) ( 10 .74 ) ( 10 .75 ) where X = A'(0). (10 .76 ) Differentiating Equation (10 .75 ) we can now obtain the higher-order deriva- tives as (10 .77 ) Using t...

Ngày tải lên: 13/08/2014, 09:21

... following eigenvalue equations [Eqs. (9. 172 - 173 ) and (9.1 87- 188)]: Eigenvalue Equations for L,, Lk, and L2 LzKm(Q, 4) = mXm(Q7 4)> L-K,(Q,~) = J(i+m)(i -m+ 1)x,m-1(Q74 )7 L+xm(Q, ... write Equation (11. 276 ) as (11. 278 ) (11. 279 ) Using the matrix elements of (Ly)n obtained in Section 11.11 .7, we write this as (11.280) dm,,(P) 1 = Jmmg -i(L...

Ngày tải lên: 13/08/2014, 09:21

... in two dimensions. 2. The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some definite integrals. 3. Infinite series, infinite ... straight line), that starts from the origin and then extends all the way to infinity. CONTINUOUS GROUPS AND SYMMETRIES 277 there corresponds a Hermitian differential operator...

Ngày tải lên: 13/08/2014, 09:21