MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps

... LORENTZ GROUP AND ITS LIE ALGEBRA 245 and and introduce the parameters (1 1.166) 8= vf8: +8; +8, 2 and (11.167) (1 1.1 68) so that we can summarize these results as L = X. 68 + v-pp ... 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. 189 ) (1 1,190...

Ngày tải lên: 13/08/2014, 09:21

... Using these results we now obtain the following categories: A) For a # 0 Equation (9 .82 ) gives (9 .87 ) Ic* =acota(z+p). (9 .88 ) Substituting this into Equation (9 .83 ) and integrating ... + . 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 )...

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 ... (1+;)sin6’}yY(B). d8 sin8 9.11 tions of the first kind: Use the factorization method to show that the spherical Hankel func- hp = j, + in, can be expressed as Hint: Introduce...

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 ... mapping. Two different points in the z-plane, zo (12 .86 ) and are mapped to the same point in the w-plane as w = 20. 2 (12 .88 ) We now consider w = ez. (12 .89 )...

Ngày tải lên: 13/08/2014, 09:21

... definite integrals encountered in physics and engineering can be evaluated by using the complex integral theorems and analytic continuation: I. Integrals of the form I = Jc R (cos 8, ... 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. De...

Ngày tải lên: 13/08/2014, 09:21

... 1 78 1 78 179 180 181 183 164 186 188 188 188 189 189 189 189 190 193 195 196 197 197 197 199 80 1 201 x CONTENTS 10.10.5 Four- Tensors in Minkowski Spacetime 10.10.6 ... derivative, 484 Caputo definition, 429 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATH...

Ngày tải lên: 13/08/2014, 09:21

... points x = fl and z = 0. If we 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 ... 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 ... sphere is cut into sections that are separated by a very thin insulating material. One section extending from 6 = 0 to 8 = 8, at potential Vo and the second section extendi...

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

... However the second matrix has j =-1 -sin8 -cos8 (I 1.52) and can be written as [ case -sin81 = [ -sin8 cOse cos6 sine ] [ A :l 1. -sin8 -cos8 (11.53) This is reflection, that ... we see that the transformation matrix has the following possible two forms: (1 1.51) cos8 sin6 ] and [ cos8 -sin8 -sin8 cos8 The first matrix is familiar; it corresponds to ro...

Ngày tải lên: 13/08/2014, 09:21