MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 16 pptx

... dk'g(k')eikfZ VG -" and Their inverse Fourier transforms are and Using these in Equation (19. 116) we get which gives us Substituting this in Equation (19.118) we obtain Writing g(k') ... Xj and take its complex conjugate as (18.97) Multiplying Equation (18.96) by Xjy;(z) and Equation (18.97) by Xiyi(z), and integrating over x in the interv...

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

... definition, 429 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING ... 11.15 Hilbert Space and Quantum Mechanics 11 .16 Continuous Groups and Symmetries 11 .16. 1 One-Parameter Point Groups and Their 11 .16. 2 Transformation of Generators and...

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

... 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

... Cancelling m on both sides and noting that (9 .166 ) (9 .167 ) and using Equation (9 .164 ) we finally write Similarly We now define the ladder operators L+ and L- for the m index of ... defined as Ladder Operators for the Spherical Harmonics Using Equation (9 .162 ) we write Using (9 .162 ) (9 .163 ) (9 .164 ) (9 .165 ) 146 STURM-LIOUVILLE SYSTEMS AND THE F...

Ngày tải lên: 13/08/2014, 09:21

... 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 aa and ... 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 Starti...

Ngày tải lên: 13/08/2014, 09:21

... Groups with infinite number of elements like R(n) are called infinite groups. A group with finite number of elements is called a finite group, where the number of elements in a finite group ... 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. Defini...

Ngày tải lên: 13/08/2014, 09:21

... GROUP AND ITS LIE ALGEBRA 245 and and introduce the parameters (1 1 .166 ) 8= vf8:+8;+8,2 and (11 .167 ) (1 1 .168 ) so that we can summarize these results as L = X.68 + v-pp (11 .169 ) ... 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‘) coordin...

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