MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 11 ppt

... Relativity 16 3 1 63 16 6 16 6 1 70 1 70 1 72 1 74 17 5 17 7 1 78 1 78 17 9 18 0 18 1 18 3 16 4 18 6 18 8 18 8 18 8 18 9 18 9 18 9 18 9 19 0 19 3 19 5 19 6 19 7 19 7 19 7 19 9 8 01 2 01 ... Hermitian Operators in Quantum Mechanics Problems 10 7 10 7 I 08 11 0 11 0 11 1 11 1 11 2 11 3 11 4 11 5 11 8 9 STURM-LIOUVILLE SYST...

Ngày tải lên: 13/08/2014, 09:21

... substitutions kin and 24 2-n (2. 74) to write (2. 75) n=O C21 (2 - n)!n! (1 - 2n)! Comparing this with the right-hand side of Equation (2. 65), which is we obtain (2. 76) (2. 77) 2. 3.3 Recursion ... write &s+l = - (4s + 3, - p2s (O) U2S +2( avo), s = 0,1 ,2, 2 (2s +2) (2. 1 32) Substituting these in Equation (2. 127 ) we finally obtai...

Ngày tải lên: 13/08/2014, 09:21

... 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 (3. 42). ... defined by setting a0 = 1 in Equation (3. 23) as 3. 2 OTHER DEFINITIONS OF LAGUERRE POLYNOMIALS 3. 2.1 The generating function of the Laguerre polynomials is defined as Gener...

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 ... following integral definitions: (6 .48 ) and 1 2 (1 - t2)n-6 cosztdt, (n > ). (6 .49 ) Jn(IL.1 = 6 .4 RECURSION RELATIONS OF THE BESSEL FUNCTIONS Using the s...

Ngày tải lên: 13/08/2014, 09:21

... + . 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

... use the approximations sin 6$ N S+, sin 6Cp N 6Cp (10. 96) cos6$ N 1, cos6Cp _N 1 to find 10 R1R2 = [ 6+ 1 =R2R,. (10.97) 6$ -64 Note that in terms of the generators ... 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...

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... 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

... 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

... 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

... DERIVATIVES AND INTEGRALS 381 situation on the applied side of this branch of mathematics is now changing rapidly, and there are now a growing number of research areas in science and engineering ... techniques in evaluating definite integrals and finding sums of infinite series. In this chapter, we introduce some of the basic properties of fractional calculus...

Ngày tải lên: 13/08/2014, 09:21

... DIFFERINTEGRALS IN SCIENCE AND ENGINEERING 427 Fig. 14.7 Probability distribution in random walk and CTRW An important area of application for fractional derivatives is that the ex- traordinary ... dPaex r*(a,z) = r(a)ecX- dx- 14.7 APPLICATIONS OF DIFFERINTEGRALS IN SCIENCE AND ENGINEERING 14.7.1 Continuous Time Random Walk (CTRW) We have seen that the...

Ngày tải lên: 13/08/2014, 09:21

... INFINITE PRODUCTS 471 Integrating Equation (15.223) gives and finally the general expression is obtained as (15.225) Applying this formula with z = x to the sine and cosine functions ... (z) are continuous functions, cannot be uniformly convergent in any interval containing a discontinuity of f (x). INFINITE PRODUCTS 469 exists, then we say the infinite pro...

Ngày tải lên: 13/08/2014, 09:21

... for the above integral is an infinite straight line passing through the point y and parallel to the imaginary axis in the complex s-plane. y is chosen such that all the singularities of ... defined as and it is usually encountered in potential energy calculations in cylindrical coordinates. Another useful integral transform is the Mellin transform: (16 .14) The Mell...

Ngày tải lên: 13/08/2014, 09:21

... 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