MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 11 ppt

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

... 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 SYSTEMS and the FACTORIZATION METHOD 12 1 9 .1 Another Form ... 11 .11 .6 Generalized Fourier Expansion in 11 .11 .7 Matrix Elements ofL,,Lv, and L, 11 .11 .8 Rotation Matrices for the Spherical 11 .11 .9 Evaluation of the di,m(,f?) Matrices 11 .19 .10 ...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf

... 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 obtain the potential outside the sphere as ... obtain (2. 61) (2. 62) (2. 63) (2. 64) thus proving the equivalence of Equations (2. 60) and (2. 59). 2. 3 .2 Generating Function Another way to define the Legendre polynomials is by using...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx

... 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 Generating ... 43 This Page Intentionally Left Blank 64 HERMl TE POLYNOMIALS Writing this as (4. 53) and defining a new parameter (4.54) 50 - = t, 2 and using the generating function [Eq....

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx

... 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 series definitions ... 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 linearly independent...

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) mmin = -1. ... m=O which converges absolutely and uniformly in all subintervals free of points of discontinuity. At the points of discontinuity this series rep resents (as in the Fourier series) the arithmetic...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

... 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 and ... 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...

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 dxp a; = (10. 277 ) In Minkowski spacetime ... (10. 378 ) we can write Equations (10. 376 ) and (10. 377 ) in covariant form as (10. 379 ) SPACETIME AND FOUR-TENSORS 203 For the Lorentz transformations [Eqs. (10.252-10.255) and (10.260-10.261)],...

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) Writing L in ... A* and C = -B*; thus u becomes ( 1 1 .80 ) (1 1 .81 ) Taking the determinant of the unitarity condition [Eq. (11.71)] and using the fact that det ut = det u, we obtain (1...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps

... 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 mechanics appear as an ... absolutely and uniformly in all subdomains free of points of discontinuity. At the points of discontinuity it represents the arithmetic mean of the right- and the left-hand limits. 00 m=O In this...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10 pps

... 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. Definition ... lines into circles in the z-plane. 12.11 Use the transformation given in Problem 12 .10 to find the equipoten- tials and the electric field lines for the electrostatics problem of two infinite...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 11 ppt

... 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 along with some mathematical ... infinitesimally close point, (a - a), to a and then continue integration on the other side from an arbitrarily close point, (a + S), to infinity, that is, define the integral I as (13.175)...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 12 doc

... 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 diffusion equation is ... and jC integrals Writing the remaining part of the $, dz integral we get (:!$I (14.82) After taking the limit we substitute this into the definition [Eq. (14.74)] to obtain (14.83)...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 13 pdf

... 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 product ... 454 INFINITE SERIES 4. n 5. 6. (15.124) (15.125) 7. In the interval [O, 11 and for s 2 1, the only zeroes of &+1(z) are 0, 4, and 1. In the same interval 0 and 1...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 14 doc

... 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 Mellin transform ... function in the infinite interval (-CO,CO). We now consider a nonperiodic function in the infinite interval (-o,co). Physically this corresponds to expressing an arbitrary signal in terms...

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 interval [a, b] we obtain two ... also he obtained by direct construction, that is, by determining the u and the solutions in Equation (19.15) as sin bx and sin ko(x - L), respectively. We can now obtain a closed...

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