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 ... a, (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 2 pdf

... 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, we obtain 9 (1) = 1 ... and for meticulously reading Chapters 1 and 9 with 14 and 20. I also thank Prof. N. K. Pak for many interesting and stimulating discussions, encouragement, and critical reading...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 13 pdf

... 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 to obtain ... (15.131) 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...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

... 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 97 199 801 201 x CONTENTS ... 429 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING PROBLEMS...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx

... 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 BlankORTHOGONALiTY OF LAGUERRE POLYNOMlALS 49 Interchanging the integral and the summation signs and integrating with re- spect to x gives ... are very useful in cosmology and quantum field theory in curved backgrounds. Both the spherical harmonics and the Gegenbauer poly- nomials are combinations of sines and cosines. Chebyshev...

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 ... 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) (6.52) X and (6.53) ... equation has three regular singular points at z = 0,l and 00; hence we can find a series solution about the origin by using the Frobenius method. Substituting the series 03 y = CarxJ+T,...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx

... 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 47) ... 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...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

... 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 these in the Taylor ... 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...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps

... 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(Lg)mm, sinp+ (L~)mm~(co~P- ... A"B + C*D = 0 (1 1 .74 ) (1 1 .75 ) (1 1 .76 ) (11 .77 ) From elementary matrix theory (Boas), the inverse of u can be found as (11 .78 ) Because for SU(2) the inverse of u is also...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps

... 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 acting on the ... useful in physics and engineering applications: 1. In the theory of complex functions there are pairs of functions called conjugate harmonic functions, which are very useful in finding solutions...

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