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 5 ppsx

... 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. ... 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 ... 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

... (10. 1%) (10. 200) (10. 201) where we have defined the Christoffel symbols of the second kind as (10. 202) Substituting Equation (10. 201) in Equation (10. 190), we get Rearranging, 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 ... 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 Starting with...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps

... where the indices i, j and lc take the values 1,2,3 and they correspond to x, y, z, respectively. SPHERICAL HARMONICS AND REPRESENTATIONS OF R(3) 261 11.11 .10 To find the inverse ... 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‘) coordinates we obtain (Fig. 11.6) ... 240 CONTINUOUS GROUPS AND REPRESENTATIONS induces the following change in a function f(v1, ~2): - f(V) = f Mff, P, Y)VI' (1 1.126) Taking the active view we define an operator...

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 ... 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 ... (12 .108 ) 0 5 62 < 2~. (12 .109 ) We now investigate the limits of the points A, B, C, D, F, G, and H in the z-plane as they approach the real axis and the corresponding points...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

... 10. 10.5 Four- Tensors in Minkowski Spacetime 10. 10.6 Four- Velocity 10. 10.7 Four-Momentum and Conservation Laws 10. 10.8 Mass of a Moving Particle 10. 10.9 Wave Four- Vector 10. 10 .10 ... 10. 10 .10 Derivative Operators in Spacetime 10. 10.11 Relative Orientation of Axes in R and 10. 10.12 Maxwell’s Equations in Minkowski 10. 10.13 Transformation of Electromagnetic 10. 10.14 ... definition, 429 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING...

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 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 respect 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 ... 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 ... 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)...

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 ... (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)], ... form as (10. 388) 214 COORDINATES AND TENSORS and ( 10. 368) (10. 369) (10. 370) If given as is moving with respect to K with 3 the transformation equations are (10. 371) (10. 372)...

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