MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 13 pdf

... 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 is convergent and ... (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 ... 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 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 7 pdf

... 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. Defining a four-potential ... +ae d (TaB) r2sinOaW +a (WE)] After simplifying, the Laplacian is obtained as r2sin6 dr 34 r2sin28 a4 ' (10.239) Here we have obtained a well-known formula in a rather straightfor-...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

... Singular Points 13. 5 Residue Theorem 13. 6 Analytic Continuation 13. 7 Complex Techniques in Taking Some Definite Integrals 13. 8 Gamma and Beta Functions 13. 8.1 Gamma Function 13. 8.2 ... definition, 429 Cauchy integral formula, 390 Griinwald definition differintegrals, 385 Laplace transforms, 396 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING ... Mappings 12.4.4 Schwarz- Christ0 ffel Trans formations Problems 12.4 Mappings 13 COMPLEX INTEGRALS and SERIES 13. 1 Complex Integral Theorems 13. 2 Taylor Series 13. 3 Laurent Series 13. 4...

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 ... 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 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. ... sin28 3 1 If we call 1 4 = (A+ -) and compare with 2m y) + {A + T(Z, m))yx”(z) = 0, we obtain (m2 - 4) T(Z,rn) = sin2z . (9 .131 ) (9 .132 ) (9 .133 ) (9 .134 ) (9 .135 ) ... 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...

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 (10.197) I0 COORDINATES and TENSORS Starting with ... 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 8 pps

... 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) ... spherical harmonics &(O, 4). LORENTZ GROUP AND ITS LIE ALGEBRA 241 and These give us the remaining generators as and (11 .138 ) (11 .139 ) (11.140) Oi satisfy the commutation relation ... 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

... 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...

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