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 Gener...

Ngày tải lên: 13/08/2014, 09:21

... and Definitions 14.1.2 The nth Derivative of a Function 2 93 2 93 295 296 297 299 30 0 31 3 31 4 31 8 32 2 32 9 33 5 33 5 33 9 34 0 34 7 34 7 34 9 35 2 36 0 36 0 36 2 36 4 36 4 36 5 ... Series 431 15.2 Absolute Convergence 432 15 .3 Convergence Tests 433 15 .3. 1 Comparison Test 433 15 .3. 2 Ratio Test 433 15 .3. 3 Cuuchy Root Test 433 15 .3....

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... points x = fl and z = 0. If we 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 ... 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. I...

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 ... 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)...

Ngày tải lên: 13/08/2014, 09:21

... exists a minimum value, mmin, thus determining X as X = mmin (9.1 43) To find mmin we equate the two expressions [Eqs. (9.141) and (9.1 43) ] 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....

Ngày tải lên: 13/08/2014, 09:21

... a13x3 22 = a2121 + a2222 + a 232 3 x3 = a31x1 + a3222 + (33 3x3 . (10. 23) - - aij are constants independent of 7 and 7. A convenient way to write Equa- tion (10. 23) ... choosing three mutually orthogonal straight lines. A point is defined by giving its coordinates, (q,z2,q), or by using the position vector 1 63 188 COORDINATES AND TENSORS Hence...

Ngày tải lên: 13/08/2014, 09:21

... be expressed in terms of the poten- tials -A' and 4 as In the Lorentz gauge 2 and 4 satisfy and (10 .37 3) (10 .37 4) (10 .37 5) (10 .37 6) (10 .37 7) respectively. Defining a four-potential ... 3. 3 = 471-p ( 10 .34 8) (10 .34 9) 3. 3=0 (10 .35 0) = 0, 1 83 VXZ+ cat (10 .35 1) determine the electric and magnetic fields for a given charge,...

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... 1 .33 2) where A and B satisfy detu=IAl 2 +lBl 2 =l. (1 1 .33 3) These act on two-dimensional vectors in the complex plane, which we show as (11 .33 4) or w =war a = 1,2. (11 .33 5) ... -L 23 0 L= 1 Introducing six independent parameters (PI, 02, j 33) and (01, 02, 03) , this can also be written as L = PiV1 + P2V2 + P3V3 +&XI + 02x2 +...

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

... leaving us with dz‘. f 1 Un(n<O) = - 2Fa z’n+l Jn ( 13. 47) Integrals over 11 and 12 can be evaluated by defining the parameters and writing ( 13. 49) ( 13. 50) MAPPINGS 31 3 Fig. ... real w-axis into the triangular region shown in Figure 12 .31 , 25 f 03 and 23 f - 03 12.14 Find the equipotentials and the electric field lines for a conducting c...

Ngày tải lên: 13/08/2014, 09:21