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 ... (r)- 43 This Page Intentionally Left Blank64 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 1 potx

... 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. 4 Integral Test 434 15 .3. 5 Raabe ... definition, 429 Cauchy integral formula, 39 0 Griinwald definition differintegrals, 38 5 Laplace transforms, 39 6 484 Fractional derivatives MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf

... 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. In fact, the number ... P2 (.) = (2) [32 - 11 P3 (z) = (k) [5z3 - 3x1 P4(z) = (i) [35 z4 - 30 x2 + 31 (2.58) P5(z) = (t) [63z5 - 70x3 + 15.1. P6(.) = (A) [ 231 z6 - 31 5~~ + 105~~ -...

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 ... 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) ... - dxl+I (5 .37 ) (5 .38 ) 5 .3. 3 Chebyshev polynomials of the second kind are defined as Chebyshev Polynomials of the Second Kind Un(x) = sin(nX), n = O,1,2 , (5 .39 ) where x...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx

... 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. 134 ) (9. 135 ) ... Substituting this into Equation (9. 83) and integrating gives us ko(z) = cacota(z +p) + . d sin+ + p) ’ (9.89) where p and d are integration constants. With these Ico and Icl...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

... 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 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 .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, and current distribution. We now introduce ... Orientation of Axes in K and K Frames A = (Ao,A1,A2,A3) (10 .33 1) (10 .33 2) In terms of another Minkowski frame, the same four-vector can be written as (10 .33 3) A=B,A , where &...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps

... 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 + 03x3, (11.158) where ... is (1 1 .34 8) (11 .34 9) (1 1 .35 0) (11 .35 1) (1 1 .35 2) (1 1 .35 3) ( 1 1 .35 4) (1 1 .35 5) (11 .35 6) ( 1 1 .35 7) In terms of the three Euler angles the four independent parameters of u...

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 ... (gv:vi)1'2. (1 1 .37 2) Note that the inner product in the complex plane is no longer symmetric, that is, -81 .32 # 3. 2 .31 , (11 .37 3) however, (11 .37 4) is true. 11.14.5 We now define a vector...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10 pps

... 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 circular cylinder held ... 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...

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