MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 1 potx

... 16 6 16 6 1 70 1 70 1 72 1 74 17 5 17 7 1 78 1 78 17 9 18 0 18 1 18 3 16 4 18 6 18 8 18 8 18 8 18 9 18 9 18 9 18 9 19 0 19 3 19 5 19 6 19 7 19 7 19 7 19 9 8 01 2 01 x CONTENTS 10 .10 .5 ... Hermitian Operators in Quantum Mechanics Problems 10 7 10 7 I 08 11 0 11 0 11 1 11 1 11 2 11 3 11 4 11 5 11 8 9 STURM-LIOUVILLE SYSTEMS and the FACTORIZATION METHOD 12 1 9 .1 Another Form ... 11 .11 .6 Generalized Fourier Expansion in 11 .11 .7 Matrix Elements ofL,,Lv, and L, 11 .11 .8 Rotation Matrices for the Spherical 11 .11 .9 Evaluation of the di,m(,f?) Matrices 11 .19 .10 ...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf

... (2 .11 1) 21 dx to obtain This gives or ( 21 - 'i! llxd [ (1 - x'),'] ( 21 - 1) N[ = - N1 -1+ - 21 2 21 (l!) ( 21 - 1) 1 Ni = ~ " -1 - -Nl 21 21 (2 .11 2) ... the 21- fold derivative of (x2 - 1) l as ( 21) !. Thus Equation (2 .10 8) becomes (2 .11 0) 1 We now write (1 - x2) as (1 - 2) 1 = (1 - x2) (1 - xy = (1 -xy + xd (1 - ... 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...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 3 docx

... - m)! (1 + m)! PFrn(Z) = ( -1) rn- P;"(z), m 2 0. 2 .13 in the interval [ -1, 1]. 2 .14 A metal sphere is cut into sections that are separated by a very thin insulating material. ... points in three dimensions 1 1 where r< and r, denote the lesser and the greater of r and r', respectively. 2 .17 Evaluate the sum & +1 03 s = c -fi(Z). l=O 1+ 1 ... where x and E are defined in terms of the radial distance r and the energy E as r E x=- and E= - Iri hW/2 vmw and 1 takes integer values 1 = 0 ,1, 2 . i) Examine the...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 4 pptx

... as X1 22 + ), x< ;1, jdx) + ( 21 + l)!! (1 - 2(2l+ 1) ( 21 - l)!! 22 f - xl+l (1 - 2 (1- 21) + ), x< ;1, 1 LIT jl(x) = ;sin(x - -), x >> 1, X 1 17 1. ndx) ... generating function 00 1 = EC2(z)t", 11 1 < 1, 1x1 5 1, x > -1/ 2, n=O (1 - 22t + t2)X HYPERGEOMETRIC SERIES 10 1 and the recursion relation (s+r- l+a)(s+r- 1+ b) ... Orthogonality and the Generating Function The generating function of the Gegenbauer polynomials is defined as M = xC;(~)t", It1 < 1, 1x1 5 1, X > -1/ 2. (5.30) 1 (1 - 2xt...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 5 ppsx

... value, mmin, thus determining X as X = mmin (9 .14 3) To find mmin we equate the two expressions [Eqs. (9 .14 1) and (9 .14 3)] for X to obtain (9 .14 4) mmin = -1. (9 .14 6) Since ... (9 .11 0) 15 2 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD This is Type F with q = -1 and m = 1. Thus we determine k(r, 1) and p (1) as 11 rI k(r,l) = - - -, 1 12 . ... eigenvalues XI as 1 4 Xt+-=X (9 .14 8) Xl+L(l+;) 2 4 (9 .14 9) Xt + - 1 = 12 +1 +- 1 (9 .15 0) 4 4 XI = 1( 1+ 1) . (9 .15 1) THEORY OF FACTORIZATION AND THE LADDER OPERATORS 12 7 a given...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 6 pps

... derivative to obtain (10 .19 8) (10 .1% ) (10 .200) (10 .2 01) where we have defined the Christoffel symbols of the second kind as (10 .202) Substituting Equation (10 .2 01) in Equation (10 .19 0), we ... general tensor can be defined with mixed indices as (10 .12 9) (10 .13 0) (10 .13 1) (10 .13 2) GENERALIZED COORDINATES AND GENERAL TENSORS 18 5 Similarly, we can define the metric tensor with ... procedure used in Path I in Section 9.6.5 to derive ct m 1 (1+ 1-m) yln,?’(8) = - C,+i,m (1 + 1 - m) J(l + m + 1) (1+ m + 2) COS~ 7- (1+ ;)sin6’}yY(B). d8 sin8 9 .11 tions...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 7 pdf

... = mo-, dT i = 1, 2,3 (10 .302) (10 .303) U1 = Using this in Equation (10 .3 01) we obtain (1 - u2/c2) or (1 - ?l2lc2 U”C2) 1 l? i po=moc 1+ In order to interpret po, we ... 99 -1 = 9 -19 = go. (1 1. 4) (11 .5) 11 .1. 1 Terminology In n dimensions the set {A} of all linear transformations with det A #O forms a group called the general linear group in ... has j = -1 -sin8 -cos8 (I 1. 52) and can be written as [ case -sin 81 = [ -sin8 cOse cos6 sine ] [ A :l 1. -sin8 -cos8 (11 .53) This is reflection, that is, (1 1.54) followed...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 8 pps

... R we get Using Equation (11 .207), this becomes We now operate on this with R to get where (11 .209) (11 . 210 ) (11 . 211 ) (11 . 212 ) (11 . 213 ) (11 . 214 ) ( 1 1. 2 15 ) ( 11 .2 16 ) SPHERICAL ... = 4i-P x a‘. (11 .18 9) (1 1 ,19 0) Writing L in Cartesian coordinates we find its components as (we set fi = 1) (1 1. 1 91) (1 1 .19 2) (1 1 .19 3) In Section 11 .3 .1 we have seen that ... write which is (1 1.348) (11 .349) (1 1. 350) (11 .3 51) (1 1. 352) (1 1. 353) ( 1 1.354) (1 1.355) (11 .356) ( 1 1.357) In terms of the three Euler angles the four independent parameters...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 9 pps

... (Fig. 12 .2) as w = f (z) = u (2, y) + iv (z, y) . (12 .7) (12 .8) (12 .9) (12 .10 ) (12 .11 ) (12 .12 ) (12 .13 ) (12 .14 ) (12 .15 ) (12 .16 ) (12 .17 ) 276 CONTINUOUS GROUPS AND REPRESENTATIONS ... (1 1. 388) (1 1. 389) (11 .390) (11 .3 91) 11 .16 CONTINUOUS GROUPS AND SYMMETRIES In everyday language the word symmetry is usually associated with familiar operations like rotations and ... vectors (11 .426) where E@ are small. so that (11 .427) where is a unit vector in the direction of e and the generators are defined as in Equation (11 . 418 ). 11 .16 .5 Infinitesimal...

MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10 pps

... Jn (13 .47) Integrals over 11 and 12 can be evaluated by defining the parameters and writing (13 .49) (13 .50) MAPPINGS 313 Fig. 12 .12 Conformal mapping 12 .4 .1 Conformal Mappings ... polar coordinates to write the modulus 20 lim At-0 and the argument (Fig. 12 .12 ) as a = lim arg[Azu] - lim arg[Az] Az-0 Ar-0 (12 .11 6) ( 12 .1 17) (12 .11 8) (12 .11 9) Since this ... TECHNIQUES IN TAKING SOME DEFINITE INTEGRALS Many of the definite integrals encountered in physics and engineering can be evaluated by using the complex integral theorems and analytic continuation:...

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