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...

Ngày tải lên: 13/08/2014, 09:21

... (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 -...

Ngày tải lên: 13/08/2014, 09:21

... - 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...

Ngày tải lên: 13/08/2014, 09:21

... 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...

Ngày tải lên: 13/08/2014, 09:21

... 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(...

Ngày tải lên: 13/08/2014, 09:21

... 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...

Ngày tải lên: 13/08/2014, 09:21

... = 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 gene...

Ngày tải lên: 13/08/2014, 09:21

... 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...

Ngày tải lên: 13/08/2014, 09:21

... (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...

Ngày tải lên: 13/08/2014, 09:21

... 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) (...

Ngày tải lên: 13/08/2014, 09:21