MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 2 pdf

... substitutions kin and 24 2-n (2. 74) to write (2. 75) n=O C21 (2 - n)!n! (1 - 2n)! Comparing this with the right-hand side of Equation (2. 65), which is we obtain (2. 76) (2. 77) 2. 3.3 Recursion ... write &s+l = - (4s + 3, - p2s (O) U2S +2( avo), s = 0,1 ,2, 2 (2s +2) (2. 1 32) Substituting these in Equation (2. 127 ) we finally obtai...

Ngày tải lên: 13/08/2014, 09:21

... div and the grad operators for the fol- i) ds2 = dr2 + p2dB2 + dz2 ii) ds2 = [ ] dr2 + r2a2 + r2 sin2 Odd2, 1 - kr2/R2 where k takes the values k = 0,1, -1. 21 6 ... 10.9): 1 V 2J = J [." - (,) x'] 1 - u2/c2 1 V -1 z - - J [-(,)."+x'] 1 - u2/c2 (10 .25 2) (10 .25 3) (10 .25 4) (10 .25 5) -2 - 2 x -x 2=...

Ngày tải lên: 13/08/2014, 09:21

... (15 .22 5) Applying this formula with z = x to the sine and cosine functions we obtain 22 00 sinz=zn (I ) nW n=l (15 .22 6) and (15 .22 7) These products are finite for all the finite values ... l1 (2k + 2) (2lc + 1) lim k - - 1 = lim k k-cc [Uztt2 ] k+oo [ (2k )2- n2 6k +2+ n2 3 k-co (21 C)Z - n2 which indicates that the series is convergent...

Ngày tải lên: 13/08/2014, 09:21

... Generators Normal Forms 25 0 25 1 25 1 25 5 25 5 25 7 25 8 26 0 26 1 26 2 26 4 26 6 26 8 26 9 27 2 27 2 27 3 27 4 27 4 27 4 27 5 27 6 27 7 27 8 27 9 28 1 28 1 28 2 28 5 Problems 28 8 ... Harmonics and Representations of R(3) 11.11.1 Angular Momentum in Quantum Mechanics 20 2 20 4 20 5 20 7 20 8 20 8 20 9 21 1 21 3 21 4 21 5 21 6 22 3...

Ngày tải lên: 13/08/2014, 09:21

... H: (2) - 22 HL (x) + 2nHn (x) = 0. Hermite Polynomials HO (2) = 1 HI (2) = 22 H2(z) = -2 + 4x2 H&) = - 122 + 823 H4 (2) = 12 - 48x2 + 16x4 HS (2) = 120 2 - ... Frobenius method to i> 2x 3 -++x2-++3y=O d2Y dY dx2 dx ii) iii) iv) v) vi) 3 d2Y 2 dY 8 x -++x -+(x3+-x)y=O 9 dx2 dx 3 d2Y 2 dY 3 x -+fx -+(x3+-x)y=O dx...

Ngày tải lên: 13/08/2014, 09:21

... C(c0sx) with a two-term recursion relation. 5 .2 Using the line element ds2 = c2dt2 - &(t )2[ dX2 + sin2 xdB2 + sin2 Xsin2 13d4~], find the spatial volume of a closed universe. ... functions are given 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 - -)...

Ngày tải lên: 13/08/2014, 09:21

... (9.138) 1 2 p(1 72) = (m - - )2. (9.139) For convenience we have taken p(0) = a2c2 rather than zero in Equation (9. 92) . 122 STURM-LIOUVILLE SYSTEMS AND THE FACTORIZATION METHOD 9.1 ... write O+(z, 2) O- (2, l)yX"(z) = [-m2 + P]i+(z) (9 .20 2) 0-(z,1+l)O+(z,I+ l)~Xm(z) = [-m2+(1+1 )2] 1/Ixm(.z). (9 .20 3) Using (9 .20 4) GENERALIZED FOURIER SE...

Ngày tải lên: 13/08/2014, 09:21

... written as - 21 = allxl f a 122 2 + a13x3 22 = a2 121 + a 222 2 + a2 323 x3 = a31x1 + a 322 2 + (333x3 . (10 .23 ) - - aij are constants independent of 7 and 7. A convenient ... obtained from SinJ-n+l /2 COSJ+n+l /2 - e 2 2 by using 2 1r2 1 YE-’ = (J + - )2 - (m - - )2 L The corresponding eigenvalues are c7 + K + 1/4 = (J...

Ngày tải lên: 13/08/2014, 09:21

... (1 1 .22 7) d icotOsin4- 84 a a ae 34 L* = L, * ZL, = e*;q*- + icot 0-), L2 = L: + L; + Lq (11 .22 8) (11 .22 9) (11 .23 0) (11 .23 1) Using Equations (11 .22 7-11 .22 9) we can ... (1 1 .22 5) Expressing the components of the angular momentum operator in spherical polar coordinates: x + zy = r sin &+a6 and z = rcosf?, (1 1 .22 6) 25 2 CO...

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) 27 6 CONTINUOUS GROUPS AND REPRESENTATIONS ... IZll + 1z21 7 I21z21 = 121 1 122 1 7 arg (21 22 ) = arg z1 + arg 22 are also very useful in calculation with complex numbers. 12....

Ngày tải lên: 13/08/2014, 09:21