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0534552080_98868

# Advanced Engineering Mathematics

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Unformatted text preview: 394 Section 21.1 Section 21.1 Limits, Continuity and Derivatives 1. “With 2 2 m—l—i—y, we find ﬁg) 2 2—4, 2 :5+-.iy—i 2 at+i(y——1}; Thus Mat, 1;} 2 :17, Mitt, y] 2 y— , K 51:, Be (321. 81} 1. For the Cauchydhmimnn equatlon, con'ipute —,——— 2 1, ‘— 2 11 and -,---w 2 U, —— 2 U; Since ‘ 0:1: (3y 6y 3.1: u, 1)? “annoy, ﬂy, '0; are all conth’mons and the Cauchy-Riemann equations hold everywhere, f is differentiable everywhere. 2. ﬁx + iv) = (:1: + is)? - in: + iv) e (3:2 :92 + y) + We; — :6}; Thus utv. y) = :62 — y? + .1 . , 8o 81; 8' 3;, M33, 3;) 2 23:3; — 17:. }‘or the Cauchy—310111511111 equations, coi'npnte ,— 2 2:5, — 2 23;, j 2 (3 d3: 3y 3y 1; 22;!) + 1, 3— 2 2y — 1; so Caucliy~Rimna11n equations hold evei'ywl'iere‘ Since we and all 3; partials are continuous, f is differentiable everywhere. 3. f(:1: + ﬁg) 2 1:1: +in 2 xxx? 4—y2; so n(s;,y) 2 “\$2 +-y2,v(1?,y) 2 O. For the Cauchy— Riernann equations, compute 5'1; _ ('31; y 31) 6o :1: Z I _ 1 D) r : ——'W—I — 33: 1/32 + 3:2 83; (ﬁg \/:::2 —§-- 3;? 8:1: We see that the Canchy~Riemann hold nowhere, and f(z) 2 lzj is differentiable nowhere. 23? + 1 + 2m: __ [(21—13 + 1) + 2in [1: —?_,_y] 0. 4-. f(x + 33,12) — 2562+ iy ' \$2 + 9'2 " — fir—2+3? ; Thus Many) 2 %,o(m, y) 2 Efﬁgy—2 For the Cauchy—Riemann equations, compute an m (\$2 +y2)(4:s + 1) — (2:132 —l— :r: + 2y2}(2\$) _n y? — 9:2 a- — (m2 + m2 """""" — W 3v _ —(:L"2 + 122) + 32(23):) _ 322 — :62 fly (x2 + 3;?)2 (CC2 + 92)” 81:. _ 4y(\$2 + y?) — 23,4232 + .‘I: -+- 23:2) _ —2:1:y a: H " (x2 +4235" — (x2 we“ 31) 2mg - 3—3: : [m2 + 3:2)? Since um, and all partial are continuous at all z 75 0, and the Cauchy-Riemann equations hold for all z 79 O, f is differentiable for all z 2 U. 5. f(:1: + ﬂy) 2 315 + iy|2 2 its}! + yg), so u(:r:,y) 2 O,v(a:,y) 2 3:2 + yi". For the Gaming— as 61; an o Riemann e nations, com Jute — 2 0, ----- — 2 2 ,— 2 0,——~ 2 2:. -= ‘c - - " c C] 1 as: at), y (’33,! <33: ”L The Cauchy Riemann equations hold only for a: 2 y 2 U, is. z 2 0 but f is not diﬁ'erentiahle at z 2 0. 6. f(:L‘ + ﬁg) 2 a: + iy + Im(\$ +1131) 2 (:1: + y) + £3}, so May) 2 :3: + y,-u(:1:, y) 2 3;. For 61:. _ 6-!) (31¢ _ 8o 2 (l. The Cauchy- the Cancl1y~RieInann equations, compute — ------ 2 1, mm 2 H— 2 l, -— 6:2: By (93; 8:1: Riemann equations hold I'iowhere, so f is not differentiable at any point. , :1; + #21,: :I: -£— iy , y) y . . 7. .1: ' 1n 2 2 21 '1 :so-u. 33 2 ]. 10:1: 2 ‘—. F ‘ .2 It " — H r J) Reﬁt + 3,9) 3: | (:1; , ( ,y) i ( ,y] :1: 01 ilit C inchy _ , 5n Eh: 1. 3-1:, 81: 1 . Riemann equations, compute — 2 0 --- — -- ——-J--. 'i‘he Cauchy—Riemann (3:1: ’5} _ ifs—y —" J8:1: 3:2 Section 21.1 ' 395 equations held nowhere, so f is not differentiable at any point. 8. ﬁg: + ty) 2 (a: + iy)3 __ 8(3: ~1— t'y) + 2 z 3:3 ~— 33:3,:2 — 8.1; + 2 + 1:83:23; w y3 — 8y), so n(:r,y) z 3:3 — 3:1:y2 _. 83: + 2, o(\$,y) 2 33:23; — ya — 8y. For the Cauchy-Riemann equations, 61L 2 2 a” 2 2 81}, av t-.—=3s: —3 —8-~w:3a: 3 8, 2 6x1, compu 0 as y , 8y y 83; J 6.1: Riemann equations hold everywhere, 1t,’U, and all partial are continuous everywhere, so f is diﬁerentiabie everywhere. = 6:33;. The Cauchy- 9. ﬁx —l— ty) : (a: — ty)2 : :1:2 — :92 — 233m, so u(e:,y) 1—, 3:2 — y2,o(\$,y) : -—--23:y. For . . 6o 8 8 6- the Cauchwaiemann equatlons, compute :9"; — 23:, (73% : —2:c, 8—: 2 —2y, 5:- : ~23}. The Cauohwaieinann equations hold only at 3; z y m U or z = 0, but f is not diﬁ‘erentiable at z x 0. 10. f(a:+iy) = i(m+iy)+l:v+iyl : —t!+\/332 +y2+ico so note) = _?}+\/E2—+?av(\$iyl = . . . 8 8 6 3:. For the Cauchy-Riemann equations, _con1pute ii 56 o = O _’u "r w~1+ a Z Two—y ,5, — 8 . . —-E——7, —U = 1. The Cauchy-Blemann equat1oris hold nowhere, so f is not differentiable Va:2 + “5:3 am at any point. . . 1 3: y 11. :* 2—4 + - 2—4‘ m ' —4 — -. f(t+%s) (93 19H m+iy \$+\$2+y2+3( 3! \$2+y2),so _ 35 _ '5‘ “(31y)“ _4m + \$2 +y21v(mly) _ “4y — \$2 +y2 For the Cauchy—Riemann equations, compute 6 2 W ,2 a 2 _ 2 u: 4+3! ....... i ..,..’v: 4+9 at, 32: (\$2 + 11,12)2 83,! \$2 + y2 31:. ——2:cy 51) 23:2; 3y 3 (\$2 + 3:2)? ’ 8:1: 2 (3:2 +y2)2' Since 15,1), and all partials are continuous at all points except 2 = U, and the Cauchy-Riemann equations hold for all z % U, f is differentiable for all z ¢ 0. m——t(y—1) __ \$2+y2-1v~2\$t 12. ‘ 2 _ - “33H?” m——i(y+1) x2+(y+1)2 ‘30 \$2 + 3,2 — 1 —23: 1L(\$, 3;) ~—- mm,v(£,y) — m. For the Cauchy-Riemann equations, compute 8o 2:1:(32 + (y + 1)?) —— 2:1:(332 + 9'21) m 43:63,: +1) 3:6 [3:2 + (s + 1W _ [c52 + (y + ilglz’ 31; 422(3) + 1) a; _[m2+(y+1)212’ 396 Seotio:121.2 a...“ _ 211962 + (2» +112) ~ 211 + no? +12 — 1) _ W 83} u [-732 ‘1‘ (y '1‘ :02]? [\$2 + (y +1)2]21 5'12 __ ﬂaﬁﬂb 43: I 2e +112 811:— [32+(y+1)2]2 [:22-1(—y—11 1) )22] Since 11,11, and ail parstiai s‘aie continuous at all points except 3— M U y— _ —1 01 3— w —z' and the Cauchy- Riemann equations holds at all such points f is differentiable at all points except at z —— —1'. Section 21.2 Power Series In Problems 1 through 8 we apply the ratio test to determine the open disk and radius of conver— gence. Cn+1(z + 3’0”“ '1. Consider the ratio -— as 111+ 00. For _ |z+3ii (11+ 2) |z+3t| Cn(z + 31‘)n 2 n + 1 2 12 + 3111 . . . . . . convergence we have 2 — < 1 or ]z + 31:! < 2. This IS an Open dlsk of radlos R — 2 centered at z = —31'. 211+? 2 1 ‘2 2. Consider the ratio C1213: 32” 2 ~ iJ2E2::3;2 > ]z 1'12 as 71. mi 00. The series converges on the open disk |z — 1| < 1 which has radius R 2 1 and center z =1. 3. Consider the ratio cn+1(z — 1 + 30”“ cn(z — 2' + i)” l 3. 1 n+1 1 n _fz—( w t)i(1—n+2) (1+5). k n ‘ _ _ _' - n+1 Since 11m (1 + e) = 6", we have M 11-400 n cn(z — 1 + 31,)11 The series converges on the open disk |z w (1 ~ 31')| < 1 of radius R : 1, center at z = 1 — 32'. (n + 1)”+1(n +1)n (n+2)1l+1 nn =12 (1 3211 —1|z-—(1—3'i)|asn—>oo. . . - 4—3—11;n+1 _ 2' 4. Consuier the ratio 6,125+ 3 _ 41;” 2 [z 1 3 + 43'“ 5 :1. 4 I2 — 1 3 + 41'“ _ . . . [6 as n "1 00. The series converges on the open their |z w (—3 + 41” < ______ of radius R— — , center at z = —3 + 41. qHﬁz + 813”“ cn(z -I- 813)" on the open disk |z + 811 -< 2 of radius R = 2, center at z : w83'. . . 2+8t , . . . . 5. Cons1der the ratio 2 I --2 IJ2| Time series 15 geometric and converges ...
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0534552080_98868 - 394 Section 21.1 Section 21.1 Limits...

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