7 - POW Series W $0WL \{H +1750. A. PM W - —‘9 (17,00...

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Unformatted text preview: POW Series W $0WL \{H +1750. A. PM W - —‘9 (17,00 =\ Pblynom/Caio OLLLX.\: 0 mix) 1—0 —For (AM/>40 aooatl fl W02 x090 W V.) i an Lx— 03“ “9‘0 fiCom \bofl {A’ dmmx Am; WSW—5 CAMCA 7/27/2010 '-'- SCAXH 2c». A” (NV/\CMB CM: 0 L/ ~20“ C 3 — My (thn-H} ’ZCD ’1. :0 C, '2. —- -" C V‘ L ’L-\ 9.1. ° ’1 .. :lc h=\ : —3:_L C\ B“ \ - - n. M CW: 10L zixcbzfl—‘w W'E a" W, _, _ “u, -7. -7. _ "\ V\. '5 C5 — _—.3 __ ’— "' \ - \ CA -L.\ 6'“ 3‘! so I r. \ \ sypwaq he w m ‘05 \eoWuxs 0d“ 0“ 4’ M" vavxm=7zlb Odinz-‘LL‘A .91». 0.; C° “L [OD C\ 01, ‘— 3.7.“ -, :7: i w \ c3 "— 3‘. C5 0‘4 _ Uc‘ 0" 1rd. " C’s —._ i0 (4, , J; 5‘. ‘ ' “Ch [:3 6" C7 —. :16 OS, _, 1%“ ‘1‘. \ (- L K (:73 _ €13 “— ——’ 0 Cu; " Cb om" Cuuh‘. \ Lecture 7: Math 2860 Summer 2010 1 Sottdwbn is 2 on x“ mu m do (DA-0L) Z: ’LV. + 2 C 'Hun 14—45 (ii-d \c—z‘o 22x ‘ 3/ 0’2 EL (3 7’“ a (5mm c, 747'“ \L—J 0V3" 0 x Ar \LT—D (Wm \ 0° L a: \L 5‘“ [1% < (:3. Wk” 2 \ X ’7 2 Co ( bb (LVN. C4 tam/UN.“ W W YL Who are these guys? L’Ion ‘Z'L Ema gUnouXM Lecture 7: Math 2860 Summer 2010 A Special Equation 1 n X\/ A Special Type of Function 00 (:le z. ink)? 3)) (xx :- QB W‘. VCL—H’Mfl W m GWA We?“ \ JG W man: rm: 36 We; [7% MN, \$‘\‘ \(MA. 045 mm 9 F GA"— W F‘ C UNA: XV“) Lecture 7: Math 2860 Summer 2010 Lecture 7: Math 2860 Summer 2010 4 Theorem 1 The general solution to Bessel’s equation with v {a Z is: Theorem 2 The general solution to Bessel’s equation with any v is: 15" y: ('4 3-)) (>6 + (.7, Y, ()0 nil/W Y9 [>33 : (,0sz Uny) » 3-9 L50 Lecture 7: Math 2860 Summer 2010 5 X \/ Examples I L ’2. 1 ’- Solve: \< Y + X)! + (x vfy I O yaig E (Maggy—.3 35M6, (mkfiWB _' $0\ui\mr\.\s \/3 c,\ 3300 'k-C.» YBCX) I I ’L Li,“ Solve: X17] 4' “7 + K ‘ M3W 3° bk 2 a Z _ 37 5 ‘56 «Wk $0“! ‘} 7 C» jaifl + L1, SJ-LLXX wa g s ‘9 “M, W X0 (MM/Wm) U»an “m1 ‘gfimwm) Lecture 7: Math 2860 Summer 2010 0L; Using Bessel Functions (ii-iv C2?" +xyl + a2x2 _v2 _ 0 dhx obesm- 01w}; M OILXZ" LAX)?“ M m LLZJX a x: : Km: $3" 0&1 o\ 4"“ Mm)‘ my} a): (1* a d1; all do in 'CDr ‘ \R'DE M1 fiuw ’ (XML 738 \I d‘“ _ 2L om fi 1 m A; Xi d dx ax ‘ Amkdvh my ~ 1 oh dw Ethel at \\ 7' 7‘ swam 4w ) fi’ b 04 ’25 \nbEj 2. X1“ (id—EC" _+ x é-g‘x +— (COLXLH 'b\\/ 2 O SWL’rdh’mh “L ,L M M l W ‘ + (QCJWL 1\+ (M’\\}«D 'L “m fl» + uéfiu 4— UHQ’N ,0 OWLX' IMHO massa, \ \ 1' \i / A Mi XL)!“ +><y + City”) 9 nyH +xyr _(x2 +v2)Y: 0 7.. l\ J L 1; a n. m x +>< + (—x~ 5 ,0 Vow u ci‘x 7 iY x73?»— ‘M 7' M: bx x C U A g A M i a di— ixL m Axv d d“, mx Ok-X (M4 ' dw ' J04 a i (C a}: - ' dfl / Au. dim J b vdm A 0‘,— « i J dim” x17“ *XY) ’V (41" “571:0 (11' _ a _M (’ fiv + L” 31“ + Lula fly I d: (h V» gay 1‘ \A 3:11 k “llvmyfl Jo W \, Uan \ gobdhm \/1 04 39(UA A'way (Va 1 C\39 (50” + LL \<>)(.b)<\ 00mp\o¥ 'r‘x M fl — T? l (\—_L H I); DA 3 9 Jy€b><\ \LVLX): 1 9 x 9" Sm (WIT) w) m mow W mm L SDWJflm WM \) ’5 Cl :3) (XB’X’CJQJ A Lecture 7: Math 2860 Summer 2010 \ H‘nO'H'W/K . 7: 1.07.. 7' [—LA 1 +1 (bLbeZo—l + 0L PL ) 3 O \l H '1'" x Y X \Mo gokmm \f: x“ [a JPCbx‘A + (.1, Yr, (5%)] Examples Solve: XLYH +—Xy\ 4r (30%“7‘TA770 I! j q Solve: Y‘] 4" ‘1’ X\/ I O n l \ A O —\’ \ , v x 7 + / 1 n “Lab—- 9 'L '1- 7-6-1 4—61" m: J?) L.) 0 X xv _’ S 9-01-7\ Zo—Lao a" 91.01,: 7/097’ \oL’\ 0”\ q '1’ o J _' e 3‘ \o —, M I: / FL? 4 ‘01 \ {av/‘27:: S0\h ‘_ \)1 X'3 LCA va?) I Lecture 7: Math 2860 Summer 2010 Laplace” © Improper Integrals S git d36 '-'- 0 Lecture 7: Math 2860 Summer 2010 1O W84” EL WM 00 t m Loam) Definition «5 _ is We 5&4:us F53 ‘ éa “L0 W “LAPMLO kwms’gym -Hms w\\\ Ra» 4v—«dhow Ki W3” Bis an H— ems |_ Examples £{1} “AA—XL (W!) “CW “3620) °° J: was FEW 5 E 65 '1 0”“ \9 _ hm S 54’: Ark ’ \o—‘bqa b l ~98 ‘9 "a Lim (PI 5 6 b 3% —s\o L 0 : \\m E“: e” + 56 ‘3 in,» W (Km 2’; , 67D Lecture 7: Math 2860 Summer 2010 11 — —- £6, uc'b ON‘ve dw'k. ._ 5 _. £{t} 3 $5 amid-l: \f: m: 5"“: '0 _Jc : 1mm S ’b5 ‘5 avg She/"skit M33260 6 4: sis ‘5 : gudv hm “35$ ‘f‘5 L " W “3”” \ _$JG .L‘Sb ‘03250 2 ’EJce _. “8&de \’ J. _‘ L ~+. \_ L..-\: —, hm [- (0-3] J, w: s .8}. 3 L —st 1, _ 20 L 2‘s“ “515ch \L 4'an70 5" ,L b 0-0 ; ‘ _\. - “S \I‘M gs : “5 tom“ Saks :J5<O\ID hem a” \ ,L 2 % 570 m s» Li?an Lecture 7: Math 2860 Summer 2010 12 Lecture 7: Math 2860 Summer 2010 13 L is a linear transform (or tSPV/mi‘O“3 iifiwflé : écdwchflxdk 3 0% 0—654: OUC + a§ 64% 3 193 Jr J35; PM» flora: 0;;ng L{3+5ekt} 3 H33 4- 31250th 3 3m Jr s M"; _; 13694’5—5121" Zs S’K Lecture 7: Math 2860 Summer 2010 14 l/ 491;? When does it exist? M'nw M [03603 L/___—_.J Q0 holes I gaff) in ohmic: JW5 50 pie;me % kg \W Mods W—EVT swan M mom i MN» lanai MCS’S 4M s>c. Lecture 7: Math 2860 Summer 2010 15 WNW/u, we» where f(t) = {0 0 S t < 9 4 6315 00 at iflw} :. g 6'“ 40a 0ch 0 Lecture 7: Math 2860 Summer 2010 16 ...
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7 - POW Series W $0WL \{H +1750. A. PM W - —‘9 (17,00...

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