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by: Alverta Blick III


Alverta Blick III
GPA 3.66

J. Dowling

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J. Dowling
Class Notes
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This 21 page Class Notes was uploaded by Alverta Blick III on Tuesday October 13, 2015. The Class Notes belongs to PHYS 4112 at Louisiana State University taught by J. Dowling in Fall. Since its upload, it has received 41 views. For similar materials see /class/222999/phys-4112-louisiana-state-university in Physics 2 at Louisiana State University.




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Date Created: 10/13/15
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1quot3 T33 3 A 1965 M47704 24m Raw1 773w 502 A l 5 AWl s ynnemlc lF 2 ED 73439 6 3 Ta TB 0 4 3 39e Trz an Ta we 0 e T13 TL 0 V 1 Q C INCE X quotX gt X 31 T g4 TM 6 aco ANY 2ND KANA quotTE950 C le 4LW9XS 135 oRII i39EY 48 a 775 60quot 0F 4 Sri maagT fc 4D 41715me 3792 TEN 50R 1 1 T4 a 6 Tm La 395 Wm LN 1 T 1 Tquot 4 41 H39 4 9 T wobboizoo IHR Fobnxogk hon do Ra wth Emlh Il Napalm uqkkmcxxfhiw n L n T gt H 1 HAD 5 H gta II HM ba fm 11 m nIFHD 4 nix 1 lllllll kkilt ou m m cm IPQDAONQ L00 dowlmxh 2m mt NbOLIUm gt A gth n D 2 ll WOmZNL vgv hxtxm Y n 2 3 1gt TuJOQOO VEOM kk u gt 0 QWJSRQ A yp olv 0le E M47 400 Sam E awK v 15039 W507 MD 219 Tans 75 ream 1 Rm 2 1 Ra t L 1415 1quot Sp 739ng 619 1 1 RJO39GLVQ TENJo S 7 Gilif 5 p T7545 H F ALL QUE L 620077wa JZW39E 5 n V Non25220 vgworzs v THE 4 lt gt 4 Wampum U T V 5 A AOIYZEZO v15 cmfi 4N1 TI15 woos N w Dr s 307 4397 60 FWth u 1 v 7mm 12 5 350M 344 MSOZ f J j 4 1 a V 050 U T V 4quot Uolt T413 3 s 9 J L 50 By DEFINTGN of Mom 76739 7 e 101ch HENC E 7 ms 371 coarsemam 50 Cd MJST P oUE 77er xf 7x79st FriM15 LA E A 775445on 4550107 U4 Til GLUEv 4 4 4550 U Miro V 14 WOFEQ 457 fZ aVf TEA564 f l 5 VA 04 UJ mamz urea Lx Mk V 39 3 ayk V P12010131 sctvzxn v 4 3 2 CH3 3 pRo E xF f L 4 I I I lNuEZJE HEMc l 39 n Q Engithn CARVQ Qgt ou h l a uamp xx 96 me SEQ lhv wb o M qu OQNKK 5 V w LI k muQWxx WNW Q RN m KfsQQ hM MW I MVbu 0 x 040E G 0 i u Hint x7 0 Ix lt2 gt r r 1239 1 Q d 1 i 0 l A 30 Jd I gt31 mt xw kaS 9 5ng w gz 3 The un nonnalized Gaussian is given by gx Exp xquot2 9x e39x2 It looks like thiS P1 t9x x 3 3 The area under the curve is area Integrategx x Infinity Infinity 7T ote this is equal to Euler Gammal2 Gamma l 2 Vzr We can define a normalized Gaussian with area one as Gx gx SqrtPi Gx 72 It looks like thiS 2 11 911 nb P1 tGxx 3 3 73 72 71 1 2 3 The area under the curve is IntegrateGx x Infinity Infinity 1 We can estimate the width as the distance between the inflection points which we find by setting the second derivative to zero SolveDDGx x x 0 x l l H I H J 2 2 The standard normal distribution is defined to have both area and width of one phix Exp xquot2 2 Sqrt2Pi Philx Integratephix x Infinity Infinity 1 SolveDDphix x x 0 x pH 1 x gtl The erf function is erfx Integrate2 Gt t 0 x This is a built in function in Mathematica ca11ed Erfx note that Erfx s antisymmetric or odd with special values Erf oo 1 Erf00 Erfcn1 II he plot is 11 911 nb 3 PlotErfx x 3 3 10 This is just the area under the curve of G from x to x PlotGx x 1 1 PlotRangea o 1 The complementary error function is Erfc with Erfc cn 2 Erfc01 Erfcm0 i 4 11 911 nb PlotErfcx x 3 3 The imaginary error function is Erfi with Erfi oo oo Erfi00 Erficnoo and the plot looks like PlotErfix x 3 3 60 As in the textbook we can expand Erfx in a Taylor Series about x0 keeping the first 10 terms SeriesErfx x 0 10 2 x 2 x3 x5 x7 x9 F 3W 5F 21W 108V This is the approximation for xltltl ox11 The asymptotic series in the book can be considered a Taylor series about on as SeriesErfx x Infinity 10 2 1 1 3 15 105 1 11 O x x 23 45x5 85x7 169 This is Boas EqlO4 with some rearrangement of terms Use this series for xgtgtl 1e39K Compare to Taylor seriex of ArcTanhx about zerouc 11 911 nb 5 PlotArc39I anx x 5 5 SeriesArcTanx x 0 10 x5 X7 Kg 11 x Ox 3 5 7 9 There is an asymptotic series for this also about on that is SeriesArcTanx x Infinity 10 7r 1 l l l l l 11 O 2 X 3 x3 5 x5 7 x7 9 x9 X Finally we can apply this to the Euler Gama to get FullSimplifySeriesGammap l p Infinity 3 n n n I I 139113 1 72 O lp P 5 144 W2 25920 W2 emwmwwm m With some simplification this also reduces to Boas Eqll5 which are the first few terms of the asymptotic series expansion for S erling39s Formula We can pick out just the first term and simplify it FullSimplifyExp l Logl p p Sqrt2 Pip p gt 0 69 p9 l 2 7r With a bit of algebra this is equal to pp e399 2 that is 39ust E lll or Sterlin Is Formula 7 P J 1 9


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