(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 81337, 3083]*) (*NotebookOutlinePosition[ 82030, 3108]*) (* CellTagsIndexPosition[ 81986, 3104]*) (*WindowFrame->Normal*) Notebook[{ Cell["Dirac Delta function", "Title"], Cell["\<\ In mathematica it is easy to define a function, lets consider the statement\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fun[x_] = x^2\)], "Input"], Cell[BoxData[ \(TraditionalForm\`x\^2\)], "Output"] }, Open ]], Cell["\<\ This statement defines a mathematica function called fun, with a dummy argument x. It is important to put the underbar on x, it means that \ x_ is a dummy argument on the lhs. \ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \("\"; \nfun[x]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`x\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fun[z]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`z\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(fun[thisisadummyvariable]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`thisisadummyvariable\^2\)], "Output"] }, Open ]], Cell["You get the idea.", "Text", Evaluatable->False], Cell["\<\ We can now plot this function (look under help bar for all the \ plotting routines available).\ \>", "Text", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[fun[x], {x, \(-1\), 1}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.476191 0.0147151 0.588604 [ [.02381 .00222 -6 -9 ] [.02381 .00222 6 0 ] [.2619 .00222 -12 -9 ] [.2619 .00222 12 0 ] [.7381 .00222 -9 -9 ] [.7381 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.4875 .13244 -18 -4.5 ] [.4875 .13244 0 4.5 ] [.4875 .25016 -18 -4.5 ] [.4875 .25016 0 4.5 ] [.4875 .36788 -18 -4.5 ] [.4875 .36788 0 4.5 ] [.4875 .4856 -18 -4.5 ] [.4875 .4856 0 4.5 ] [.4875 .60332 -6 -4.5 ] [.4875 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .01472 m .02381 .02097 L s [(-1)] .02381 .00222 0 1 Mshowa .2619 .01472 m .2619 .02097 L s [(-0.5)] .2619 .00222 0 1 Mshowa .7381 .01472 m .7381 .02097 L s [(0.5)] .7381 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .21429 .01472 m .21429 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .40476 .01472 m .40476 .01847 L s .45238 .01472 m .45238 .01847 L s .54762 .01472 m .54762 .01847 L s .59524 .01472 m .59524 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .78571 .01472 m .78571 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .5 .13244 m .50625 .13244 L s [(0.2)] .4875 .13244 1 0 Mshowa .5 .25016 m .50625 .25016 L s [(0.4)] .4875 .25016 1 0 Mshowa .5 .36788 m .50625 .36788 L s [(0.6)] .4875 .36788 1 0 Mshowa .5 .4856 m .50625 .4856 L s [(0.8)] .4875 .4856 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(1)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .04415 m .50375 .04415 L s .5 .07358 m .50375 .07358 L s .5 .10301 m .50375 .10301 L s .5 .16187 m .50375 .16187 L s .5 .1913 m .50375 .1913 L s .5 .22073 m .50375 .22073 L s .5 .27959 m .50375 .27959 L s .5 .30902 m .50375 .30902 L s .5 .33845 m .50375 .33845 L s .5 .39731 m .50375 .39731 L s .5 .42674 m .50375 .42674 L s .5 .45617 m .50375 .45617 L s .5 .51503 m .50375 .51503 L s .5 .54446 m .50375 .54446 L s .5 .57389 m .50375 .57389 L s .25 Mabswid .5 0 m .5 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .60332 m .06244 .51168 L .10458 .42058 L .14415 .34341 L .18221 .27686 L .22272 .21429 L .26171 .16211 L .30316 .11529 L .34309 .07863 L .3815 .05117 L .40095 .04018 L .42237 .03036 L .44268 .02324 L .45178 .02075 L .46172 .01852 L .4671 .01752 L .4721 .01674 L .47727 .01606 L .48196 .01556 L .48658 .01518 L .4887 .01505 L .49093 .01493 L .49332 .01483 L .49438 .0148 L .49552 .01477 L .49675 .01474 L .49789 .01473 L .49859 .01472 L .49925 .01472 L .50049 .01472 L .50163 .01472 L .50286 .01474 L .50401 .01476 L .50508 .01478 L .50754 .01486 L .51014 .01498 L .51268 .01513 L .51504 .0153 L .5204 .0158 L .5293 .01694 L .53882 .01863 L .54906 .02096 L .56016 .02411 L .58032 .03146 L .60019 .04077 L .62123 .05287 L .65912 .08044 L .69946 .11799 L .73829 .1621 L .77956 .21759 L Mistroke .81932 .2794 L .85757 .34659 L .89827 .42644 L .93745 .51144 L .97619 .60332 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg>75aaXhhSSR>SZZZZTaaWYVCMe9?@=;@`jJKHmWD_5UD=BLW9g=c Lh20P4aSYVIVJ>SXjf][KNg]l71`Lc[Zj_Cdm6UTBO[`Dhb6HYT0I_ooc020P800082006IV c3c/C4a;Zj^[6a/JRX Z:2PX89mIN7Oeda9>eJ0Zk75fBa1EjFOOgl07/K3^4=1=:NUWRHU7WmoOg=^FLc9^bDT7B]0EADO:/gI iZ_0eD1PP:VXX3dk<861PL6L`H:2PWEeM@00?`00_`0o000o?`0oO`0o_`0oo`1o001o?`1oO`1o_`1o o`2o002o?`2oO`2o_`2oo`3o?`3oO`3o_cl003l0?cl0Ocl0_cl0oclo03lo?cloOclo_cloocmo03mo ?cmoOcmo_cmoocno03no?cnoOcno_cnoocoo03oo?cooOcoo_cooogl007l0?gl0Ogl0_gl0oglo07lo ?gloOglo_gloogmo07mo?gmo_gmoogno07no?gnoOgno_gnoogoo07oo?gooOgoo_goookl00;l0?kl0 Okl0_kl0oklo0;lo?kloOklo_klookmo0;mo?kmoOkmo_kmookno0;no?knoOkno_knookoo0;oo?koo Okoo_kooool0?ol0Ool0_olo0?lo?oloOolo_olooomo0?mo?omoOomo_omooono0?no?onoOono_ono oooo?oooOooo_gAdM:NWYl;2`_mVc>SXj79bLYVIc9W<09W<1`D0D@L00`071`1D1`<0 3PL505<70`0F1`00e`L00`071`3F1`00e`L00`071`3F1`00e`L00`071`3F1`00e`L00`071`3F1`00 2`Oo09X030L001@700<01`L04@L00`071`0@1`0300L7014700<01`L040L00`071`0A1`0300L70107 00<01`L04@L00`071`0@1`0300L7014700<01`L00PL800H700<01`L010L700D700<01`L04@L00`07 1`0@1`0300L7014700<01`L040L00`071`0A1`0300L7010700<01`L04@L00`071`0@1`0300L70147 00<01`L04`L001@700<01`L04@L00`071`0@1`0300L7014700<01`L040L00`071`0A1`0300L70107 00<01`L04@L00`071`0@1`0300L701471@0>1`0300L700/710000`L01`0B1`0300L7010700<01`L0 4@L00`071`0@1`0300L7014700<01`L040L00`071`0A1`0300L7010700<01`L04@L00`071`0C1`00 50L00`071`1O1`0300L704T70`0B1`0300L700l71@1:1`0300L705l700<01`L04`L00;l70`0E1`03 00L701@7102n1`00_0L301P700<01`L060L20;`7002i1`<06`L00`071`0J1`<0^@L00;H70`0N1`03 00L701d70P2g1`00]0L2024700<01`L07`L20;D7002b1`808`L00`071`0Q1`80/`L00;070P0U1`03 00L702<70P2a1`00[PL202L700<01`L09@L20:l7002/1`80:@L302L70P2]1`00ZPL202/700<01`L0 :@L20:/7002X1`80;@L00`071`0[1`80Z@L00:L700<01`L0;@L00`071`0]1`80Y`L00:D70P0`1`03 00L702l700<01`L0Y0L00:@700<01`L0<0L00`071`0`1`80Y0L00:870P0c1`0300L7038700<01`L0 X@L00:4700<01`L0<`L00`071`0c1`0300L70:07002O1`80=PL00`071`0d1`80X0L009h700<01`L0 =PL00`071`0f1`0300L709d7002M1`0300L703L700<01`L0=`L00`071`2L1`00V`L203X700<01`L0 >0L209`7002J1`0300L703X70`0j1`0300L709T7002H1`80?@L00`071`0k1`0300L709P7002G1`03 00L703d700<01`L0?0L209P7002F1`0300L703h700<01`L0?PL00`071`2E1`00U@L00`071`0o1`03 00L703l700<01`L0U0L009@700<01`L0@0L00`071`101`0300L709<7002C1`0300L7044700<01`L0 @@L00`071`2B1`00TPL00`071`121`0300L704870P2B1`00T@L00`071`131`0300L704@700<01`L0 S`L008l70P161`0300L704D700<01`L0SPL008h700<01`L0APL00`071`161`0300L708d7002=1`03 00L704L700<01`L0A`L00`071`2<1`00S0L00`071`181`<0B0L00`071`2;1`00R`L00`071`191`03 00L704T700<01`L0RPL008X700<01`L0BPL00`071`1:1`80RPL008T700<01`L0B`L00`071`1<1`03 00L708L700281`0300L704`700<01`L0C@L00`071`261`00Q`L00`071`1=1`0300L704h700<01`L0 Q@L008H700<01`L0CPL00`071`1?1`0300L708@700251`0300L704l700<01`L0D0L00`071`231`00 Q0L00`071`1@1`0300L7054700<01`L0PPL008<700<01`L0D@L00`071`1A1`0300L7088700231`03 00L703@70`071`8010L600L700<01`L0DPL00`071`211`00PPL00`071`0d1`800PL00`071`041`80 1@L01@071`L000L700<01`L0D`L00`071`201`00P@L00`071`0e1`0300L7008700<01`L02`L00`07 1`081`@0D`L00`071`1o1`00P0L00`071`0f1`0300L7008700<01`L030L00`071`071`0300L705D7 00<01`L0OPL007l700<01`L0=`L00`071`021`0300L700d700<01`L01PL00`071`1F1`0300L707d7 001n1`0300L703P700<01`L00PL00`071`0>1`0300L700D700<01`L0EPL00`071`1m1`00O@L00`07 1`0i1`0300L7008700<01`L03PL00`071`051`0300L705L700<01`L0O0L007`700<01`L0>PL01@07 1`L000`700<01`L00PL00`071`051`0300L705P700<01`L0N`L007/700<01`L0?0L300d71@081`03 00L705T700<01`L0NPL007X700<01`L0FPL00`071`1J1`0300L707T7001j1`0300L705X700<01`L0 F`L00`071`1h1`00N@L00`071`1K1`0300L705/700<01`L0N0L007P700<01`L0G0L00`071`1L1`03 00L707L7001g1`0300L705d700<01`L0G@L00`071`1f1`00MPL00`071`1N1`<0GPL00`071`1e1`00 M@L00`071`1O1`0300L705l700<01`L0M0L007D700<01`L0G`L00`071`1O1`0300L707@7001d1`03 00L7060700<01`L0H0L00`071`1c1`00L`L00`071`1Q1`0300L7064700<01`L0LPL0078700<01`L0 HPL00`071`1R1`0300L70747001b1`0300L7068700<01`L0HPL00`071`1a1`00L@L00`071`1S1`03 00L706<700<01`L0L0L0070700<01`L0I0L00`071`1T1`0300L706l7001_1`0300L706D700<01`L0 I@L00`071`1^1`00K`L00`071`1U1`0300L706D700<01`L0KPL006h700<01`L0IPL00`071`1V1`03 00L706d7001]1`0300L706L70`1W1`0300L706`7001]1`0300L706L700<01`L0J0L00`071`1[1`00 K0L00`071`1X1`0300L706P700<01`L0J`L006/700<01`L0J@L00`071`1Y1`0300L706X7001Z1`03 00L706X700<01`L0JPL00`071`1Y1`00JPL00`071`1Z1`0300L706/700<01`L0J0L006T700<01`L0 J`L00`071`1[1`0300L706P7001X1`0300L706`700<01`L0K0L00`071`1W1`00I`L00`071`1]1`03 00L706d700<01`L0IPL006L700<01`L0K@L00`071`1^1`0300L706D7001V1`0300L706h700<01`L0 KPL00`071`1U1`00I@L00`071`1_1`0300L706l700<01`L0I0L006D700<01`L0K`L3070700<01`L0 H`L006@700<01`L0L0L00`071`1`1`0300L706<7001S1`0300L7074700<01`L0L@L00`071`1R1`00 H`L00`071`1a1`0300L7078700<01`L0H@L0068700<01`L0LPL00`071`1b1`0300L70647001Q1`03 00L707<700<01`L0L`L00`071`1P1`00H@L00`071`1c1`0300L707<700<01`L0H0L0060700<01`L0 M0L00`071`1d1`0300L705l7001O1`0300L707D700<01`L0M@L00`071`1N1`00G`L00`071`1e1`03 00L707D700<01`L0GPL005h700<01`L0F@L300L70P061`D01PL00`071`1f1`0300L705d7001M1`03 00L705T70P021`0300L700@70P081`0300L700H700<01`L0M`L00`071`1L1`00G0L00`071`1J1`03 00L7008700<01`L02@L700H7101f1`0300L705`7001L1`0300L705X700<01`L00PL00`071`091`05 00L71`0020L00`071`1h1`0300L705/7001K1`0300L705/700<01`L00PL00`071`0:1`0400L700P7 00<01`L0N@L00`071`1J1`00FPL00`071`1L1`0300L7008700<01`L02`L00`0700081`0300L707T7 00<01`L0FPL005X700<01`L0G0L00`071`021`0300L700/700<01`0020L00`071`1j1`0300L705T7 001I1`0300L705d700D01`L7000?1`8020L00`071`1k1`0300L705P7001H1`0300L705l70`0A1`03 00L700H700<01`L0N`L00`071`1H1`00F0L00`071`1l1`0300L707`700<01`L0E`L005L700<01`L0 O@L00`071`1l1`0300L705L7001F1`0300L707h700<01`L0O@L00`071`1F1`00EPL00`071`1n1`03 00L707h700<01`L0E@L005D700<01`L0O`L00`071`1n1`0300L705D7001D1`0300L708070`1o1`03 00L705@7001D1`0300L7080700<01`L0P0L00`071`1C1`00D`L00`071`211`0300L7080700<01`L0 D`L005<700<01`L0P@L00`071`211`0300L70587001B1`0300L7088700<01`L0P@L00`071`1B1`00 DPL00`071`221`0300L7088700<01`L0D@L0054700<01`L0P`L00`071`221`0300L70547001A1`03 00L708<700<01`L0P`L00`071`1@1`00D0L00`071`241`0300L708@700<01`L0C`L0050700<01`L0 Q0L00`071`241`0300L704l7001?1`0300L708D700<01`L0Q@L00`071`1>1`00C`L00`071`251`03 00L708D700<01`L0CPL004h700<01`L0QPL308H700<01`L0C@L004h700<01`L0QPL00`071`261`03 00L704d7001=1`0300L708L700<01`L0Q`L00`071`1<1`00C0L00`071`281`0300L708P700<01`L0 B`L004`700<01`L0R0L00`071`281`0300L704/7001;1`0300L708T700<01`L0R@L00`071`1:1`00 B`L00`071`291`0300L708T700<01`L0BPL004X700<01`L0RPL00`071`2:1`0300L704T7001:1`03 00L708X700<01`L0RPL00`071`191`00B@L00`071`2;1`0300L708/700<01`L0B0L004T700<01`L0 R`L00`071`2<1`0300L704L700181`0300L708`700<01`L0S0L00`071`171`00B0L00`071`2<1`03 00L708d700<01`L0APL004L700<01`L0S@L308d700<01`L0APL004L700<01`L0S@L00`071`2>1`03 00L704D700161`0300L708h700<01`L0SPL00`071`151`00A@L00`071`2?1`0300L708l700<01`L0 A0L004D700<01`L0S`L00`071`2@1`0300L704<700141`0300L7090700<01`L0T0L00`071`131`00 A0L00`071`2@1`0300L7094700<01`L0@PL004<700<01`L0T@L00`071`2A1`0300L7048700131`03 00L7094700<01`L0TPL00`071`111`00@PL00`071`1e1`<01`L200H710071`0300L7098700<01`L0 @@L0048700<01`L0M0L2008700<01`L010L200D700@01`L70P061`0300L709<700<01`L0@0L00447 00<01`L0M@L00`071`021`0300L700X700<01`L00PL00`071`041`0300L709<700<01`L0@0L00447 00<01`L0M@L00`071`021`0300L700X700<01`L00PL00`071`041`@0T`L00`071`0o1`00@0L00`07 1`1f1`0300L7008700<01`L02PL01@071`L000L700<01`L0U0L00`071`0o1`00?`L00`071`1g1`03 00L7008700<01`L02PL400P700<01`L0U@L00`071`0n1`00?`L00`071`1g1`0300L7008700<01`L0 2PL00`071`091`0300L709D700<01`L0?PL003h700<01`L0N0L01@071`L000h700<01`L020L00`07 1`2F1`0300L703d7000n1`0300L707T70`0@1`@01PL00`071`2F1`0300L703d7000m1`0300L709L7 00<01`L0U`L00`071`0l1`00?@L00`071`2G1`0300L709L700<01`L0?0L003`700<01`L0V0L00`07 1`2H1`0300L703/7000l1`0300L709P700<01`L0V@L00`071`0j1`00>`L00`071`2I1`0300L709T7 00<01`L0>PL003X700<01`L0VPL00`071`2J1`0300L703T7000j1`0300L709X70`2J1`0300L703T7 000i1`0300L709/700<01`L0V`L00`071`0h1`00>@L00`071`2K1`0300L709/700<01`L0>0L003P7 00<01`L0W0L00`071`2L1`0300L703L7000h1`0300L709`700<01`L0W0L00`071`0g1`00=`L00`07 1`2M1`0300L709d700<01`L0=PL003L700<01`L0W@L00`071`2M1`0300L703H7000f1`0300L709h7 00<01`L0WPL00`071`0e1`00=PL00`071`2N1`0300L709h700<01`L0=@L003D700<01`L0W`L00`07 1`2O1`0300L703@7000e1`0300L709l700<01`L0W`L00`071`0d1`00=0L00`071`2P1`0300L70:07 00<01`L0<`L003@700<01`L0X0L30:0700<01`L0<`L003<700<01`L0X@L00`071`2Q1`0300L70387 000c1`0300L70:4700<01`L0X@L00`071`0b1`00"], ImageRangeCache->{{{0, 431}, {265.875, 0}} -> {-1.1075, -0.0773627, 0.00770883, 0.00623658}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Getting back to our lesson. Lets define the following function\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[x_, epsilon_] = 1/epsilon\ 1/Sqrt[Pi]\ Exp[\(-\((x/epsilon)\)^2\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\[ExponentialE]\^\(-\(x\^2\/epsilon\^2\)\)\/\(epsilon\ \@\[Pi]\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[f[x, 0.1], {x, \(-3\), 3}, PlotRange -> All]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.15873 0.0147151 0.10434 [ [.02381 .00222 -6 -9 ] [.02381 .00222 6 0 ] [.18254 .00222 -6 -9 ] [.18254 .00222 6 0 ] [.34127 .00222 -6 -9 ] [.34127 .00222 6 0 ] [.65873 .00222 -3 -9 ] [.65873 .00222 3 0 ] [.81746 .00222 -3 -9 ] [.81746 .00222 3 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.4875 .11906 -6 -4.5 ] [.4875 .11906 0 4.5 ] [.4875 .2234 -6 -4.5 ] [.4875 .2234 0 4.5 ] [.4875 .32774 -6 -4.5 ] [.4875 .32774 0 4.5 ] [.4875 .43208 -6 -4.5 ] [.4875 .43208 0 4.5 ] [.4875 .53642 -6 -4.5 ] [.4875 .53642 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .01472 m .02381 .02097 L s [(-3)] .02381 .00222 0 1 Mshowa .18254 .01472 m .18254 .02097 L s [(-2)] .18254 .00222 0 1 Mshowa .34127 .01472 m .34127 .02097 L s [(-1)] .34127 .00222 0 1 Mshowa .65873 .01472 m .65873 .02097 L s [(1)] .65873 .00222 0 1 Mshowa .81746 .01472 m .81746 .02097 L s [(2)] .81746 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(3)] .97619 .00222 0 1 Mshowa .125 Mabswid .05556 .01472 m .05556 .01847 L s .0873 .01472 m .0873 .01847 L s .11905 .01472 m .11905 .01847 L s .15079 .01472 m .15079 .01847 L s .21429 .01472 m .21429 .01847 L s .24603 .01472 m .24603 .01847 L s .27778 .01472 m .27778 .01847 L s .30952 .01472 m .30952 .01847 L s .37302 .01472 m .37302 .01847 L s .40476 .01472 m .40476 .01847 L s .43651 .01472 m .43651 .01847 L s .46825 .01472 m .46825 .01847 L s .53175 .01472 m .53175 .01847 L s .56349 .01472 m .56349 .01847 L s .59524 .01472 m .59524 .01847 L s .62698 .01472 m .62698 .01847 L s .69048 .01472 m .69048 .01847 L s .72222 .01472 m .72222 .01847 L s .75397 .01472 m .75397 .01847 L s .78571 .01472 m .78571 .01847 L s .84921 .01472 m .84921 .01847 L s .88095 .01472 m .88095 .01847 L s .9127 .01472 m .9127 .01847 L s .94444 .01472 m .94444 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .5 .11906 m .50625 .11906 L s [(1)] .4875 .11906 1 0 Mshowa .5 .2234 m .50625 .2234 L s [(2)] .4875 .2234 1 0 Mshowa .5 .32774 m .50625 .32774 L s [(3)] .4875 .32774 1 0 Mshowa .5 .43208 m .50625 .43208 L s [(4)] .4875 .43208 1 0 Mshowa .5 .53642 m .50625 .53642 L s [(5)] .4875 .53642 1 0 Mshowa .125 Mabswid .5 .03558 m .50375 .03558 L s .5 .05645 m .50375 .05645 L s .5 .07732 m .50375 .07732 L s .5 .09819 m .50375 .09819 L s .5 .13992 m .50375 .13992 L s .5 .16079 m .50375 .16079 L s .5 .18166 m .50375 .18166 L s .5 .20253 m .50375 .20253 L s .5 .24426 m .50375 .24426 L s .5 .26513 m .50375 .26513 L s .5 .286 m .50375 .286 L s .5 .30687 m .50375 .30687 L s .5 .3486 m .50375 .3486 L s .5 .36947 m .50375 .36947 L s .5 .39034 m .50375 .39034 L s .5 .41121 m .50375 .41121 L s .5 .45294 m .50375 .45294 L s .5 .47381 m .50375 .47381 L s .5 .49468 m .50375 .49468 L s .5 .51555 m .50375 .51555 L s .5 .55728 m .50375 .55728 L s .5 .57815 m .50375 .57815 L s .5 .59902 m .50375 .59902 L s .25 Mabswid .5 0 m .5 .61803 L s .5 Mabswid .02381 .01472 m .04262 .01472 L .0522 .01472 L .05746 .01472 L .06244 .01472 L .06371 .01472 L .06506 .01472 L .06576 .01472 L .06653 .01472 L .06724 .01472 L .0679 .01472 L .06914 .01472 L .07029 .01472 L .0716 .01472 L .07286 .01472 L .07402 .01472 L .07526 .01472 L .07643 .01472 L .07749 .01472 L .07874 .01472 L .07945 .01472 L .0801 .01472 L .08129 .01472 L .08255 .01472 L .08326 .01472 L .08392 .01472 L .08517 .01472 L .08587 .01472 L .08652 .01472 L .08723 .01472 L .08798 .01472 L .08923 .01472 L .09039 .01472 L .09167 .01472 L .09239 .01472 L .09305 .01472 L .09373 .01472 L .09444 .01472 L .09574 .01472 L .09653 .01472 L .09727 .01472 L .09794 .01472 L .09867 .01472 L .09938 .01472 L .10016 .01472 L .10098 .01472 L .10174 .01472 L .10246 .01472 L .10311 .01472 L .10386 .01472 L Mistroke .10458 .01472 L .10583 .01472 L .10653 .01472 L .10718 .01472 L .10837 .01472 L .10962 .01472 L .11075 .01472 L .1118 .01472 L .11296 .01472 L .11419 .01472 L .11551 .01472 L .11626 .01472 L .11694 .01472 L .11821 .01472 L .11884 .01472 L .11955 .01472 L .12076 .01472 L .12206 .01472 L .1233 .01472 L .12442 .01472 L .1255 .01472 L .12665 .01472 L .12773 .01472 L .12871 .01472 L .12987 .01472 L .13112 .01472 L .1323 .01472 L .13339 .01472 L .13452 .01472 L .13558 .01472 L .13674 .01472 L .13798 .01472 L .13928 .01472 L .14051 .01472 L .14162 .01472 L .14282 .01472 L .14404 .01472 L .14515 .01472 L .14637 .01472 L .14766 .01472 L .14879 .01472 L .14983 .01472 L .15098 .01472 L .15221 .01472 L .15291 .01472 L .15355 .01472 L .15425 .01472 L .155 .01472 L .15626 .01472 L .15697 .01472 L Mistroke .15764 .01472 L .15888 .01472 L .16002 .01472 L .16133 .01472 L .16258 .01472 L .16326 .01472 L .1639 .01472 L .1651 .01472 L .16639 .01472 L .16712 .01472 L .16778 .01472 L .16891 .01472 L .17014 .01472 L .17142 .01472 L .17262 .01472 L .1733 .01472 L .17404 .01472 L .17474 .01472 L .17539 .01472 L .17661 .01472 L .17791 .01472 L .17858 .01472 L .1793 .01472 L .1806 .01472 L .18136 .01472 L .18215 .01472 L .18283 .01472 L .18357 .01472 L .1849 .01472 L .18611 .01472 L .18675 .01472 L .18745 .01472 L .18818 .01472 L .18887 .01472 L .19012 .01472 L .19127 .01472 L .19191 .01472 L .1926 .01472 L .19386 .01472 L .19509 .01472 L .19621 .01472 L .1975 .01472 L .19871 .01472 L .19983 .01472 L .20087 .01472 L .202 .01472 L .2032 .01472 L .20451 .01472 L .2057 .01472 L .20636 .01472 L Mistroke .20708 .01472 L .20838 .01472 L .20959 .01472 L .21088 .01472 L .21199 .01472 L .2132 .01472 L .21393 .01472 L .21462 .01472 L .21592 .01472 L .21665 .01472 L .21733 .01472 L .21807 .01472 L .21886 .01472 L .21954 .01472 L .22029 .01472 L .22099 .01472 L .22163 .01472 L .22286 .01472 L .22416 .01472 L .22537 .01472 L .22606 .01472 L .2267 .01472 L .22787 .01472 L .22911 .01472 L .23022 .01472 L .23126 .01472 L .2324 .01472 L .23363 .01472 L .23486 .01472 L .23556 .01472 L .23621 .01472 L .23752 .01472 L .23826 .01472 L .23893 .01472 L .24012 .01472 L .24142 .01472 L .24265 .01472 L .24378 .01472 L .24484 .01472 L .24597 .01472 L .24704 .01472 L .24802 .01472 L .24917 .01472 L .25042 .01472 L .25159 .01472 L .25269 .01472 L .25392 .01472 L .25508 .01472 L .25613 .01472 L .25725 .01472 L Mistroke .25849 .01472 L .25979 .01472 L .2609 .01472 L .26212 .01472 L .26342 .01472 L .26463 .01472 L .26526 .01472 L .26596 .01472 L .26669 .01472 L .26737 .01472 L .26862 .01472 L .26976 .01472 L .27102 .01472 L .27234 .01472 L .27346 .01472 L .27465 .01472 L .27577 .01472 L .27679 .01472 L .27799 .01472 L .27929 .01472 L .28052 .01472 L .28165 .01472 L .28286 .01472 L .28416 .01472 L .28538 .01472 L .2865 .01472 L .28781 .01472 L .28855 .01472 L .28923 .01472 L .29049 .01472 L .29112 .01472 L .29182 .01472 L .2925 .01472 L .29321 .01472 L .29449 .01472 L .2952 .01472 L .29586 .01472 L .29659 .01472 L .29737 .01472 L .29804 .01472 L .29877 .01472 L .29946 .01472 L .3001 .01472 L .3013 .01472 L .30258 .01472 L .30378 .01472 L .3051 .01472 L .30634 .01472 L .3075 .01472 L .30864 .01472 L Mistroke .30967 .01472 L .31087 .01472 L .31199 .01472 L .31323 .01472 L .31453 .01472 L .31564 .01472 L .31686 .01472 L .31817 .01472 L .31891 .01472 L .31958 .01472 L .32088 .01472 L .32212 .01472 L .32317 .01472 L .32433 .01472 L .32554 .01472 L .32667 .01472 L .32773 .01472 L .3287 .01472 L .32977 .01472 L .33089 .01472 L .33203 .01472 L .33325 .01472 L .3344 .01472 L .33544 .01472 L .33667 .01472 L .33736 .01472 L .338 .01472 L .33918 .01472 L .34042 .01472 L .34163 .01472 L .34295 .01472 L .3442 .01472 L .34536 .01472 L .3465 .01472 L .34754 .01472 L .34874 .01472 L .34986 .01472 L .3511 .01472 L .3524 .01472 L .35352 .01472 L .35474 .01472 L .35605 .01472 L .35679 .01472 L .35746 .01472 L .35877 .01472 L .36 .01472 L .36106 .01472 L .36222 .01472 L .36343 .01472 L .36457 .01472 L Mistroke .36563 .01472 L .3666 .01472 L .36772 .01472 L .36878 .01472 L .36992 .01472 L .37114 .01472 L .37229 .01472 L .37334 .01472 L .37457 .01472 L .37526 .01472 L .3759 .01472 L .37708 .01472 L .37832 .01472 L .37953 .01472 L .38022 .01472 L .38086 .01472 L .38203 .01472 L .38327 .01472 L .38437 .01472 L .38541 .01472 L .38655 .01472 L .38778 .01472 L .38901 .01472 L .38971 .01472 L .39036 .01472 L .39167 .01472 L .39241 .01472 L .39308 .01472 L .39428 .01472 L .39558 .01472 L .3968 .01472 L .39794 .01472 L .39899 .01472 L .40012 .01472 L .40119 .01472 L .40217 .01472 L .40332 .01472 L .40456 .01472 L .40574 .01472 L .40684 .01472 L .40807 .01472 L .40923 .01472 L .41027 .01472 L .4114 .01472 L .41263 .01472 L .41393 .01472 L .41505 .01472 L .41626 .01472 L .41757 .01472 L .41878 .01472 L Mistroke .41941 .01472 L .42011 .01472 L .42083 .01472 L .42151 .01472 L .42276 .01472 L .42391 .01472 L .42516 .01472 L .42648 .01472 L .4276 .01472 L .42879 .01472 L .42991 .01472 L .43093 .01472 L .43213 .01472 L .43343 .01472 L .43466 .01472 L .43579 .01472 L .437 .01472 L .4383 .01472 L .43952 .01472 L .44064 .01472 L .44194 .01472 L .44268 .01472 L .44337 .01472 L .44462 .01472 L .44533 .01472 L .44596 .01472 L .44664 .01472 L .44735 .01472 L .44863 .01473 L .44934 .01474 L .45 .01474 L .45073 .01475 L .4515 .01477 L .45218 .01478 L .45291 .0148 L .4536 .01483 L .45423 .01486 L .45544 .01494 L .45672 .01506 L .45792 .01524 L .45923 .01552 L .46048 .01591 L .46163 .01642 L .46277 .01712 L .4638 .01796 L .465 .01927 L .46612 .02091 L .46736 .02329 L .46866 .02664 L .46977 .03036 L Mistroke .47099 .03556 L .4723 .04272 L .47371 .05258 L .47503 .06424 L .47625 .07743 L .4786 .11034 L .48116 .15865 L .48582 .27969 L .49037 .42216 L .49236 .4818 L .49455 .53784 L .49564 .56055 L .4968 .57998 L .49803 .59442 L .49918 .60181 L .49982 .60332 L .50052 .60275 L .50179 .59593 L .50303 .58233 L .50417 .56412 L .50626 .51858 L .50856 .45485 L .51321 .3093 L .51834 .16969 L .52122 .11331 L .52258 .09248 L .52385 .07629 L .52629 .05258 L .52758 .04345 L .52896 .03582 L .53027 .03022 L .5315 .02618 L .53261 .02336 L .53381 .02102 L .53505 .0192 L .53619 .01797 L .5375 .01693 L .53874 .01624 L .53988 .01578 L .54094 .01547 L .5421 .01523 L .54334 .01506 L .54457 .01494 L .54526 .01489 L .5459 .01485 L .54707 .0148 L .5483 .01477 L .54896 .01476 L .54966 .01475 L .55092 .01474 L Mistroke .55169 .01473 L .55242 .01473 L .55307 .01472 L .55378 .01472 L .5551 .01472 L .55632 .01472 L .55703 .01472 L .55767 .01472 L .55841 .01472 L .55911 .01472 L .55979 .01472 L .56052 .01472 L .56181 .01472 L .56253 .01472 L .56321 .01472 L .56395 .01472 L .56475 .01472 L .56598 .01472 L .56667 .01472 L .56732 .01472 L .56863 .01472 L .56936 .01472 L .57002 .01472 L .57122 .01472 L .57251 .01472 L .57373 .01472 L .57486 .01472 L .57611 .01472 L .57729 .01472 L .57836 .01472 L .57952 .01472 L .58078 .01472 L .58148 .01472 L .58212 .01472 L .58336 .01472 L .58453 .01472 L .58564 .01472 L .5868 .01472 L .5878 .01472 L .58889 .01472 L .59006 .01472 L .59132 .01472 L .59251 .01472 L .5936 .01472 L .59487 .01472 L .59552 .01472 L .59623 .01472 L .59749 .01472 L .59867 .01472 L .59935 .01472 L Mistroke .60006 .01472 L .60133 .01472 L .60204 .01472 L .6027 .01472 L .60342 .01472 L .60419 .01472 L .60549 .01472 L .60622 .01472 L .6069 .01472 L .60814 .01472 L .60945 .01472 L .61062 .01472 L .61172 .01472 L .61293 .01472 L .61422 .01472 L .61491 .01472 L .61556 .01472 L .61677 .01472 L .61808 .01472 L .61881 .01472 L .61949 .01472 L .62062 .01472 L .62186 .01472 L .62316 .01472 L .62436 .01472 L .62505 .01472 L .6258 .01472 L .6265 .01472 L .62716 .01472 L .62839 .01472 L .6297 .01472 L .63038 .01472 L .6311 .01472 L .63242 .01472 L .63322 .01472 L .63398 .01472 L .63466 .01472 L .63541 .01472 L .63606 .01472 L .63677 .01472 L .63806 .01472 L .63931 .01472 L .64046 .01472 L .6417 .01472 L .64241 .01472 L .64307 .01472 L .64377 .01472 L .6444 .01472 L .64514 .01472 L .64583 .01472 L Mistroke .647 .01472 L .64824 .01472 L .6493 .01472 L .65047 .01472 L .65172 .01472 L .65243 .01472 L .65307 .01472 L .65426 .01472 L .65552 .01472 L .65675 .01472 L .65744 .01472 L .65809 .01472 L .65926 .01472 L .66051 .01472 L .66163 .01472 L .66267 .01472 L .66382 .01472 L .66505 .01472 L .66636 .01472 L .6671 .01472 L .66778 .01472 L .66904 .01472 L .66974 .01472 L .67038 .01472 L .67158 .01472 L .67288 .01472 L .67411 .01472 L .67524 .01472 L .67631 .01472 L .67745 .01472 L .67853 .01472 L .67951 .01472 L .68066 .01472 L .68191 .01472 L .68309 .01472 L .68418 .01472 L .6853 .01472 L .68635 .01472 L .68751 .01472 L .68875 .01472 L .68999 .01472 L .69129 .01472 L .6924 .01472 L .69361 .01472 L .69425 .01472 L .69493 .01472 L .69615 .01472 L .69746 .01472 L .69821 .01472 L .6989 .01472 L Mistroke .70016 .01472 L .70131 .01472 L .70195 .01472 L .70264 .01472 L .7039 .01472 L .70502 .01472 L .70625 .01472 L .70754 .01472 L .70875 .01472 L .70988 .01472 L .71091 .01472 L .71205 .01472 L .71324 .01472 L .71455 .01472 L .71575 .01472 L .71641 .01472 L .71712 .01472 L .71842 .01472 L .71964 .01472 L .72093 .01472 L .72203 .01472 L .72325 .01472 L .72398 .01472 L .72467 .01472 L .72597 .01472 L .7267 .01472 L .72737 .01472 L .72812 .01472 L .7289 .01472 L .72959 .01472 L .73034 .01472 L .73104 .01472 L .73168 .01472 L .73291 .01472 L .7342 .01472 L .73542 .01472 L .73611 .01472 L .73675 .01472 L .73792 .01472 L .73916 .01472 L .74027 .01472 L .74131 .01472 L .74245 .01472 L .74368 .01472 L .74492 .01472 L .74562 .01472 L .74627 .01472 L .74758 .01472 L .74832 .01472 L .74898 .01472 L Mistroke .75018 .01472 L .75148 .01472 L .75271 .01472 L .75384 .01472 L .7549 .01472 L .75604 .01472 L .75711 .01472 L .75809 .01472 L .75923 .01472 L .76048 .01472 L .76166 .01472 L .76275 .01472 L .76399 .01472 L .76515 .01472 L .76619 .01472 L .76732 .01472 L .76855 .01472 L .76985 .01472 L .77096 .01472 L .77218 .01472 L .77349 .01472 L .7747 .01472 L .77533 .01472 L .77603 .01472 L .77676 .01472 L .77744 .01472 L .77869 .01472 L .77983 .01472 L .78109 .01472 L .78241 .01472 L .78353 .01472 L .78472 .01472 L .78584 .01472 L .78686 .01472 L .78807 .01472 L .78937 .01472 L .7906 .01472 L .79173 .01472 L .79294 .01472 L .79424 .01472 L .79546 .01472 L .79658 .01472 L .79789 .01472 L .79863 .01472 L .79931 .01472 L .80057 .01472 L .8012 .01472 L .8019 .01472 L .80258 .01472 L .80329 .01472 L Mistroke .80457 .01472 L .80528 .01472 L .80594 .01472 L .80667 .01472 L .80745 .01472 L .80812 .01472 L .80885 .01472 L .80954 .01472 L .81018 .01472 L .81138 .01472 L .81266 .01472 L .81387 .01472 L .81518 .01472 L .81643 .01472 L .81758 .01472 L .81872 .01472 L .81976 .01472 L .82095 .01472 L .82208 .01472 L .82331 .01472 L .82462 .01472 L .82573 .01472 L .82695 .01472 L .82826 .01472 L .829 .01472 L .82967 .01472 L .83097 .01472 L .83221 .01472 L .83326 .01472 L .83442 .01472 L .83563 .01472 L .83677 .01472 L .83783 .01472 L .8388 .01472 L .83986 .01472 L .84098 .01472 L .84212 .01472 L .84334 .01472 L .84449 .01472 L .84554 .01472 L .84677 .01472 L .84746 .01472 L .8481 .01472 L .84927 .01472 L .85052 .01472 L .85173 .01472 L .85305 .01472 L .8543 .01472 L .85545 .01472 L .8566 .01472 L Mistroke .85764 .01472 L .85884 .01472 L .85996 .01472 L .8612 .01472 L .86251 .01472 L .86362 .01472 L .86484 .01472 L .86616 .01472 L .8669 .01472 L .86757 .01472 L .86887 .01472 L .87011 .01472 L .87116 .01472 L .87232 .01472 L .87354 .01472 L .87467 .01472 L .87573 .01472 L .8767 .01472 L .87783 .01472 L .87889 .01472 L .88003 .01472 L .88125 .01472 L .88241 .01472 L .88345 .01472 L .88468 .01472 L .88537 .01472 L .88601 .01472 L .88719 .01472 L .88843 .01472 L .88964 .01472 L .89033 .01472 L .89097 .01472 L .89214 .01472 L .89338 .01472 L .89449 .01472 L .89553 .01472 L .89667 .01472 L .8979 .01472 L .89913 .01472 L .89983 .01472 L .90048 .01472 L .90179 .01472 L .90253 .01472 L .9032 .01472 L .9044 .01472 L .9057 .01472 L .90692 .01472 L .90806 .01472 L .90911 .01472 L .91025 .01472 L Mistroke .91132 .01472 L .9123 .01472 L .91344 .01472 L .91469 .01472 L .91587 .01472 L .91696 .01472 L .91819 .01472 L .91936 .01472 L .9204 .01472 L .92153 .01472 L .92276 .01472 L .92406 .01472 L .92517 .01472 L .92639 .01472 L .9277 .01472 L .92891 .01472 L .92954 .01472 L .93023 .01472 L .93096 .01472 L .93164 .01472 L .93289 .01472 L .93404 .01472 L .93529 .01472 L .93662 .01472 L .94593 .01472 L .96686 .01472 L .97619 .01472 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg>75aaXhhSSR>SZZZZTaaWYVCMe9?@=;@`jJKHmWD_5UD=BLW9g=c Lh20P4aSYVIVJ>SXjf][KNg]l71`Lc[Zj_Cdm6UTBO[`Dhb6HYT0I_ooc020P800082006IV c3c/C4a;Zj^[6a/JRX Z:2PX89mIN7Oeda9>eJ0Zk75fBa1EjFOOgl07/K3^4=1=:NUWRHU7WmoOg=^FLc9^bDT7B]0EADO:/gI iZ_0eD1PP:VXX3dk<861PL6L`H:2PWEeM@00?`00_`0o000o?`0oO`0o_`0oo`1o001o?`1oO`1o_`1o o`2o002o?`2oO`2o_`2oo`3o?`3oO`3o_cl003l0?cl0Ocl0_cl0oclo03lo?cloOclo_cloocmo03mo ?cmoOcmo_cmoocno03no?cnoOcno_cnoocoo03oo?cooOcoo_cooogl007l0?gl0Ogl0_gl0oglo07lo ?gloOglo_gloogmo07mo?gmo_gmoogno07no?gnoOgno_gnoogoo07oo?gooOgoo_goookl00;l0?kl0 Okl0_kl0oklo0;lo?kloOklo_klookmo0;mo?kmoOkmo_kmookno0;no?knoOkno_knookoo0;oo?koo Okoo_kooool0?ol0Ool0_olo0?lo?oloOolo_olooomo0?mo?omoOomo_omooono0?no?onoOono_ono oooo?oooOooo_gAdM:NWYl;2`_mVc>SXj79bLYVIc9W<09W<`L00`071`0o1`0300L707/700<01`L0 ?@L00`071`111`0300L70147000J1`0300L703`700<01`L0?PL00`071`1k1`0300L703h700<01`L0 @0L00`071`0A1`0030L700H70P0b1`L01PL00`071`0a1`L01@L00`071`1k1`0300L703l700<01`L0 ?PL201<7000H1`80@@L00`071`0l1`0300L707/700<01`L0@0L00`071`0l1`8050L001X700<01`L0 ?PL00`071`0l1`0300L707/700<01`L0@0L00`071`0n1`0300L70147000J1`0300L703T700<01`L0 0PL00`071`0l1`0300L703X700<01`L0?PL00`071`0k1`0300L7008700<01`L0?PL00`071`0A1`00 5@L503`71@0m1`<0?0L00`071`0l1`<0?@L503`71@0D1`00e`L00`071`3F1`00e`L00`071`3F1`00 e`L00`071`3F1`00e`L00`071`3F1`002POo09`02`L001@700<01`L02PL00`071`0:1`0300L700X7 00<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`07 1`0:1`0300L700X700<01`L02PL00`071`061`<000<700L02`L00`071`0:1`0300L0008020L00`07 1`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X7 00<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`071`0:1`0300L701<7000D1`0300L700X7 00<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`07 1`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X700<01`L020L300`700<01`L02PL300X7 00<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`07 1`0:1`0300L700X700<01`L02PL00`071`0:1`0300L700X700<01`L02PL00`071`0C1`0050L00`07 1`0n1`0300L703h700<01`L0<0L00`071`0;1`0300L700/700<01`L0<0L00`071`0n1`0300L703h7 00<01`L04`L001`0300L700H7 00<01`L01PL00`071`3=1`00cPL00`071`061`0300L700H700<01`L0c@L001`0300L700H700<01`L01PL00`071`3=1`00cPL00`071`061`0300L700H7 00<01`L0c@L001`00c`L00`071`051`0300L700D700<01`L0cPL001`00c`L00`07 1`051`0300L700D700<01`L0cPL001`00c`L00`071`051`<01@L00`071`3>1`00c`L00`071`051`0300L700D7 00<01`L0cPL001`00c`L00`071`051`0300L700D700<01`L0cPL001`00c`L00`071`051`0300L700D700<01`L0cPL001`00c`L00`071`051`0300L700D700<01`L0cPL00=0700<01`L010L00`07 1`051`0300L701`00d0L00`071`041`0300L700@7 00<01`L0c`L00=0700<01`L010L00`071`041`0300L701`0300L000H700<01`L010L00`07 1`3?1`00c`L200H700<01`L010L00`071`3?1`00c`L200H700<01`L010L00`071`3?1`00bPL00`07 1`021`801PL00`071`041`0300L70"], ImageRangeCache->{{{0, 431}, {265.875, 0}} -> {-3.31278, -0.435493, 0.0230588, 0.0350788}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Lets Integrate this function (numerically) for different values of \ epsilon. for epsilon=0.2 we get\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[f[x, 0.2], {x, \(-3\), 3}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1.00000000000006661`\)], "Output"] }, Open ]], Cell[TextData[{ "How about this integral ", Cell[BoxData[ \(TraditionalForm \`\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\(\( g(x)\)\ \(f(x, epsilon)\)\) \[DifferentialD]x\)]], ", ", "for different functions g[x], if f is a dirac delta this\nintegral should \ approach the value g[x=0]" }], "Text"], Cell["lets choose ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(g[x_] = 2.3\ Cos[x]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox["2.29999999999999982`", StyleBoxAutoDelete->True, PrintPrecision->2], " ", \(cos(x)\)}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[g[x]\ f[x, 1], {x, \(-3\), 3}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1.79129207291287091`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ NIntegrate[g[x]\ f[x, epsilon], {x, \(-3\), 3}], {epsilon, 1, 0.1, \(-0.1\)}]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`{1.79129207291287091`, 1.87838445427546378`, 1.95993097473595927`, 2.03482358446744715`, 2.10204172612787543`, 2.16065004447039132`, 2.2098157100507465`, 2.24882784554478476`, 2.27711461762330991`, 2.29425718151432933`}\)], "Output"] }, Open ]], Cell["(This is a table of integrals)", "Text"], Cell["Note the integral appears to converge to the value g(0)", "Text"], Cell["So we have shown that", "Text"], Cell[BoxData[ \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity] g \((x)\) \(\[ExponentialE]\^\(-\(x\^2\/\[Epsilon]\^2\)\)\/\(\[Epsilon] \@ \[Pi]\)\) \[DifferentialD]x = g \((0)\)\ \ \ as\ \[Epsilon] -> 0, \ or\ \[ExponentialE]\^\(-\(x\^2\/\[Epsilon]\^2\)\)\/\(\[Epsilon] \@ \[Pi]\)\ behaves\ like\ \[Delta] \((x)\)\ when\ \ \[Epsilon] -> 0\)], "Input", Background->RGBColor[0, 1, 0]], Cell["This representation is not unique, lets define", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f2[x_, epsilon_] = \ 1/Pi\ Sin[x/epsilon]/x\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(sin(x\/epsilon)\)\/\(\[Pi]\ x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[f2[x, 0.1], {x, \(-10\), 10}, PlotRange -> All]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0476191 0.119752 0.152037 [ [.02381 .10725 -9 -9 ] [.02381 .10725 9 0 ] [.2619 .10725 -6 -9 ] [.2619 .10725 6 0 ] [.7381 .10725 -3 -9 ] [.7381 .10725 3 0 ] [.97619 .10725 -6 -9 ] [.97619 .10725 6 0 ] [.4875 .27179 -6 -4.5 ] [.4875 .27179 0 4.5 ] [.4875 .42383 -6 -4.5 ] [.4875 .42383 0 4.5 ] [.4875 .57586 -6 -4.5 ] [.4875 .57586 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .11975 m .02381 .126 L s [(-10)] .02381 .10725 0 1 Mshowa .2619 .11975 m .2619 .126 L s [(-5)] .2619 .10725 0 1 Mshowa .7381 .11975 m .7381 .126 L s [(5)] .7381 .10725 0 1 Mshowa .97619 .11975 m .97619 .126 L s [(10)] .97619 .10725 0 1 Mshowa .125 Mabswid .07143 .11975 m .07143 .1235 L s .11905 .11975 m .11905 .1235 L s .16667 .11975 m .16667 .1235 L s .21429 .11975 m .21429 .1235 L s .30952 .11975 m .30952 .1235 L s .35714 .11975 m .35714 .1235 L s .40476 .11975 m .40476 .1235 L s .45238 .11975 m .45238 .1235 L s .54762 .11975 m .54762 .1235 L s .59524 .11975 m .59524 .1235 L s .64286 .11975 m .64286 .1235 L s .69048 .11975 m .69048 .1235 L s .78571 .11975 m .78571 .1235 L s .83333 .11975 m .83333 .1235 L s .88095 .11975 m .88095 .1235 L s .92857 .11975 m .92857 .1235 L s .25 Mabswid 0 .11975 m 1 .11975 L s .5 .27179 m .50625 .27179 L s [(1)] .4875 .27179 1 0 Mshowa .5 .42383 m .50625 .42383 L s [(2)] .4875 .42383 1 0 Mshowa .5 .57586 m .50625 .57586 L s [(3)] .4875 .57586 1 0 Mshowa .125 Mabswid .5 .15016 m .50375 .15016 L s .5 .18057 m .50375 .18057 L s .5 .21097 m .50375 .21097 L s .5 .24138 m .50375 .24138 L s .5 .3022 m .50375 .3022 L s .5 .3326 m .50375 .3326 L s .5 .36301 m .50375 .36301 L s .5 .39342 m .50375 .39342 L s .5 .45423 m .50375 .45423 L s .5 .48464 m .50375 .48464 L s .5 .51505 m .50375 .51505 L s .5 .54546 m .50375 .54546 L s .5 .08934 m .50375 .08934 L s .5 .05894 m .50375 .05894 L s .5 .02853 m .50375 .02853 L s .5 .60627 m .50375 .60627 L s .25 Mabswid .5 0 m .5 .61803 L s .5 Mabswid .02381 .1173 m .02499 .11635 L .02605 .11565 L .02729 .11511 L .02846 .11487 L .02954 .11492 L .03053 .11518 L .03163 .1157 L .03279 .11648 L .03523 .11871 L .03786 .12142 L .03913 .12261 L .04035 .12356 L .04143 .12421 L .04262 .12466 L .04386 .1248 L .04499 .12464 L .04629 .12411 L .04751 .12332 L .04969 .12135 L .05205 .11884 L .05468 .11629 L .05601 .11536 L .05676 .11497 L .05746 .11471 L .05872 .11453 L .05987 .11468 L .0605 .11489 L .06119 .11521 L .06244 .11605 L .06508 .11856 L .06792 .12168 L .06922 .12296 L .0706 .12406 L .07177 .12472 L .07306 .12511 L .07376 .12516 L .0744 .1251 L .07514 .12491 L .07583 .12462 L .07707 .12385 L .07842 .12269 L .08123 .11962 L .08273 .11792 L .08433 .11629 L .08504 .11568 L .08571 .11518 L .08697 .11449 L .08816 .11418 L .08945 .11423 L Mistroke .09016 .11443 L .09081 .11471 L .09156 .11516 L .09226 .11569 L .09487 .11832 L .09729 .12119 L .09863 .12268 L .09989 .12389 L .10107 .12476 L .10235 .12536 L .10351 .12556 L .10458 .12545 L .10566 .12504 L .10685 .12427 L .10809 .12317 L .10925 .12192 L .11192 .11866 L .11309 .11727 L .11437 .11591 L .11563 .11483 L .11679 .11413 L .11803 .11375 L .11934 .11379 L .12004 .11399 L .12079 .11435 L .1215 .11481 L .12215 .11534 L .1247 .11809 L .12694 .12096 L .12817 .12248 L .12934 .12377 L .13043 .12475 L .13161 .12553 L .13269 .12594 L .13368 .12604 L .13473 .12585 L .13588 .1253 L .13708 .12437 L .13821 .12323 L .14078 .11999 L .14318 .11685 L .14435 .11552 L .14545 .11449 L .14672 .11365 L .14787 .11325 L .14915 .11326 L .15033 .11367 L .15162 .11456 L .15228 .11516 L .153 .11593 L Mistroke .15798 .12258 L .15942 .12429 L .16012 .12499 L .16078 .12554 L .16201 .12628 L .16265 .1265 L .16334 .1266 L .16458 .12641 L .16528 .12611 L .16592 .12571 L .1671 .12468 L .16835 .12327 L .1707 .12002 L .17289 .11686 L .17495 .1144 L .17611 .11342 L .17716 .11284 L .17831 .11259 L .17956 .11279 L .18074 .11342 L .18184 .11435 L .18302 .11569 L .18426 .11737 L .18648 .12073 L .18773 .1226 L .18907 .1244 L .19033 .12577 L .19147 .12667 L .19254 .12715 L .1937 .12725 L .19479 .12695 L .19579 .12633 L .19689 .12532 L .19809 .12387 L .20052 .12021 L .20315 .11607 L .20445 .11434 L .20565 .11309 L .20632 .11256 L .20703 .11215 L .20782 .1119 L .20853 .11186 L .20919 .11197 L .20989 .11226 L .21117 .11319 L .21237 .11452 L .21369 .11636 L .21636 .12077 L .21756 .12276 L .21884 .12468 L Mistroke .22002 .12617 L .22112 .12721 L .22232 .1279 L .22296 .12806 L .22365 .12806 L .22438 .12787 L .22508 .12752 L .22639 .1264 L .22762 .12488 L .22877 .12314 L .23137 .11859 L .23369 .1147 L .23483 .11315 L .23588 .11204 L .23709 .11122 L .2384 .11095 L .23951 .11124 L .24069 .11205 L .24199 .1135 L .24318 .11525 L .24585 .12007 L .24835 .12462 L .24945 .12629 L .25066 .12775 L .25197 .12875 L .25269 .12902 L .25336 .12909 L .25409 .12894 L .25477 .12862 L .25551 .12805 L .25629 .12724 L .25761 .1254 L .25884 .12329 L .26162 .11778 L .26407 .11331 L .26531 .1116 L .26601 .11086 L .26667 .11033 L .26738 .10996 L .26804 .10982 L .26879 .10988 L .26951 .11018 L .27077 .11121 L .27148 .11207 L .27214 .11303 L .27482 .11809 L .27735 .12347 L .27845 .12561 L .27962 .12756 L .28089 .12918 L Mistroke .28207 .13009 L .28314 .13038 L .28429 .13009 L .28537 .12927 L .28636 .12808 L .28752 .12622 L .28878 .12374 L .29107 .11858 L .29348 .11332 L .2947 .11117 L .29541 .11017 L .29605 .10943 L .29719 .10857 L .29843 .10836 L .29951 .1088 L .30065 .10989 L .30189 .11172 L .30323 .11434 L .30567 .12012 L .30784 .12535 L .30898 .12777 L .31022 .12988 L .31092 .13079 L .31155 .13142 L .3128 .13205 L .31347 .13205 L .3142 .13176 L .31489 .13123 L .31553 .13052 L .31666 .12879 L .3179 .12627 L .32043 .1199 L .32264 .11405 L .32385 .11124 L .32498 .10908 L .3262 .10738 L .32685 .1068 L .32755 .10644 L .32878 .10649 L .3299 .10731 L .33104 .10887 L .33209 .11088 L .33448 .11695 L .33669 .12342 L .3388 .12903 L .33997 .13147 L .34124 .13337 L .34239 .13428 L .34346 .13438 L .34473 .13353 L Mistroke .34589 .13189 L .34653 .13066 L .34722 .12909 L .34847 .12574 L .35279 .11221 L .35398 .10898 L .35524 .10621 L .35644 .10443 L .35753 .10362 L .35882 .10374 L .35954 .10434 L .36022 .10521 L .36145 .10757 L .36276 .11102 L .3656 .12063 L .36828 .12981 L .36938 .13293 L .37054 .13551 L .37181 .13731 L .37298 .13789 L .37426 .13727 L .37545 .13554 L .37676 .13246 L .37816 .12804 L .38314 .10875 L .38435 .10484 L .38565 .10162 L .38639 .10035 L .38707 .09958 L .38776 .09922 L .38841 .09927 L .3891 .09974 L .38984 .10072 L .39054 .10209 L .39117 .10366 L .39272 .10877 L .39419 .11479 L .39678 .12666 L .39819 .13275 L .39951 .13755 L .40068 .14077 L .40192 .14288 L .40263 .14341 L .4033 .14343 L .40456 .14219 L .40572 .13963 L .40679 .13616 L .4092 .12525 L .41182 .11123 L .41428 .09959 L Mistroke .41551 .09535 L .41621 .09363 L .41684 .09251 L .41753 .09183 L .41829 .09175 L .419 .09233 L .41967 .09343 L .42439 .11456 L .42669 .12943 L .42879 .14192 L .42993 .14732 L .43101 .15113 L .43215 .15351 L .43339 .15396 L .43466 .15201 L .43531 .15009 L .43602 .14729 L .43727 .14072 L .43844 .13295 L .44092 .11352 L .44354 .09318 L .4449 .08465 L .44568 .08088 L .4464 .07821 L .44767 .07571 L .44838 .07565 L .44903 .07645 L .45028 .08033 L .45143 .08656 L .4527 .09601 L .45404 .10845 L .45651 .135 L .45877 .15853 L .45998 .16886 L .46131 .17712 L .46203 .18001 L .4627 .18153 L .46335 .18185 L .46403 .18095 L .46467 .17892 L .46536 .17541 L .46661 .16568 L .46783 .15232 L .46895 .13714 L .47153 .0954 L .4743 .05029 L .47511 .03909 L .47589 .03002 L .47658 .02343 L .47733 .0182 L Mistroke .47805 .01533 L .4788 .01472 L .47945 .01633 L .48016 .02051 L .48143 .03449 L .4826 .0548 L .48388 .08523 L .48522 .12578 L .48997 .31745 L .49237 .42144 L .49492 .51699 L .49639 .55863 L .49708 .574 L .49772 .58546 L .49837 .59426 L .49906 .60059 L .50033 .60332 L .50158 .59483 L .50278 .57677 L .50384 .55281 L .50501 .51924 L .51004 .3171 L .51249 .21116 L .5148 .12498 L .51687 .06634 L .51804 .04276 L .51911 .02745 L .52032 .01741 L .52164 .01476 L .52231 .01648 L .52302 .02038 L .5243 .03211 L .52502 .0408 L .52579 .05163 L .5272 .07389 L .52983 .11827 L .53233 .15436 L .5336 .16763 L .53432 .17332 L .53499 .17737 L .53619 .18139 L .53729 .18154 L .53855 .17778 L .53976 .17075 L .541 .16063 L .54235 .14736 L .54478 .12078 L .54696 .09901 L .54811 .0897 L .54934 .08211 L Mistroke .55004 .07905 L .55068 .07708 L .55193 .07555 L .5526 .076 L .55333 .07744 L .55402 .07968 L .55466 .08243 L .55579 .08872 L .55704 .09733 L .55956 .11748 L .56178 .13456 L .563 .14226 L .56412 .14789 L .56535 .15204 L .566 .15333 L .56669 .15401 L .56792 .15341 L .56904 .15097 L .57019 .14681 L .57124 .14174 L .57362 .12741 L .57584 .11316 L .57795 .1016 L .57911 .09682 L .58039 .09332 L .58155 .09183 L .58262 .09193 L .58389 .09385 L .58505 .09711 L .58638 .10238 L .58764 .10842 L .59205 .13134 L .59312 .13581 L .59427 .1396 L .59553 .14234 L .59671 .14343 L .59737 .14341 L .5981 .14286 L .59941 .14056 L .60069 .13688 L .60188 .13247 L .60663 .11132 L .60782 .10681 L .6091 .10295 L .60984 .1013 L .61051 .1002 L .61183 .09921 L .61311 .09975 L .61384 .1007 L .61452 .10196 L Mistroke .61589 .10553 L .61735 .1105 L .61992 .12074 L .62234 .12972 L .62355 .13329 L .62485 .13608 L .62559 .13711 L .62629 .1377 L .62761 .13773 L .62831 .13719 L .62905 .13622 L .62968 .13508 L .63038 .13353 L .6334 .1242 L .63618 .11438 L .6374 .11058 L .63873 .10718 L .63999 .10491 L .64116 .10375 L .6424 .1036 L .6431 .10398 L .64373 .10462 L .64442 .10562 L .64506 .10679 L .6465 .11022 L .64912 .11821 L .65157 .12587 L .65277 .12907 L .65387 .13144 L .65502 .13325 L .65626 .13428 L .6569 .13443 L .65761 .13428 L .65889 .13322 L .66376 .1221 L .66632 .11474 L .66772 .11128 L .66903 .10875 L .67017 .10724 L .6714 .10643 L .6721 .10636 L .67276 .10656 L .674 .10762 L .67514 .10928 L .67619 .11134 L .67857 .11721 L .68133 .12446 L .68268 .12751 L .68391 .12974 L .68521 .13132 L Mistroke .68589 .13181 L .68661 .13206 L .68734 .13202 L .68803 .13172 L .68878 .13112 L .68958 .13017 L .69087 .12805 L .69227 .12512 L .6948 .11895 L .69608 .11588 L .69729 .11326 L .69846 .11113 L .69955 .10965 L .70068 .10868 L .70191 .10834 L .70314 .10875 L .70445 .10998 L .70568 .1118 L .70685 .11398 L .70902 .11879 L .71146 .12424 L .71279 .12676 L .71403 .1286 L .71468 .12932 L .71538 .12989 L .71615 .13027 L .71685 .13038 L .71749 .13027 L .71818 .12995 L .71944 .12882 L .72074 .127 L .72196 .12483 L .72475 .119 L .72626 .11588 L .72769 .1133 L .72897 .1115 L .7297 .11075 L .73038 .11024 L .73161 .10982 L .73273 .11001 L .73403 .11089 L .73525 .1123 L .73656 .11434 L .73776 .11656 L .74049 .122 L .74175 .12433 L .74294 .12623 L .74401 .12758 L .74518 .12859 L .74645 .12907 L Mistroke .74709 .12907 L .74779 .12887 L .74904 .12803 L .75021 .12675 L .75132 .12515 L .75249 .12317 L .75458 .11925 L .75705 .11488 L .75813 .11335 L .7593 .11206 L .76057 .1112 L .76177 .11095 L .76243 .11105 L .76314 .11134 L .76441 .11231 L .76558 .11366 L .76667 .11525 L .76916 .11954 L .77044 .1218 L .77183 .12408 L .77306 .12577 L .77436 .12711 L .77516 .12766 L .77588 .12796 L .7766 .12808 L .77729 .12801 L .77857 .12743 L .77929 .12686 L .77996 .1262 L .78466 .11903 L .78591 .11699 L .78726 .115 L .78853 .11349 L .78971 .1125 L .79081 .11197 L .79202 .11188 L .79328 .11231 L .79447 .11319 L .79556 .11435 L .79656 .11566 L .79881 .11911 L .80113 .12278 L .80241 .12451 L .80362 .12583 L .80431 .1264 L .80505 .12687 L .80635 .12726 L .80767 .12708 L .80888 .12643 L .81015 .1253 L Mistroke .81133 .12389 L .81403 .11995 L .8154 .11787 L .81686 .11585 L .81821 .11431 L .81948 .11327 L .82062 .11273 L .82185 .1126 L .82251 .11273 L .82321 .11301 L .82447 .11387 L .82512 .11447 L .82581 .11522 L .82706 .1168 L .8299 .1209 L .83118 .12267 L .83257 .12434 L .83387 .12555 L .83508 .12628 L .83617 .12658 L .83737 .12649 L .83862 .12595 L .83979 .12507 L .84094 .1239 L .84218 .12236 L .84446 .11922 L .8467 .11628 L .84767 .11521 L .84873 .11428 L .84997 .11353 L .85114 .11321 L .85179 .11321 L .85248 .11334 L .85371 .11389 L .85485 .11475 L .85592 .1158 L .85834 .11881 L .86077 .12202 L .8621 .12356 L .86334 .12474 L .86399 .12521 L .86469 .12561 L .86546 .1259 L .86617 .12603 L .86745 .1259 L .86809 .12568 L .86879 .1253 L .87009 .12431 L .87127 .12312 L .87392 .11985 L Mistroke .8764 .11679 L .87749 .11566 L .87869 .11468 L .87984 .11404 L .88092 .11374 L .88205 .11376 L .88327 .11416 L .88391 .11451 L .8846 .115 L .88587 .11614 L .88824 .11886 L .8905 .12163 L .89175 .123 L .89291 .12408 L .89401 .12485 L .89519 .12538 L .89626 .12556 L .89725 .12547 L .89852 .12499 L .89972 .1242 L .90079 .12326 L .90196 .12204 L .90441 .11916 L .90575 .11762 L .90701 .11633 L .90819 .11534 L .90926 .11468 L .91046 .11425 L .91174 .11417 L .91276 .1144 L .91387 .11492 L .91503 .11574 L .91611 .11671 L .91859 .11941 L .92126 .12238 L .92249 .1235 L .92379 .12442 L .9249 .12492 L .92612 .12515 L .92685 .12512 L .92751 .12498 L .92828 .1247 L .92901 .12431 L .93031 .12337 L .93171 .12206 L .93416 .1194 L .93544 .11801 L .93679 .11669 L .93796 .11575 L .93923 .115 L Mistroke .94039 .11462 L .94147 .11453 L .94258 .11473 L .94359 .11514 L .94476 .11587 L .94587 .11676 L .9484 .11932 L .95108 .1221 L .95218 .12307 L .95337 .12391 L .95467 .12453 L .95589 .12479 L .95714 .12471 L .95784 .12452 L .9585 .12424 L .9597 .12353 L .96097 .12251 L .96563 .11786 L .967 .11665 L .96772 .11611 L .96848 .11563 L .96915 .11531 L .96988 .11505 L .97117 .11486 L .97242 .11502 L .97313 .11526 L .97377 .11556 L .97501 .11634 L .97619 .1173 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg>75aaXhhSSR>SZZZZTaaWYVCMe9?@=;@`jJKHmWD_5UD=BLW9g=c Lh20P4aSYVIVJ>SXjf][KNg]l71`Lc[Zj_Cdm6UTBO[`Dhb6HYT0I_ooc020P800082006IV c3c/C4a;Zj^[6a/JRX Z:2PX89mIN7Oeda9>eJ0Zk75fBa1EjFOOgl07/K3^4=1=:NUWRHU7WmoOg=^FLc9^bDT7B]0EADO:/gI iZ_0eD1PP:VXX3dk<861PL6L`H:2PWEeM@00?`00_`0o000o?`0oO`0o_`0oo`1o001o?`1oO`1o_`1o o`2o002o?`2oO`2o_`2oo`3o?`3oO`3o_cl003l0?cl0Ocl0_cl0oclo03lo?cloOclo_cloocmo03mo ?cmoOcmo_cmoocno03no?cnoOcno_cnoocoo03oo?cooOcoo_cooogl007l0?gl0Ogl0_gl0oglo07lo ?gloOglo_gloogmo07mo?gmo_gmoogno07no?gnoOgno_gnoogoo07oo?gooOgoo_goookl00;l0?kl0 Okl0_kl0oklo0;lo?kloOklo_klookmo0;mo?kmoOkmo_kmookno0;no?knoOkno_knookoo0;oo?koo Okoo_kooool0?ol0Ool0_olo0?lo?oloOolo_olooomo0?mo?omoOomo_omooono0?no?onoOono_ono oooo?oooOooo_gAdM:NWYl;2`_mVc>SXj79bLYVIc9W<09W<1`0400L700L700<01`L01@L00`07003<1`00cPL010071`071`0300L700@7 00@01`L0c0L001`0400L700L700<01`L010L01007 1`3<1`00cPL010071`071`0300L700@700@01`L0c0L001`0400L700L700<01`L010L01@071`L001`0400L700L700<01`L010L01@071`L001`0400L700L700<01`L010L01@071`L001`0400L700L700<01`L010L01@071`L001`0400L700L700<01`L010L01@071`L001`0400L700L70P051`0500L71`00b`L000L200X700@01`L01`L00`071`031`0300L700@700<01`L0 10L01@071`L000P700D01`L700091`0300L004<700<01`L0F`L00`071`041`0300L7008700<01`L0 0PL000031`0000D01@L00`071`041`0300L7008700<01`L0C`L700<700D01`L7000k1`802PL01007 1`071`0300L700<700<01`L010L00`071`031`0300L700<700<01`L01@L01@071`L000P700@01`L0 ?PL01@071`L005h700<01`L010L00`071`021`0300L70087000=1`0300L700@700<01`L00PL00`07 1`1I1`@0;`L300T700@01`L020L01@071`L000L700<01`L00`L00`071`041`0300L700<700<01`L0 0`L00`071`051`0500L71`0020L010071`0:1`801`8020L30=D7 003@1`0300L700D70`3E1`00d0L00`071`051`<0e@L00"], ImageRangeCache->{{{0, 431}, {265.875, 0}} -> {-10.663, -0.79962, 0.0736533, 0.0230688}}], Cell[BoxData[ FormBox[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Try plotting this for different values of epsilon, note this function doesnt look anything like f, but we show it has the same properties when epsilon->0. Lets perform our integral.\ \>", "Text", FormatType->TextForm], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[g[x]\ f2[x, 0.01], {x, \(-Infinity\), Infinity}, Method -> Oscillatory]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`2.29999953706920479`\)], "Output"] }, Open ]], Cell["\<\ Note: because of the rapid oscillations it is difficult to \ numerically integrate this function (it easy analytically).\ \>", "Text"], Cell["\<\ The above representation is commonly used and is equivalent to\ \>", "Text"], Cell[BoxData[ \(\[Delta][x] = \(\(1\/\(2 \[Pi]\)\) \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\( e\^\(i\ p\ x\)\) \[DifferentialD]p\) = \ \(Lim\+\(L -> \[Infinity]\)\ \(1\/\(2 \[Pi]\)\) \(\[Integral]\_\(-L\)\%L\( e\^\(i\ p\ x\)\) \[DifferentialD]p\) = \ \(Lim\+\(L -> \[Infinity]\)\ \(\ sin \((L\ x)\)\)\/\(\[Pi]\ x\) = \ Lim\+\(\[Epsilon] -> 0\)\ \(\ sin \((x\/\[Epsilon])\)\)\/\(\[Pi]\ x\)\)\)\)\)], "Input", Background->RGBColor[0, 1, 0]], Cell[BoxData[{ \(\nThere\ are\ several\ representations\ of\ the\ dirac\ delta.\nShow, as\ above, \ that\ yet\ another\ representation\ is\ given\), \(by\n\), \(\[Delta][x] = 1\/\[Pi]\ \[Epsilon]\/\(\[Epsilon]\^2 + x\^2\)\ as\ \[Epsilon] -> 0. \)}], "Input"] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowSize->{919, 782}, WindowMargins->{{Automatic, 137}, {Automatic, 38}}, Magnification->1.5, StyleDefinitions -> "Classroom.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 37, 0, 84, "Title"], Cell[1749, 51, 99, 3, 73, "Text"], Cell[CellGroupData[{ Cell[1873, 58, 46, 1, 69, "Input"], Cell[1922, 61, 55, 1, 74, "Output"] }, Open ]], Cell[1992, 65, 224, 5, 73, "Text"], Cell[CellGroupData[{ Cell[2241, 74, 57, 1, 92, "Input"], Cell[2301, 77, 55, 1, 74, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2393, 83, 39, 1, 69, "Input"], Cell[2435, 86, 55, 1, 74, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2527, 92, 58, 1, 69, "Input"], Cell[2588, 95, 74, 1, 74, "Output"] }, Open ]], Cell[2677, 99, 55, 1, 38, "Text"], Cell[2735, 102, 140, 4, 38, "Text"], Cell[CellGroupData[{ Cell[2900, 110, 61, 1, 69, "Input"], Cell[2964, 113, 16214, 413, 279, 3523, 252, "GraphicsData", "PostScript", "Graphics"], Cell[19181, 528, 167, 4, 70, "Output"] }, Open ]], Cell[19363, 535, 89, 3, 38, "Text"], Cell[CellGroupData[{ Cell[19477, 542, 104, 2, 69, "Input"], Cell[19584, 546, 123, 3, 124, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19744, 554, 82, 1, 69, "Input"], Cell[19829, 557, 27374, 1236, 279, 16617, 1099, "GraphicsData", "PostScript", "Graphics"], Cell[47206, 1795, 167, 4, 70, "Output"] }, Open ]], Cell[47388, 1802, 124, 4, 73, "Text"], Cell[CellGroupData[{ Cell[47537, 1810, 70, 1, 69, "Input"], Cell[47610, 1813, 71, 1, 70, "Output"] }, Open ]], Cell[47696, 1817, 332, 9, 75, "Text"], Cell[48031, 1828, 28, 0, 38, "Text"], Cell[CellGroupData[{ Cell[48084, 1832, 52, 1, 69, "Input"], Cell[48139, 1835, 196, 5, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48372, 1845, 74, 1, 69, "Input"], Cell[48449, 1848, 71, 1, 70, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[48557, 1854, 132, 3, 69, "Input"], Cell[48692, 1859, 296, 5, 70, "Output"] }, Open ]], Cell[49003, 1867, 46, 0, 38, "Text"], Cell[49052, 1869, 71, 0, 38, "Text"], Cell[49126, 1871, 37, 0, 38, "Text"], Cell[49166, 1873, 446, 8, 128, "Input"], Cell[49615, 1883, 62, 0, 38, "Text"], Cell[CellGroupData[{ Cell[49702, 1887, 76, 1, 69, "Input"], Cell[49781, 1890, 84, 1, 108, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49902, 1896, 85, 1, 69, "Input"], Cell[49990, 1899, 29614, 1127, 279, 14063, 930, "GraphicsData", "PostScript", "Graphics"], Cell[79607, 3028, 167, 4, 70, "Output"] }, Open ]], Cell[79789, 3035, 229, 5, 108, "Text"], Cell[CellGroupData[{ Cell[80043, 3044, 122, 2, 92, "Input"], Cell[80168, 3048, 71, 1, 70, "Output"] }, Open ]], Cell[80254, 3052, 145, 4, 73, "Text"], Cell[80402, 3058, 89, 3, 38, "Text"], Cell[80494, 3063, 544, 10, 113, "Input"], Cell[81041, 3075, 292, 6, 215, "Input"] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)