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startGroup; ] Functions Defined by Integrals :[font = section; inactive; preserveAspect; startGroup; ] Introduction :[font = input; preserveAspect; ] ln[z_] := Integrate[1/x, {x, 1, z}] :[font = input; preserveAspect; ] P[z_] := Sqrt[2/Pi] Integrate[Exp[-x^2/2], {x, 0, z}] :[font = input; preserveAspect; endGroup; ] erf[z_] := 2/Sqrt[Pi] Integrate[Exp[-t^2], {t, 0, z}] :[font = section; inactive; preserveAspect; startGroup; ] Exercise 6 :[font = input; preserveAspect; startGroup; ] Log[1] :[font = output; output; inactive; preserveAspect; endGroup; ] 0 ;[o] 0 :[font = input; preserveAspect; startGroup; ] D[Log[z], z] :[font = output; output; inactive; preserveAspect; endGroup; ] z^(-1) ;[o] 1 - z :[font = input; preserveAspect; startGroup; ] Log'[z] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] z^(-1) ;[o] 1 - z :[font = section; inactive; preserveAspect; startGroup; ] Exercise 7 :[font = input; preserveAspect; startGroup; ] Table[{z, ln[z], Log[z]}, {z, 1.0, 2.0, 0.1}] //TableForm :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] TableForm[{{1., 0, 0.}, {1.1, 0.0953101798043249, 0.0953101798043249}, {1.2, 0.1823215567939548, 0.1823215567939548}, {1.3, 0.2623642644674913, 0.2623642644674913}, {1.4, 0.3364722366212132, 0.3364722366212132}, {1.5, 0.4054651081081647, 0.4054651081081647}, {1.6, 0.4700036292457359, 0.4700036292457359}, {1.700000000000001, 0.5306282510621707, 0.5306282510621707}, {1.800000000000001, 0.5877866649021195, 0.5877866649021195}, {1.900000000000001, 0.6418538861723953, 0.6418538861723953}, {2.000000000000001, 0.6931471805599458, 0.6931471805599458}}] ;[o] 1. 0 0. 1.1 0.0953102 0.0953102 1.2 0.182322 0.182322 1.3 0.262364 0.262364 1.4 0.336472 0.336472 1.5 0.405465 0.405465 1.6 0.470004 0.470004 1.7 0.530628 0.530628 1.8 0.587787 0.587787 1.9 0.641854 0.641854 2. 0.693147 0.693147 :[font = section; inactive; preserveAspect; startGroup; ] Exercise 8 :[font = input; preserveAspect; startGroup; ] Erf[z] + O[z]^7 :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] SeriesData[z, 0, {2/Pi^(1/2), 0, -2/(3*Pi^(1/2)), 0, 1/(5*Pi^(1/2))}, 1, 7, 1] ;[o] 3 5 2 z 2 z z 7 -------- - ---------- + ---------- + O[z] Sqrt[Pi] 3 Sqrt[Pi] 5 Sqrt[Pi] :[font = section; inactive; preserveAspect; startGroup; ] Exercise 9 :[font = input; preserveAspect; startGroup; ] p[z_] = Erf[z] + O[z]^7 //Normal//N; Table[{z, erf[z], Erf[z], p[z]}, {z, 0.0, 2.0, 0.25}] // TableForm :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] TableForm[{{0., 0., 0., 0.}, {0.25, 0.2763263901682369, 0.2763263901682369, 0.2763280102232923}, {0.5, 0.5204998778130465, 0.5204998778130465, 0.52069996981595}, {0.75, 0.7111556336535151, 0.7111556336535151, 0.7143830215117392}, {1., 0.842700792949715, 0.842700792949715, 0.865090694773226}, {1.25, 0.922900128256458, 0.922900128256458, 1.02020609915618}, {1.5, 0.966105146475311, 0.966105146475311, 1.280005117673972}, {1.75, 0.986671671219182, 0.986671671219182, 1.810879600161975}, {2., 0.995322265018953, 0.995322265018953, 2.858560556641966}}] ;[o] 0. 0. 0. 0. 0.25 0.276326 0.276326 0.276328 0.5 0.5205 0.5205 0.5207 0.75 0.711156 0.711156 0.714383 1. 0.842701 0.842701 0.865091 1.25 0.9229 0.9229 1.02021 1.5 0.966105 0.966105 1.28001 1.75 0.986672 0.986672 1.81088 2. 0.995322 0.995322 2.85856 :[font = section; inactive; preserveAspect; startGroup; ] Exercise 10 :[font = input; preserveAspect; startGroup; ] erf[Infinity] :[font = output; output; inactive; preserveAspect; endGroup; ] 1 ;[o] 1 :[font = input; preserveAspect; startGroup; ] Erf[Infinity] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] 1 ;[o] 1 :[font = section; inactive; preserveAspect; startGroup; ] Exercise 11 :[font = input; preserveAspect; startGroup; ] Plot[Erf[x], {x, -5, 5}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! 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