PK VmXńB–Hmimetypetext/x-wxmathmlPK VmXQdBV55 format.txt This file contains a wxMaxima session in the .wxmx format. .wxmx files are .xml-based files contained in a .zip container like .odt or .docx files. After changing their name to end in .zip the .xml and eventual bitmap files inside them can be extracted using any .zip file viewer. The reason why part of a .wxmx file still might still seem to make sense in a ordinary text viewer is that the text portion of .wxmx by default isn't compressed: The text is typically small and compressing it would mean that changing a single character would (with a high probability) change big parts of the whole contents of the compressed .zip archive. Even if version control tools like git and svn that remember all changes that were ever made to a file can handle binary files compression would make the changed part of the file bigger and therefore seriously reduce the efficiency of version control wxMaxima can be downloaded from https://github.com/wxMaxima-developers/wxmaxima. It also is part of the windows installer for maxima (https://wxmaxima-developers.github.io/wxmaxima/). If a .wxmx file is broken but the content.xml portion of the file can still be viewed using an text editor just save the xml's text as "content.xml" and try to open it using a recent version of wxMaxima. If it is valid XML (the XML header is intact, all opened tags are closed again, the text is saved with the text encoding "UTF8 without BOM" and the few special characters XML requires this for are properly escaped) chances are high that wxMaxima will be able to recover all code and text from the XML file. PK VmXĄyčć!ć! content.xml Kedvenc Matematikai KĂ­sĂ©rleteim Alapok Halmazok Ă©s fĂŒggvĂ©nyek SzĂĄmok HatĂĄrĂ©rtĂ©k DifferenciĂĄlszĂĄmĂ­tĂĄs IntegrĂĄlszĂĄmĂ­tĂĄs LineĂĄris algrebra TöbbvĂĄltozĂłs fĂŒggvĂ©nyek MĂ©rtĂ©k Ă©s valĂłszĂ­nƱsĂ©g Fourier-elmĂ©let Fourier-sorok ÁltalĂĄnosĂ­tott Pitagorasz-tĂ©tel Fourier-sor TĂ©tel TĂ©tel DefinĂ­ciĂł Klasszikus Fourier-sorok PĂ©ldĂĄk TĂ©tel KövetkezmĂ©ny Weierstrass mĂĄsodik approximĂĄciĂłs tĂ©tele Lemma load(orthopoly)$ plot2d([chebyshev_t(0,x),chebyshev_t(1,x), chebyshev_t(2,x),chebyshev_t(3,x)],[x,-1,1])$ plot2d([chebyshev_u(0,x),chebyshev_u(1,x), chebyshev_u(2,x),chebyshev_u(3,x)],[x,-1,1])$ Weierstrass elsƑ approximĂĄciĂłs tĂ©tele MegjegyzĂ©s taylor(sin(x),x,0,7); (%o50)/T/ x−x36+x5120−x75040+... f:%/x; (%o51)/T/ 1−x26+x4120−x65040+... f:taytorat(%); (%o52)/R/ −x6−42*x4+840*x2−50405040 taylor(sin(x),x,0,9); (%o53)/T/ x−x36+x5120−x75040+x9362880+... g:%/x; (%o54)/T/ 1−x26+x4120−x65040+x8362880+... g:taytorat(%); (%o55)/R/ x8−72*x6+3024*x4−60480*x2+362880362880 chebyshev_t(8,x); (%o56) −64*

1−x

+128*

1−x

8−1024*

1−x

7+3328*

1−x

6−5632*

1−x

5+5280*

1−x

4−2688*

1−x

3+672*

1−x

2+1 expand(%); (%o57) 128*x8−256*x6+160*x4−32*x2+1 h:g-%/362880/128; (%o58)/R/ −8960*x6−386912*x4+7741408*x2−4644863946448640 plot2d([sin(x)/x-f,sin(x)/x-g,sin(x)/x-h],[x,0,%pi/4])$ plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: expression evaluates to non−numeric value somewhere in plotting range. TĂ©tel load(interpol)$ L:makelist([k/4,abs(k/4)],k,-4,4); (%o64) [[−1,1],[−34,34],[−12,12],[−14,14],[0,0],[14,14],[12,12],[34,34],[1,1]] p:lagrange(L); (%o65) 512*

x−34

*

x−12

*

x−14

*x*

x+14

*

x+12

*

x+34

*

x+1

315−1024*

x−1

*

x−12

*

x−14

*x*

x+14

*

x+12

*

x+34

*

x+1

105+1024*

x−1

*

x−34

*

x−14

*x*

x+14

*

x+12

*

x+34

*

x+1

45−1024*

x−1

*

x−34

*

x−12

*x*

x+14

*

x+12

*

x+34

*

x+1

45−1024*

x−1

*

x−34

*

x−12

*

x−14

*x*

x+12

*

x+34

*

x+1

45+1024*

x−1

*

x−34

*

x−12

*

x−14

*x*

x+14

*

x+34

*

x+1

45−1024*

x−1

*

x−34

*

x−12

*

x−14

*x*

x+14

*

x+12

*

x+1

105+512*

x−1

*

x−34

*

x−12

*

x−14

*x*

x+14

*

x+12

*

x+34

315 LC:makelist([cos(%pi*(2*k-1)/18),abs(cos(%pi*(2*k-1)/18))],k,1,9); (%o82) [[cos

%pi18

,cos

%pi18

],[32,32],[cos

5*%pi18

,cos

5*%pi18

],[cos

7*%pi18

,cos

7*%pi18

],[0,0],[cos

11*%pi18

,−cos

11*%pi18

],[cos

13*%pi18

,−cos

13*%pi18

],[−32,32],[cos

17*%pi18

,−cos

17*%pi18

]] LC:float(%), numer; (%o83) [[0.984807753012208,0.984807753012208],[0.8660254037844386,0.8660254037844386],[0.6427876096865394,0.6427876096865394],[0.3420201433256688,0.3420201433256688],[0.0,0.0],[−0.3420201433256687,0.3420201433256687],[−0.6427876096865394,0.6427876096865394],[−0.8660254037844386,0.8660254037844386],[−0.984807753012208,0.984807753012208]] pC:lagrange(LC); (%o84) 4.864286482853958*

x−0.8660254037844386

*

x−0.6427876096865394

*

x−0.3420201433256688

*x*

x+0.3420201433256687

*

x+0.6427876096865394

*

x+0.8660254037844386

*

x+0.984807753012208

−12.31680574271203*

x−0.984807753012208

*

x−0.6427876096865394

*

x−0.3420201433256688

*x*

x+0.3420201433256687

*

x+0.6427876096865394

*

x+0.8660254037844386

*

x+0.984807753012208

+14.00615470950696*

x−0.984807753012208

*

x−0.8660254037844386

*

x−0.3420201433256688

*x*

x+0.3420201433256687

*

x+0.6427876096865394

*

x+0.8660254037844386

*

x+0.984807753012208

−9.141868226653005*

x−0.984807753012208

*

x−0.8660254037844386

*

x−0.6427876096865394

*x*

x+0.3420201433256687

*

x+0.6427876096865394

*

x+0.8660254037844386

*

x+0.984807753012208

−9.141868226653001*

x−0.984807753012208

*

x−0.8660254037844386

*

x−0.6427876096865394

*

x−0.3420201433256688

*x*

x+0.6427876096865394

*

x+0.8660254037844386

*

x+0.984807753012208

+14.00615470950696*

x−0.984807753012208

*

x−0.8660254037844386

*

x−0.6427876096865394

*

x−0.3420201433256688

*x*

x+0.3420201433256687

*

x+0.8660254037844386

*

x+0.984807753012208

−12.31680574271202*

x−0.984807753012208

*

x−0.8660254037844386

*

x−0.6427876096865394

*

x−0.3420201433256688

*x*

x+0.3420201433256687

*

x+0.6427876096865394

*

x+0.984807753012208

+4.864286482853958*

x−0.984807753012208

*

x−0.8660254037844386

*

x−0.6427876096865394

*

x−0.3420201433256688

*x*

x+0.3420201433256687

*

x+0.6427876096865394

*

x+0.8660254037844386

 plot2d([abs(x),p,pC],[x,-1,1])$ Feladat kill(all); (%o0) done taylor(sin(x),x,0,7); (%o1)/T/ x−x36+x5120−x75040+... f:%/x; (%o2)/T/ 1−x26+x4120−x65040+... (%pi/2)^6/7!; (%o3) %pi6322560 float(%), numer; (%o4) 0.002980497251907565 Mivel a sor alternĂĄlĂł, a hiba legfeljebb az elsƑ elhagyott tag hibĂĄja. Δ:0.003; (%o5) 0.003 f:taylor(sin(x),x,0,5); (%o7)/T/ x−x36+x5120+... f0:taytorat(f)/x; (%o8)/R/ x4−20*x2+120120 load("orthopoly"); (%o9) /usr/share/maxima/5.43.2/share/orthopoly/orthopoly.lisp chebyshev_t(4,y); (%o10) −16*

1−y

+8*

1−y

4−32*

1−y

3+40*

1−y

2+1 T4:subst(y=%pi*x/2,%); (%o11) 8*

1−%pi*x2

4−32*

1−%pi*x2

3+40*

1−%pi*x2

2−16*

1−%pi*x2

+1 expand(%); (%o12) %pi4*x42−2*%pi2*x2+1 f1:f0-T4*2/%pi^4/5!; (%o18)/R/ −

10*%pi4−2*%pi2

*x2−60*%pi4+160*%pi4 expand(%); (%o19) x230*%pi2−x26−160*%pi4+1 Δ:Δ+2/%pi^4/5!; (%o20) 130*%pi4+0.003 float(%), numer; (%o16) 0.003171099704244739 Feladat taylor(2^x,x,1/2,8); (%o7)/T/ 2+2*log

2

*

x−12

+2*log

2

2*

x−12

22+2*log

2

3*

x−12

36+2*log

2

4*

x−12

424+2*log

2

5*

x−12

5120+2*log

2

6*

x−12

6720+2*log

2

7*

x−12

75040+2*log

2

8*

x−12

840320+... A hiba pĂ©ldĂĄul mĂ©rtani sorral becsĂŒlhetƑ, legfeljebb az elsƑ elhagyott tag kĂ©tszerese. Δ:2*sqrt(2)*log(2)^5*(1/2)^5/120; (%o23) log

2

515*2132 float(%), numer; (%o24) 1.178531172835583*10−4 f:taylor(2^x,x,1/2,4); (%o26)/T/ 2+2*log

2

*

x−12

+2*log

2

2*

x−12

22+2*log

2

3*

x−12

36+2*log

2

4*

x−12

424+... f0:expand(taytorat(f)); (%o29) log

2

4*x43*252−log

2

4*x33*232+log

2

3*x33*2+log

2

4*x2272−log

2

3*x2232+log

2

2*x22−log

2

4*x3*272+log

2

3*x252−log

2

2*x2+2*log

2

*x+log

2

43*2132−log

2

33*272+log

2

2252−log

2

2+2 float(%), numer; (%o30) 0.01360208862866361*x4+0.05129048598389343*x3+0.2423927222642392*x2+0.692596512402056*x+1.000055684318729 Feladat PĂ©ldĂĄul a sin esetĂ©n sin(x)/x_et közelĂ­tjĂŒk |x|<=%pi/2-re, Ă©s a fokszĂĄmot csökkentjĂŒk, mint fent lĂĄttuk. A többi esetet erre vezetjĂŒk vissza. A kezdeti közelĂ­tĂ©s duplapontos esetben jĂłval pontosabb legyen, mint az 1 nagysĂĄgrendƱ eredmĂ©ny 2^(-54) nagysĂĄgrendƱ hibĂĄja. Nagy pontossĂĄggal szĂĄmoljuk az egyĂŒtthatĂłkat! for i thru 12 do print(float((%pi/2)^(2*i)/(2*i+1)!/2^(-54))); 7.408124450506706*1015 9.139407210067335*1014 5.36918652522788*1013 1.83999121387704*1012 4.127269405101497*1010 6.527967353376793*108 7670054.20488753 69577.57420694108 501.9759741300782 2.948990644949606 0.01438012403565512 5.91358897793803*10−5 (%o3) done UnicitĂĄsi tĂ©tel KövetkezmĂ©ny Riemann-Lebesgue-lemma Dirichlet-formula Dini-kritĂ©rium KövetkezmĂ©ny: Lipschitz-kritĂ©rium KövetkezmĂ©ny PĂ©lda Lemma Dirichlet-Jordan-tĂ©tel KövetkezmĂ©ny: Riemann-fĂ©le lokalizĂĄciĂłs tĂ©tel TöbbdimenziĂłs Fourier-sorok Klasszikus ortogonĂĄlis polinomok load(orthopoly)$ Feladat A pĂĄratlan fĂŒggvĂ©nyek Feladat Az 1, t-1/2 ortogonĂĄlisak. integrate((t-1/2)^2,t,0,1); (%o87) 112 c0:integrate(1*exp(t),t,0,1); (%o88) %e−1 c1:integrate(exp(t)*(t-1/2)/sqrt(12),t,0,1); (%o89) 32−%e22*3 c0*1+c1*(t-1/2)/sqrt(12); (%o90)

32−%e2

*

t−12

12+%e−1 expand(%); (%o91) −%e*t24+t8+49*%e48−1716 Feladat c0:integrate(x^2*1,x,0,1); (%o96) 13 c1:integrate(x^2*(x-1/2)/sqrt(12),x,0,1); (%o97) 18*332 x^2-c0-c1*(x-1/2)/sqrt(12); (%o103) x2−x−12144−13 expand(%); (%o104) x2−x144−95288 Feladat N0:integrate(1,x,-1,1); (%o106) 2 f0:1/sqrt(N0); (%o108) 12 c0:integrate(f0*x,x,-1,1); (%o109) 0 N1:integrate(x^2,x,-1,1); (%o110) 23 f1:x/sqrt(N1); (%o111) 3*x2 c0:integrate(f0*x^2,x,-1,1); (%o112) 23 c1:integrate(f1*x^2,x,-1,1); (%o113) 0 f2:x^2-c0*f0; (%o114) x2−13 N2:integrate(f2^2,x,-1,1); (%o115) 845 f2:f2/sqrt(N2); (%o116) 3*5*

x2−13

232 Feladat (1) N2:integrate(x^4,x,0,1); (%o119) 15 f2:x^2/sqrt(N2); (%o120) 5*x2 c2:integrate(f2*x^3,x,0,1); (%o121) 56 f3:x^3-c2*f2; (%o122) x3−5*x26 N3:integrate(f3^2,x,0,1); (%o123) 1252 f3:f3/sqrt(N3); (%o124) 6*7*

x3−5*x26

 c2:integrate(x*f2,x,0,1); (%o125) 54 c3:integrate(x*f3,x,0,1); (%o126) −720 f1:x-c2*f2-c3*f3; (%o127) 21*

x3−5*x26

10−5*x24+x N:integrate(f1^2,x,0,1); (%o128) 1300 sqrt(N); (%o129) 110*3 (2) sqrt(2*%pi); (%o131) 2*%pi (3) f0:1; (%o132) 1 f1:(x-1/2)^2; (%o133)

x−12

2 N1:integrate(f1^2,x,0,1); (%o134) 180 f1:f1/sqrt(N1); (%o135) 4*5*

x−12

2 c0:integrate(x^2,x,0,1); (%o136) 13 c1:integrate(x^2*f1,x,0,1); (%o137) 23*5 f2:x^2-c0*f0-c1*f1; (%o138) x2−8*

x−12

23−13 N2:integrate(f2^2,x,0,1); (%o139) 427 f2:f2/sqrt(N2); (%o140) 332*

x2−8*

x−12

23−13

2 c0:integrate(x^3*f0,x,0,1); (%o141) 14 c1:integrate(x^3*f1,x,0,1); (%o142) 712*5 c2:integrate(x^3*f2,x,0,1); (%o143) 140*3 f:x^3-c0*f0-c1*f1-c2*f2; (%o146) x3−3*

x2−8*

x−12

23−13

80−7*

x−12

23−14 f:expand(%); (%o147) x3−109*x248+67*x30−191240 N:integrate(f^2,x,0,1); (%o148) 10109100800 sqrt(N); (%o149) 10109120*7 (4) N1:integrate(x^2,x,0,1); (%o150) 13 f1:x/sqrt(N1); (%o151) 3*x c1:integrate(x^2*f1,x,0,1); (%o153) 34 f2:x^2-c1*f1; (%o154) x2−3*x4 N2:integrate(f2^2,x,0,1); (%o155) 180 f2:f2/sqrt(N2); (%o156) 4*5*

x2−3*x4

 c1:integrate(x^3*f1,x,0,1); (%o157) 35 c2:integrate(x^3*f2,x,0,1); (%o158) 13*5 f:x^3-c1*f1-c2*f2; (%o159) x3−4*

x2−3*x4

3−3*x5 N:integrate(f^2,x,0,1); (%o160) 11575 sqrt(N); (%o161) 115*7 Feladat (1) a0:2/%pi*integrate(sin(x),x,0,%pi); (%o2) 4%pi an:2/%pi*integrate(sin(x)*cos(n*x),x,0,%pi); (%o3) 2*

−cos

%pi*n

n2−1−1n2−1

%pi subst((-1)^n,cos(%pi*n),%); (%o4) 2*

−

−1

nn2−1−1n2−1

%pi ratsimp(%); (%o5) −2*

−1

n+2%pi*n2−%pi (2) sin(x)^4; (%o8) sin

x

4 trigrat(%); (%o9) cos

4*x

−4*cos

2*x

+38 (3) a0:1; (%o10) 1 an:2/%pi*integrate(cos(n*x),x,0,%pi/2); (%o11) 2*sin

%pi*n2

%pi*n (4) assume(abs(a)<1); (%o14) [a<1] a0:2/%pi*integrate((1-a^2)/(1-2*a*cos(x)+a^2),x,0,%pi); (%o15) −2*

1−a2

a2−1 an:2/%pi*integrate((1-a^2)*cos(n*x)/(1-2*a*cos(x)+a^2),x,0,%pi); (%o17) 2*

1−a2

*0%picos

n*x

−2*a*cos

x

+a2+1dx%pi (5) a0:2/%pi*integrate((1-a*cos(x))/(1-2*a*cos(x)+a^2),x,0,%pi); (%o18) 2 an:2/%pi*integrate((1-a*cos(x))*cos(n*x)/(1-2*a*cos(x)+a^2),x,0,%pi); (%o20) 2*0%pi

1−a*cos

x

*cos

n*x

−2*a*cos

x

+a2+1dx%pi (6) plot2d(entier(2*x)-2*entier(x),[x,-4,4],[y,-.2,1.2])$ a0:1; (%o3) 1 bn:4*integrate(1/2*sin(2*%pi*n*x),x,0,1/2); (%o4) 2*

12*%pi*n−cos

%pi*n

2*%pi*n

 (7) plot2d(entier(4*x)-3*entier(2*x)+2*entier(x),[x,-4,4],[y,-1.2,1.2])$ a0:0; (%o9) 0 bn:4*integrate(sin(2*%pi*n*x),x,1/4,1/2); (%o10) 4*

cos

%pi*n2

2*%pi*n−cos

%pi*n

2*%pi*n

 (8) Tagonként integrålunk (9) Tagonként integrålunk (10) an:integrate(cos(n*x)*log(sin(x/2)),x,0,%pi); (%o11) 0%pilog

sin

x2

*cos

n*x

dx Ez egy konjugĂĄlt Fourier-sor összege Feladat etc. Feladat (1) MindenĂŒtt a fĂŒggvĂ©ny, Lipschitz-kritĂ©rium (2) MindenĂŒtt a fĂŒggvĂ©ny, analitikus Feladat assume(ÎŽ>0); (%o1) [ÎŽ>0] an:integrate(cos(n*x),x,0,ÎŽ)*2/%pi; (%o2) 2*sin

n*ÎŽ

%pi*n Feladat a0:integrate(sin(x),x,0,%pi); (%o1) 2 b3:integrate(sin(x)*sin(3*x),x,0,%pi); (%o3) 0 a3:integrate(sin(x)*cos(3*x),x,0,%pi); (%o6) 0 Feladat a0:2/%pi*integrate(sin(x),x,0,%pi); (%o9) 4%pi t:2*integrate(sin(x)^2,x,0,%pi)/2/%pi; (%o11) 12 (a0/2)^2/(1/2); (%o14) 8%pi2 Feladat A fĂŒggvĂ©ny pĂĄros, x-1 ha x<2 Ă©s 2-x, ha x<4. Az a1-bƑl lesz a sin. a1:2*integrate((x-1)*cos(%pi*x/4),x,0,2) +2*integrate((2-x)*cos(%pi*x/4),x,2,4); (%o20) 2*

4%pi−16%pi2

+32%pi2 ratsimp(%); (%o21) 8%pi Ennek a nĂ©gyzetĂ©t kell hasonlĂ­tani az integrĂĄl teljesĂ­tmĂ©nyĂ©hez. Feladat A π szerint periĂłdikus, a 0 Ă©s π között identikus fĂŒggvĂ©ny sin-sorĂĄbĂłl kivonjuk a 0-tĂłl c-ig nulla, onnan π-ig lineĂĄrisan nullĂĄig csökkenƑ fĂŒggvĂ©ny sin-sorĂĄt. Pl. declare(n,integer); (%o29) done bn:2*integrate((%pi/2-(x-%pi/2))*sin(n*x),x,%pi/2,%pi); (%o30) 2*sin

%pi*n2

+%pi*n*cos

%pi*n2

n2 trigsimp(%); (%o31) 2*sin

%pi*n2

+%pi*n*cos

%pi*n2

n2 Feladat (1) KiszĂĄmoljuk a belsƑ szorzatokat, pl. b11:integrate(x^2*exp(-x),x,0,inf); (%o22) 2 etc. PK VmXńB–HmimetypePK VmXQdBV55 5format.txtPK VmXĄyčć!ć! ’content.xmlPK§ (