PK0ilXńB–Hmimetypetext/x-wxmathmlPK0ilXQdBV55 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. PK0ilXV|ą|R|R content.xml Kedvenc Matematikai KĂ­sĂ©rleteim Alapok Halmazok Ă©s fĂŒggvĂ©nyek SzĂĄmok HatĂĄrĂ©rtĂ©k DifferenciĂĄlszĂĄmĂ­tĂĄs IntegrĂĄlszĂĄmĂ­tĂĄs LineĂĄris algebra TöbbvĂĄltozĂłs fĂŒggvĂ©nyek MĂ©rtĂ©k Ă©s valĂłszĂ­nƱsĂ©ge Fourier-elmĂ©lete Komplex fĂŒggvĂ©nytan Holomorf fĂŒggvĂ©nyek DefinĂ­ciĂł MegjegyzĂ©s Feladat kill(all); (%o0) done exp(x+%i*y); (%o1) %e%i*y+x realpart(%); (%o2) %ex*cos

y

 imagpart(%); (%o3) 0 plot3d(abs(exp(x+%i*y)),[x,-5,5],[y,-5,5]); (%o4) [/tmp/maxout113834.gnuplot_pipes] plot3d(realpart(exp(x+%i*y)),[x,-5,5],[y,-5,5]); (%o5) [/tmp/maxout113834.gnuplot_pipes] plot3d(imagpart(exp(x+%i*y)),[x,-5,5],[y,-5,5]); (%o6) [/tmp/maxout113834.gnuplot_pipes] VĂ­zszintes vonalak kĂ©pei L:create_list([parametric,realpart(exp(t+j*%pi/6*%i)),imagpart(exp(t+j*%pi/6*%i)),[t,-3,3]],j,-5,6); (%o8) [[parametric,−3*%et2,−%et2,[t,−3,3]],[parametric,−%et2,−3*%et2,[t,−3,3]],[parametric,0,−%et,[t,−3,3]],[parametric,%et2,−3*%et2,[t,−3,3]],[parametric,3*%et2,−%et2,[t,−3,3]],[parametric,%et,0,[t,−3,3]],[parametric,3*%et2,%et2,[t,−3,3]],[parametric,%et2,3*%et2,[t,−3,3]],[parametric,0,%et,[t,−3,3]],[parametric,−%et2,3*%et2,[t,−3,3]],[parametric,−3*%et2,%et2,[t,−3,3]],[parametric,−%et,0,[t,−3,3]]] plot2d(L,same_xy); (%o9) [/tmp/maxout113834.gnuplot_pipes] FĂŒggƑleges vonalak kĂ©pei L:create_list([parametric,realpart(exp(j/2+%i*t)),imagpart(exp(j/2+%i*t)),[t,-3,3]],j,-4,4); (%o38) [[parametric,%e−2*cos

t

,%e−2*sin

t

,[t,−3,3]],[parametric,%e−32*cos

t

,%e−32*sin

t

,[t,−3,3]],[parametric,%e−1*cos

t

,%e−1*sin

t

,[t,−3,3]],[parametric,cos

t

%e,sin

t

%e,[t,−3,3]],[parametric,cos

t

,sin

t

,[t,−3,3]],[parametric,%e*cos

t

,%e*sin

t

,[t,−3,3]],[parametric,%e*cos

t

,%e*sin

t

,[t,−3,3]],[parametric,%e32*cos

t

,%e32*sin

t

,[t,−3,3]],[parametric,%e2*cos

t

,%e2*sin

t

,[t,−3,3]]] plot2d(L,same_xy); (%o39) [/tmp/maxout126963.gnuplot_pipes] Az inverz fĂŒggvĂ©nyt is hasznĂĄlhatjuk ĂĄbrĂĄzolĂĄsra. A ,,nĂ©gyszögesedĂ©st" a nem elĂ©g jĂł felbontĂĄs okozza. contour_plot(min(3,max(-3,realpart(log(x+%i*y)))),[x,-9,9],[y,-9,9],[grid,209,209], [gnuplot_preamble,"set cntrparam levels 12"],[yx_ratio,1]); (%o3) [/tmp/maxout4933.gnuplot_pipes] A hamis vonalakat a kis x, y Ă©rtĂ©kek generĂĄljĂĄk. contour_plot(imagpart(log(x+%i*y)),[x,-9,9],[y,-9,9],[grid,209,209], [gnuplot_preamble,"set cntrparam levels 12"],[yx_ratio,1]); atan2: atan2(0,0) is undefined.(%o4) [/tmp/maxout4933.gnuplot_pipes] Az inverz fĂŒggvĂ©ny differenciĂĄlĂĄsi szabĂĄlya komplexben KövetkezmĂ©ny TĂ©tel Cauchy-Riemann-egyenletek A zĂ©rushelyek izolĂĄltsĂĄgĂĄnak elve Az analitikus folytatĂĄs elve A nyĂ­lt lekĂ©pezĂ©sek tĂ©tele KövetkezmĂ©ny: maximumelv Feladat solve(z^5=w,z); (%o1) [z=%e2*%i*%pi5*w15,z=%e4*%i*%pi5*w15,z=%e−4*%i*%pi5*w15,z=%e−2*%i*%pi5*w15,z=w15] Feladat Feladat Egyik sem. Feladat Ha k=-1, akkor nincs, egyĂ©bkĂ©nt van. Feladat Nem. Feladat Feladat Feladat Feladat Riemann-felĂŒletek Feladat Feladat solve(z=(a*z+b)/(c*z+d),z); (%o22) [z=−d2−2*a*d+4*b*c+a2+d−a2*c,z=d2−2*a*d+4*b*c+a2−d+a2*c] Feladat kill(all); (%o0) done Ez %i-t az origĂłba viszi: f(z):=(z-%i)/(z+%i); (%o17) f

z

:=z−%iz+%i Ha a f a z0-at az origĂłba viszi, akkor z0 konjugĂĄltjĂĄt a vĂ©gtelenbe, mivel a valĂłsak az egysĂ©gkörbe, pl. declare(z0,complex); (%o15) done f(z):=(z-z0)/(z-conjugate(z0)); (%o18) f

z

:=z−z0z−z0 Mivel a valĂłsak egysĂ©gnyi abszolĂșt Ă©rtĂ©kĂŒbe mennek, az ĂĄltalĂĄnos esetben lehet mĂ©g egy forgatĂĄs: f(z):=α*(z-z0)/(z-conjugate(z0)); (%o1) f

z

:=α*

z−z0

z−z0 ahol cabs(α)=1. Feladat ForgatĂĄsok. Az összes: ha z0 az origĂłba megy, akkor a tĂŒkörkĂ©pe, a konjugĂĄltjĂĄnak reciproka a ∞-be, Ă­gy f(z):=α*(z-z0)/(z-1/conjugate(z0)); (%o3) f

z

:=α*

z−z0

z−1z0 azaz f(z):=ÎČ*(z-z0)/(1-z*conjugate(z0)); (%o4) f

z

:=ÎČ*

z−z0

1−z*z0 declare(ÎČ,complex); (%o5) done Mivel exp(%i*φ) kĂ©pe egysĂ©gnyi abszolĂșt Ă©rtĂ©kƱ, cabs(ÎČ)=1; (%o6) ÎČ=1 Feladat kill(all); (%o0) done f(z):=(z+1/z)/2; (%o1) f

z

:=z+1z2 NyilvĂĄn z Ă©s 1/z kĂ©pe ugyan az, ezĂ©rt az egysĂ©gkör belsejĂ©t vesszĂŒk. A körök kĂ©pe: L:create_list([parametric, realpart(f(j/8*(cos(t)+%i*sin(t)))), imagpart(f(j/8*(cos(t)+%i*sin(t)))),[t,-%pi,%pi]],j,1,8)$ plot2d(L,same_xy); (%o3) [/tmp/maxout158490.gnuplot_pipes] A sugarak kĂ©pe: L:create_list([parametric, realpart(f(t*exp(%i*j*%pi/6))), imagpart(f(t*exp(%i*j*%pi/6))),[t,.1,1]],j,-5,6)$ plot2d(L,same_xy); (%o5) [/tmp/maxout158490.gnuplot_pipes] Feladat etc. Feladat (1) kill(all); (%o0) done f(z):=sin(z); (%o1) f

z

:=sin

z

 A vĂ­zszintes vonalak kĂ©pe: L:create_list([parametric,realpart(f(t+j/4*%i)),imagpart(f(t+j/4*%i)),[t,-3,3]],j,0,11)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: all values were clipped.(%o3) [/tmp/maxout60809.gnuplot_pipes] L:create_list([parametric, realpart(f(j/8*(cos(t)+%i*sin(t)))), imagpart(f(j/8*(cos(t)+%i*sin(t)))),[t,-%pi,%pi]],j,1,8)$ A fĂŒggƑleges vonalak kĂ©pe: L:create_list([parametric,realpart(f(j/2+%i*t)),imagpart(f(j/2+%i*t)),[t,-3,3]],j,-4,4)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.(%o8) [/tmp/maxout60809.gnuplot_pipes] etc. Feladat etc. Feladat f(z):=z/(1-z)^2; (%o6) f

z

:=z

1−z

2 A körök kĂ©pe: L:create_list([parametric, realpart(f(j/8*(cos(t)+%i*sin(t)))), imagpart(f(j/8*(cos(t)+%i*sin(t)))),[t,-%pi,%pi]],j,1,8)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.(%o13) [/tmp/maxout158490.gnuplot_pipes] A sugarak kĂ©pe: L:create_list([parametric, realpart(f(t*exp(%i*j*%pi/6))), imagpart(f(t*exp(%i*j*%pi/6))),[t,.1,1]],j,-5,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.(%o15) [/tmp/maxout158490.gnuplot_pipes] Cauchy-fĂ©le alaptĂ©tel TĂ©tel ElƑkĂ©szĂ­tƑ pĂ©lda DefinĂ­ciĂł Lemma TĂ©tel Feladat Feladat A körĂ­v mentĂ©n z:R*(cos(t)+%i*sin(t)); (%o63) R*

%i*sin

t

+cos

t

 diff(z,t); (%o93) R*

%i*cos

t

−sin

t

 %/(2*%i*R(cos(t)+%i*sin(t))); (%o94) −%i*R*

%i*cos

t

−sin

t

2*R

%i*sin

t

+cos

t

 factor(%); (%o95) %i*R*

sin

t

−%i*cos

t

2*R

%i*sin

t

+cos

t

 exponentialize(%); (%o96) %i*R*

−%i*

%e%i*t+%e−%i*t

2−%i*

%e%i*t−%e−%i*t

2

2*R

%e%i*t+%e−%i*t2+%e%i*t−%e−%i*t2

 trigreduce(%); (%o97) R*%e%i*t2*R

%e%i*t

 ami 1/2. Így 0=2*'integrate((exp(%i*t)-1)/(2*%i*t),t,0,R)+ 'integrate((exp(%i*R*(cos(t)+%i*sin(t)))-1)/2,t,0,%pi); (%o101) 0=0%pi%e%i*R*

%i*sin

t

+cos

t

−1dt2−%i*0R%e%i*t−1tdt Az elsƑ integrálban exp(%i*R*(cos(t)+%i*sin(t))); (%o105) %e%i*R*

%i*sin

t

+cos

t

 ratsimp(%); (%o106) %e%i*R*cos

t

−R*sin

t

 cabs(%); (%o107) %e−R*sin

t

 ami tart nullĂĄhoz, ha R tart vĂ©gtelenhez. Így a -1/2 integrĂĄlja marad, ami -π/2. A mĂĄsik integrĂĄl kĂ©pzetes rĂ©sze tehĂĄt π/2. KözvetlenĂŒl: integrate(sin(x)/x,x,0,inf); (%o4) %pi2 Feladat kill(all); (%o0) done A körĂ­ven az integrĂĄl 'integrate(exp(-R^2*exp(2*%i*t))*R*%i*exp(%i*t),t,0,%pi/4); (%o108) %i*R*0%pi4%e%i*t−R2*%e2*%i*tdt 0;cabs(exp(-R^2*exp(2*%i*t))*R*%i*exp(%i*t)); (%o110) 0(%o111) R*%e−R2*cos

2*t

 tart 0-hoz, ha R tart vĂ©gtelenhez. Így a kĂ©t fĂ©legyenesen vett integrĂĄl hatĂĄrĂ©rtĂ©ke ugyanaz. A ferdĂ©n: z:t*(1+%i)/sqrt(2); (%o52)

%i+1

*t2 diff(z,t); (%o53) %i+12 -z^2; (%o54) −

%i+1

2*t22 ratsimp(%); (%o55) −%i*t2 0;I:'integrate(exp(-z^2)*(1+%i)/sqrt(2),t,0,inf); (%o56) 0(%o57)

%i+1

*0inf%e−

%i+1

2*t22dt2 0;I:trigreduce(I); (%o58) 0(%o59) %i*0inf%e−%i*t2dt2+0inf%e−%i*t2dt2 0;I:exponentialize(I); (%o60) 0(%o61) %i*0inf%e−%i*t2dt2+0inf%e−%i*t2dt2 I:demoivre(%); (%o62) %i*0infcos

t2

−%i*sin

t2

dt2+0infcos

t2

−%i*sin

t2

dt2 I:ratsimp(%); (%o63)

2*%i+2

*0infcos

t2

−%i*sin

t2

dt2 Ir:realpart(I); (%o64) 2*0infsin

t2

dt+2*0infcos

t2

dt2 Ii:imagpart(I); (%o65) 2*0infcos

t2

dt−2*0infsin

t2

dt2 A kettƑ egyenlƑ, összegĂŒk integrate(exp(-t^2),t,0,inf)*sqrt(2); (%o67) %pi2 KözvetlenĂŒl: integrate(cos(x^2),x,0,inf); (%o2) %pi232 integrate(sin(x^2),x,0,inf); (%o3) %pi232 Feladat TĂ©tel Cauchy-fĂ©le integrĂĄlformulĂĄk Morera tĂ©tele Cauchy-egyenlƑtlensĂ©gek Liouville tĂ©tele Az algebra alaptĂ©tele MegjegyzĂ©s Feladat diff(asinh(x),x); (%o27) 1x2+1 taylor(asinh(x),x,0,10); (%o25)/T/ x−x36+3*x540−5*x7112+35*x91152+... Schwarz-lemma TĂ©tel DefinĂ­ciĂł TĂ©tel Konform ekvivalencia Riemann tĂ©tele Feladat A bevĂĄgott sĂ­kon a fĂŒggvĂ©ny a 2*log(z) valĂłs rĂ©sze, Ă­gy a harmonikus tĂĄrs konstanstĂłl eltekintve 2*arg(z), Ă­gy itt is csak ez lehetne, de nem folytonos. MĂĄskĂ©nt a harmonikus tĂĄrs kiszĂĄmolĂĄsĂĄsra hasznĂĄlt fĂŒggvĂ©ny körintegrĂĄlja nem nulla, Ă­gy az integrĂĄl fĂŒgg az ĂșttĂłl. Feladat (1) u:x^2-y^2+2*x; (%o1) −y2+x2+2*x ux:diff(u,x); (%o2) 2*x+2 uy:diff(u,y); (%o3) −2*y g:ux-%i*uy; (%o16) 2*%i*y+2*x+2 gx:subst(x=t,%); (%o17) 2*%i*y+2*t+2 gx:subst(y=0,gx); (%o18) 2*t+2 Ix:integrate(gx,t,0,x); (%o19) x2+2*x gy:subst(y=t,g); (%o20) 2*x+2*%i*t+2 Iy:integrate(gy*%i,t,0,y); (%o21) %i*

%i*y2+

2*x+2

*y

 G:ratsimp(Ix+Iy); (%o22) −y2+

2*%i*x+2*%i

*y+x2+2*x w:realpart(G); (%o23) −y2+x2+2*x v:imagpart(G); (%o24)

2*x+2

*y wx:diff(w,x); wy:diff(w,y); (%o25) 2*x+2(%o26) −2*y vx:diff(v,x); vy:diff(v,y); (%o28) 2*y(%o29) 2*x+2 etc. Feladat (1) kill(all); (%o0) done c:3.1; (%o23) 3.1 A töltĂ©sek az 1 Ă©s -1 pontban. A komplex potenciĂĄl: p:%i*log(x+%i*y-1)-%i*log(x+%i*y+1); (%o24) %i*log

%i*y+x−1

−%i*log

%i*y+x+1

 Az ekvipotenciĂĄlis vonalak (körök): contour_plot(min(c,max(-c,imagpart(p))),[x,-3,3],[y,-3,3],[grid,209,209], [gnuplot_preamble,"set cntrparam levels 13"]); rat: replaced 3.1 by 31/10 = 3.1rat: replaced −3.1 by −31/10 = −3.1rat: replaced 3.1 by 31/10 = 3.1(%o27) [/tmp/maxout4933.gnuplot_pipes] Az erƑvonalak: contour_plot(realpart(p),[x,-3,3],[y,-3,3],[grid,209,209], [gnuplot_preamble,"set cntrparam levels 13"]); (%o26) [/tmp/maxout4933.gnuplot_pipes] (2) kill(all); (%o0) done A kĂ©t ponttöltĂ©s komplex potenciĂĄljĂĄt hasznĂĄlhatjuk, ha az r sugarĂș c közĂ©pponjĂș körhöz a p-t Ășgy vĂĄlasztjuk, hogy a p>0 pontban lĂ©vƑ töltĂ©s a körre valĂł tĂŒkrözĂ©snĂ©l a -p pontba menjen: r^2=(c-p)*(c+p); (%o3) r2=

c−p

*

p+c

 solve(%,p); (%o4) [p=−c2−r2,p=c2−r2] (3) c1^2-r1^2=c2^2-r2^2; (%o5) c12−r12=c22−r22 solve(%,c2); (%o6) [c2=−r22−r12+c12,c2=r22−r12+c12] Feladat A hƑáramlĂĄs olyan mint az elektromos ĂĄramlĂĄs. Ha p=u+%i*v a komplex potenciĂĄl, akkor a potenciĂĄl negatĂ­v gradiense, a tĂ©rerƑssĂ©g E=-vx-%i*vy=%i*conjugate(p'(z)). Az koncentrikus kört vigyĂŒk ĂĄt excentrikus körbe. Feladat kill(all); (%o0) done f(z):=z+exp(z); (%o1) f

z

:=z+exp

z

 L1:create_list([parametric,realpart(f(t+%pi*j/6*%i)),imagpart(f(t+%pi*j/6*%i)),[t,-5,3]],j,-6,6)$ plot2d(L1,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.(%o3) [/tmp/maxout25336.gnuplot_pipes] L2:create_list([parametric,realpart(f(j+t*%i)),imagpart(f(j+t*%i)),[t,-%pi,%pi]],j,-5,5)$ plot2d(L2,[x,-5.1,5.1],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: all values were clipped.plot2d: all values were clipped.plot2d: all values were clipped.(%o6) [/tmp/maxout25336.gnuplot_pipes] Az f inverze összerakva a vĂ©gtelen sĂ­kkondenzĂĄtor p(w)=w potenciĂĄljĂĄval a fĂ©lsĂ­k kondenzĂĄtor potenciĂĄljĂĄt adja, Ă­gy a fenti ĂĄbrĂĄk annak az ekvipotenciĂĄlis vonalait illetve erƑvonalait mutatjĂĄk. L:append(L1,L2)$ plot2d(L,[x,-5.1,5.1],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: all values were clipped.plot2d: all values were clipped.plot2d: all values were clipped.(%o9) [/tmp/maxout25336.gnuplot_pipes] Feladat kill(all); (%o0) done Ha a feszĂŒltsĂ©g egyezƑ elƑjelƱ lenne, az f(z):=z+1/z; (%o1) f

z

:=z+1z lekĂ©pezĂ©st hasznĂĄlhatnĂĄnk. kill(all); (%o0) done TekintsĂŒk a F(z):=(z+a)/(z-a); (%o1) F

z

:=z+az−a lekĂ©pezĂ©st. Ez a kĂ©tszeresen felhasĂ­tott sĂ­kot az egyszeresen (a nemnegatĂ­v fĂ©legyenessel) felhasĂ­tott sĂ­kba viszi, sqrt(-%i); (%o2) −%i rectform(%); (%o3) 12−%i2 Ezt a nĂ©gyetgyökvonĂĄs a felsƑ fĂ©lsĂ­kba viszi, de elƑbb el kell forgatni, majd vissza: G(z):=%i*sqrt(-z); (%o4) G

z

:=%i*−z G(F(z)); (%o5) %i*−z+az−a a kĂ©pe legyen -1, -a kĂ©pe legyen 1: H(z):=(z-1)/(z+1); (%o6) H

z

:=z−1z+1 H(G(F(z))); (%o7) %i*−z+az−a−1%i*−z+az−a+1 e:radcan(%); (%o8) −a−z−%i*z+a%i*z+a+a−z expand(%); (%o9) %i*z+a%i*z+a+a−z−a−z%i*z+a+a−z ratsimp(%); (%o28) −a%i*a−z*z+a−z a:1; (%o29) 1 f:log(((x+%i*y)+%i*sqrt(1-(x+%i*y)^2)))-%pi*%i/2; (%o14) log

%i*1−

%i*y+x

2+%i*y+x

−%i*%pi2 Az ekvipotenciális vonalak: contour_plot(imagpart(f),[x,-3,3],[y,-3,3],[grid,209,209], [gnuplot_preamble,"set cntrparam levels 15"]); (%o16) [/tmp/maxout63527.gnuplot_pipes] Az erƑvonalak: contour_plot(realpart(f),[x,-3,3],[y,-3,3],[grid,209,209], [gnuplot_preamble,"set cntrparam levels 15"]); (%o15) [/tmp/maxout63527.gnuplot_pipes] Feladat (1/1) kill(all); (%o0) done f(z):=(z-1)/(z+1); (%o1) f

z

:=z−1z+1 L:create_list([parametric,realpart(f(t+j/4*%i)),imagpart(f(t+j/4*%i)),[t,-3,3]],j,-6,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.(%o5) [/tmp/maxout172749.gnuplot_pipes] A fĂŒggƑleges vonalak kĂ©pe: L:create_list([parametric,realpart(f(j/2+%i*t)),imagpart(f(j/2+%i*t)),[t,-3,3]],j,-6,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.(%o9) [/tmp/maxout172749.gnuplot_pipes] (1/2) kill(all); (%o0) done f(z):=z^2; (%o1) f

z

:=z2 L:create_list([parametric,realpart(f(t+j/4*%i)),imagpart(f(t+j/4*%i)),[t,-3,3]],j,-6,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.(%o3) [/tmp/maxout172749.gnuplot_pipes] A fĂŒggƑleges vonalak kĂ©pe: L:create_list([parametric,realpart(f(j/2+%i*t)),imagpart(f(j/2+%i*t)),[t,-3,3]],j,-6,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: all values were clipped.plot2d: all values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: all values were clipped.plot2d: all values were clipped.(%o5) [/tmp/maxout172749.gnuplot_pipes] (1/3) kill(all); (%o0) done z=(w-2)/(w+2); (%o1) z=w−2w+2 solve(%,w); (%o2) [w=−2*z+2z−1] f(z):=-(2*z+2)/(z-1); (%o4) f

z

:=−

2*z+2

z−1 L:create_list([parametric,realpart(f(t+j/4*%i)),imagpart(f(t+j/4*%i)),[t,-3,3]],j,-6,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.(%o6) [/tmp/maxout172749.gnuplot_pipes] A fĂŒggƑleges vonalak kĂ©pe: L:create_list([parametric,realpart(f(j/2+%i*t)),imagpart(f(j/2+%i*t)),[t,-3,3]],j,-6,6)$ plot2d(L,[x,-5,5],[y,-5,5],same_xy); plot2d: some values were clipped.plot2d: expression evaluates to non−numeric value somewhere in plotting range.plot2d: some values were clipped.plot2d: some values were clipped.plot2d: some values were clipped.(%o8) [/tmp/maxout172749.gnuplot_pipes] (2) g(z):=z^2; (%o10) g

z

:=z2 h(z):=(z-1)/(z+1); (%o11) h

z

:=z−1z+1 f(g(h(z))); (%o12) −2*

z−1

2

z+1

2−2

z−1

2

z+1

2−1 ratsimp(%); (%o13) z2+1z (3) Ha a kör közĂ©ppontja az %ih pontban van, akkor a kĂ©pe f(z):=z+1/z; (%o23) f

z

:=z+1z h:0.3; (%o41) 0.3 r:sqrt(1+h^2); (%o42) 1.044030650891055 z:r*cos(t)+%i*r*sin(t); (%o43) 1.044030650891055*%i*sin

t

+1.044030650891055*cos

t

 plot2d([parametric,realpart(f(r*cos(t)+%i*(h+r*sin(t)))), imagpart(f(r*cos(t)+%i*(h+r*sin(t)))),[t,-%pi,%pi]], [x,-3,3],[y,-3,3],same_xy); (%o47) [/tmp/maxout172749.gnuplot_pipes] a kĂŒlsejĂ©nek a kĂ©pe ennek a kĂŒlseje. h:0.0; (%o48) 0.0 r:sqrt(1+h^2); (%o49) 1.0 plot2d([parametric,realpart(f(r*cos(t)+%i*(h+r*sin(t)))), imagpart(f(r*cos(t)+%i*(h+r*sin(t)))),[t,-%pi,%pi]], [x,-3,3],[y,-3,3],same_xy); (%o50) [/tmp/maxout172749.gnuplot_pipes] (4) Legyen a nagyobb kör közĂ©ppontja d tĂĄvolsĂĄgban a kisebb kör közĂ©ppontjĂĄtĂłl, az u+%i*v pontban. kill(h,r,d,u,v); (%o78) done h:0.3; (%o79) 0.3 d:0.1; (%o80) 0.1 r:d+sqrt(1+h^2); (%o81) 1.144030650891055 v:h*(1-u); (%o82) 0.3*

1−u

 u:1-r/sqrt(1+h^2); (%o83) −0.09578262852211528 v:h*(1-u); (%o84) 0.3287347885566345 plot2d([parametric,realpart(f(u+r*cos(t)+%i*(v+r*sin(t)))), imagpart(f(u+r*cos(t)+%i*(v+r*sin(t)))),[t,-%pi,%pi]], [x,-3,3],[y,-3,3],same_xy); (%o85) [/tmp/maxout172749.gnuplot_pipes] Ha h=0, a Zsukovszkij-kormĂĄnyt kapjuk: kill(h,r,d,u,v); (%o86) done h:0; (%o87) 0 d:0.1; (%o93) 0.1 r:d+sqrt(1+h^2); (%o94) 1.1 v:h*(1-u); (%o95) 0 u:1-r/sqrt(1+h^2); (%o96) −0.1 v:h*(1-u); (%o97) 0 plot2d([parametric,realpart(f(u+r*cos(t)+%i*(v+r*sin(t)))), imagpart(f(u+r*cos(t)+%i*(v+r*sin(t)))),[t,-%pi,%pi]], [x,-3,3],[y,-3,3],same_xy); (%o98) [/tmp/maxout172749.gnuplot_pipes] Feladat (1) NĂ©gyzetreemelĂ©ssel a felsƑ fĂ©lsĂ­kba, az pedig az egysĂ©gkörre. (2) ForgatĂĄs, gyökvonĂĄs, forgatĂĄs. (3) ElƑször x-et nullĂĄba, majd nĂ©gyzetgyökkel fĂ©lkörbe, a fĂ©lkört forgassuk el. A felsƑ fĂ©lkört a sin a felsƑ fĂ©lsĂ­kba viszi. (4) Az z+1/z lekĂ©pezĂ©s a kör belsejĂ©t a sĂ­kba visz, a kerĂŒletĂ©t a -2,2 szakaszba. A bevĂĄgĂĄsok a szakasz hosszabbĂ­tĂĄsĂĄsba mennek. Egy konstansszoros inverzĂ©vel ezt visszavisszĂŒk a körbe. (5) A nĂ©gyzetreemelĂ©s bemetszett sĂ­kba viszi. EltolĂĄs utĂĄn nĂ©gyzetgyökvonĂĄs a felsƑ fĂ©lsĂ­kba. (6) EltolĂĄs, majd mint az egyik kondenzĂĄtornĂĄl fĂ©lsĂ­kba. (7) exp (4)-be viszi. Feladat Feladat Weierstrass tĂ©tele Feladat PK0ilXńB–HmimetypePK0ilXQdBV55 5format.txtPK0ilXV|ą|R|R ’content.xmlPK§7Y