PKdlXńB–Hmimetypetext/x-wxmathmlPKdlXQdBV55 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dlXBË{]ú]ú 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©g Fourier-elmĂ©let Komplex fĂŒggvĂ©nytan Holomorf fĂŒggvĂ©nyek Meromorf fĂŒggvĂ©nyek Laurent-sor FĂŒggvĂ©ny rendje egy pontban ReziduumtĂ©tel MegjegyzĂ©s Meromorf fĂŒggvĂ©nyek MegjegyzĂ©s Argumentum-elv KövetkezmĂ©ny RouchĂ© tĂ©tele A gamma-fĂŒggvĂ©ny Feladat Feladat kill(all); (%o0) done (1) -4*%pi*%i*residue(exp(z)/z^2,z,0); (%o1) −4*%i*%pi (2) solve(z^4+1); (%o4) [z=

−1

14
*%i,z=−

−1

14
,z=−

−1

14
*%i,z=

−1

14
]
residue(1/(z^4+1),z,rhs(%[1]))+residue(1/(z^4+1),z,rhs(%[4])); (%o5) −%i+1252−%i−1252 2*%pi*%i*%; (%o6) 2*%i*

−%i+1252−%i−1252

*%pi
ratsimp(%); (%o7) %pi2 etc. Feladat Feladat (1) etc. (2) IntegĂĄljunk a negatĂ­v fĂ©legyenes felsƑ Ă©s alsĂł partjĂĄn. Feladat kill(all); (%o0) done f:x^(2*m)-1; (%o82) x2*m−1 g:x^(2*n)-1; (%o83) x2*n−1 f/g; (%o84) x2*m−1x2*n−1 Mivel a tört egyszerƱsĂ­thetƑ x-1-el, az elƑzƑ feladat eredmĂ©nye hasynĂĄlhatĂł, de a reziduumokat az exp(%pi*%i*k/n), k=1,...,n-1 helyeken az eredeti alakbĂłl Ă©rdemes szĂĄmolni. diff(g,x); (%o85) 2*n*x2*n−1 h:f/%; (%o86) x1−2*n*

x2*m−1

2*n
subst(x=exp(%pi*%i*k/n),h); (%o87)

%e2*%i*%pi*k*mn−1

*%e%i*%pi*k*

1−2*n

n
2*n
expand(%); (%o88) %e2*%i*%pi*k*mn+%i*%pi*kn−2*%i*%pi*k2*n−%e%i*%pi*kn−2*%i*%pi*k2*n ratsimp(%); (%o89) %e−2*%i*%pi*k*

%e2*%i*%pi*k*mn+%i*%pi*kn−%e%i*%pi*kn

2*n
A szorzĂł tĂ©nyezƑ 1, mivel k egĂ©sz. KĂ©t mĂ©rtani sor összegĂ©t kell kiszĂĄmolni. Az elsƑ: q:exp((2*m+1)*%pi*%i/n); (%o42) %e%i*%pi*

2*m+1

n
sm:(q^n-q)/(q-1); (%o90) −%e%i*%pi*

2*m+1

n
−%e2*%i*%pi*m
%e%i*%pi*

2*m+1

n
−1
f0:first(sm); g0:second(sm); (%o91) −%e%i*%pi*

2*m+1

n
−%e2*%i*%pi*m(%o92) %e%i*%pi*

2*m+1

n
−1
Szorzunk a komplex konjugálttal: sm0:%/(conjugate(q)-1); (%o44) −%e%i*%pi*

2*m+1

n
−%e2*%i*%pi*m

%e−%i*%pi*

2*m+1

n
−1

*

%e%i*%pi*

2*m+1

n−1

f1:first(sm0)*(conjugate(q)-1); (%o120)

%e−%i*%pi*

2*m+1

n−1

*

−%e%i*%pi*

2*m+1

n−%e2*%i*%pi*m

g1:g0*conjugate(q-1); (%o107)

%e−%i*%pi*

2*m+1

n−1

*

%e%i*%pi*

2*m+1

n−1

realpart(%); (%o108) sin

%pi*

2*m+1

n

2
+

cos

%pi*

2*m+1

n

−1

2
expand(%); (%o109) sin

2*%pi*mn+%pin

2
+cos

2*%pi*mn+%pin

2
−2*cos

2*%pi*mn+%pin

+1
g2:trigsimp(%); (%o110) 2−2*cos

2*%pi*m+%pin

Fel fogjuk hasznĂĄlni az alĂĄbbi azonossĂĄgot: 1-cos(2*α); (%o139) 1−cos

2*α

trigsimp(%); (%o140) 1−cos

2*α

trigexpand(%); (%o141) sin

α

2
−cos

α

2
+1
trigsimp(%); (%o142) 2*sin

α

2
Innen a nevezƑ: g3:4*sin(%pi*(2*m+1)/2/n)^2; (%o143) 4*sin

%pi*

2*m+1

2*n

2
ValĂłban, trigreduce(%); (%o144) 4*

12−cos

2*%pi*mn+%pin

2

ratsimp(%); (%o145) 2−2*cos

2*%pi*m+%pin

A szĂĄmlĂĄlĂł: f1; (%o146)

%e−%i*%pi*

2*m+1

n−1

*

−%e%i*%pi*

2*m+1

n−%e2*%i*%pi*m

expand(f1); (%o147) %e2*%i*%pi*mn+%i*%pin−%e−2*%i*%pi*mn−%i*%pin+2*%i*%pi*m+%e2*%i*%pi*m−1 NekĂŒnk csak a kĂ©pzetes rĂ©sz kell: f2:imagpart(%); (%o148) sin

2*%pi*mn+%pin−2*%pi*m

+sin

2*%pi*mn+%pin

+sin

2*%pi*m

Mivel m egész, ez 2*sin(2*α); (%o149) 2*sin

2*α

trigexpand(%); (%o150) 4*cos

α

*sin

α

azaz f3:subst(α=%pi*(2*m+1)/n,%); (%o151) 4*cos

%pi*

2*m+1

n

*sin

%pi*

2*m+1

n

Innen az összeg képzetes része f3/g3; (%o157) cos

%pi*

2*m+1

n

*sin

%pi*

2*m+1

n

sin

%pi*

2*m+1

2*n

2
ami cot(%pi*(2*m+1)/n); (%o155) cot

%pi*

2*m+1

n

trigsimp(%); (%o156) cos

2*%pi*m+%pin

sin

2*%pi*m+%pin

A mĂĄsik összeg az m=0 speciĂĄlis eset, Ă­gy adĂłdik az eredmĂ©ny. Feladat (1) kill(all); (%o0) done assume(a>0); (%o8) [a>0] 2*%pi*%i*residue(z^2/(z^2+a^2)^2,z,%i*a); (%o10) %pi2*a etc. (4) kill(all); (%o0) done 2*%pi*%i*residue(1/(z^2+1),z,%i); (%o1) %pi 2*%pi*%i*residue(1/(z^2+1)^2,z,%i); (%o2) %pi2 2*%pi*%i*residue(1/(z^2+1)^3,z,%i); (%o3) 3*%pi8 TetszƑleges n-re 1/(z+%i)^n n-1-edik derivĂĄltja /(n-1)! az %i helyen a reziduum, azaz (-n)(-n-1)*...*(-2*n+1)/(n-1)!/(2*%i)^(2*n-1) (5) kill(all); (%o0) done rs:solve(z^3+1); (%o1) [z=−3*%i−12,z=3*%i+12,z=−1] z0:rhs(rs[3]); z1:rhs(rs[2]); z2:rhs(rs[1]); (%o2) −1(%o3) 3*%i+12(%o4) −3*%i−12 f:log(-z)/(z^3+1); (%o5) log

−z

z3+1
r0:residue(f,z,z0); (%o7) 0 r1:residue(f,z,z1); (%o8) −3*%i*

log

3*%i+1

−log

2

+log

−1

+log

3*%i+1

−log

2

+log

−1

6
r2:residue(f,z,z2); (%o9) −log

3*%i−1

+3*%i*

log

2

−log

3*%i−1

−log

2

6
r0 O.K., de nem vlågos, hogy log(-1) mi? Måsként: f1:log(-z)/(z-z0)/(z-z2); (%o10) log

−z

z+1

*

z+3*%i−12

r1:subst(z=z1,f1); (%o11) log

−3*%i+12

3*%i+12+1

*

3*%i+12+3*%i−12

carg(-z1); (%o12) −2*%pi3 r1:subst(log(-z1)=carg(-z1),r1); (%o13) −2*%pi3*

3*%i+12+1

*

3*%i+12+3*%i−12

f2:log(-z)/(z-z0)/(z-z1); (%o14) log

−z

z+1

*

z−3*%i+12

r2:subst(z=z2,f2); (%o15) log

3*%i−12

1−3*%i−12

*

−3*%i+12−3*%i−12

r2:subst(log(-z2)=carg(-z2),r2); (%o16) 2*%pi3*

1−3*%i−12

*

−3*%i+12−3*%i−12

r:-%i*(r1+r2); (%o17) −%i*

2*%pi3*

1−3*%i−12

*

−3*%i+12−3*%i−12

−2*%pi3*

3*%i+12+1

*

3*%i+12+3*%i−12

ratsimp(%); (%o18) 2*%pi332 integrate(1/(x^3+1),x,0,inf); (%o19) 2*%pi332 (6) PĂĄros, Ă©s a szĂĄmlĂĄlĂł x^(2*m1-1)-(x^(2*m2)-1). (7) PĂĄros Ă©s a nevezƑnek nincs valĂłs zĂ©rushelye. Feladat (1) kill(all); (%o0) done f:exp(a*t)/(1+exp(t)); (%o1) %ea*t%et+1 ff:subst(t=t+2*%pi*%i,f); (%o4) %ea*

t+2*%i*%pi

%et+1
f-ff; (%o36) %ea*t%et+1−%ea*

t+2*%i*%pi

%et+1
ratsimp(%); (%o37) −

%e2*%i*%pi*a−1

*%ea*t
%et+1
residue(f,t,%i*%pi); (%o54) −%e%i*%pi*a %*2*%pi*%i=I*(1-exp(2*%pi*%i*a)); (%o55) −2*%i*%pi*%e%i*%pi*a=I*

1−%e2*%i*%pi*a

solve(%,I); (%o56) [I=2*%i*%pi*%e%i*%pi*a%e2*%i*%pi*a−1] realpart(%); (%o57) [I=2*%pi*

cos

%pi*a

*sin

2*%pi*a

−sin

%pi*a

*

cos

2*%pi*a

−1

sin

2*%pi*a

2
+

cos

2*%pi*a

−1

2
]
demoivre(%); (%o58) [I=2*%pi*

cos

%pi*a

*sin

2*%pi*a

−sin

%pi*a

*

cos

2*%pi*a

−1

sin

2*%pi*a

2
+

cos

2*%pi*a

−1

2
]
trigreduce(%); (%o59) [I=−2*%pi*sin

%pi*a

cos

2*%pi*a

−1
]
trigexpand(%); (%o60) [I=−2*%pi*sin

%pi*a

−sin

%pi*a

2
+cos

%pi*a

2
−1
]
trigsimp(%); (%o61) [I=%pisin

%pi*a

]
(2) kill(all); (%o0) done f:(exp(a1*t)-exp(a2*t))/(1-exp(t)); (%o1) %ea1*t−%ea2*t1−%et limit(f,t,0); (%o2) a2−a1 ff:subst(t=t+%pi*%i,f); (%o3) %ea1*

t+%i*%pi

−%ea2*

t+%i*%pi

%et+1
Kicsit hanyagul: I-ff=0; (%o4) I−%ea1*

t+%i*%pi

−%ea2*

t+%i*%pi

%et+1
=0
demoivre(%); (%o5) I−

%i*sin

%pi*a1

+cos

%pi*a1

*%ea1*t−

%i*sin

%pi*a2

+cos

%pi*a2

*%ea2*t%et+1
=0
subst(exp(a1*t)=%pi*(exp(t)+1)/sin(%pi*a1),%); (%o6) I−%pi*

%i*sin

%pi*a1

+cos

%pi*a1

*

%et+1

sin

%pi*a1

−

%i*sin

%pi*a2

+cos

%pi*a2

*%ea2*t%et+1=0
subst(exp(a2*t)=%pi*(exp(t)+1)/sin(%pi*a2),%); (%o7) I−%pi*

%i*sin

%pi*a1

+cos

%pi*a1

*

%et+1

sin

%pi*a1

−%pi*

%i*sin

%pi*a2

+cos

%pi*a2

*

%et+1

sin

%pi*a2

%et+1=0
solve(%,I); (%o8) [I=%pi*cos

%pi*a1

*sin

%pi*a2

−%pi*sin

%pi*a1

*cos

%pi*a2

sin

%pi*a1

*sin

%pi*a2

]
Feladat (1) x=exp(t) helyettesítéssel (2) kill(all); (%o0) done Kiszåmítjuk a 'integrate((cos(A*x)-cos(B*x))/x^2,x,0,inf); (%o23) 0infcos

A*x

−cos

B*x

x2
dx
integrĂĄlt, majd legyen A=0, B=2. Ennek a fele a keresett integrĂĄl. Az integrĂĄl kiszĂĄmĂ­tĂĄsĂĄra integrĂĄljunk egy 0 közĂ©ppontĂș kis Ă©s egy nagy fĂ©lkörön meg kĂ©t egyenes szakaszon: assume(A>=0); (%o44) [A>=0] assume(B>=0); (%o45) [B>=0] f:(exp(%i*A*z)-exp(%i*B*z))/z^2; (%o31) %e%i*A*z−%e%i*B*zz2 w:R*exp(%i*t); (%o32) R*%e%i*t dw:diff(w,t); (%o33) %i*R*%e%i*t 0;dw*subst(z=R*exp(%i*t),exp(%i*A*z)/z^2); (%o37) 0(%o38) %i*%e%i*A*R*%e%i*t−%i*tR cabs(%); (%o39) %e−A*R*sin

t

R
A nagy félkörön vett integrål nullåhoz tart. taylor(exp(A*%i*z)-exp(B*%i*z),z,0,3); (%o42)/T/

%i*A−B*%i

*z−

A2−B2

*z2
2
−

%i*A3−B3*%i

*z3
6
+...
%/z^2; (%o43)/T/

A−B

*%i
z
−A2−B22−

A3−B3

*%i*z
6
+...
A kis félkörön vett integrål (A-B)*%i-szer 1/z integråljåhoz tart, ami -%pi%i. Tehåt a keresett integrål (B-A)*%pi/2; (%o46) %pi*

B−A

2
Röviden: integrate((sin(x))^2/x^2,x,0,inf); (%o20) %pi2 (3) Az elƑzƑbƑl, felhasznĂĄlva, hogy cos((b-a)*x)-cos((b+a)*x); (%o58) cos

b−a

*x

−cos

b+a

*x

trigexpand(%); (%o59) cos

b−a

*x

−cos

b+a

*x

expand(%); (%o60) cos

b*x−a*x

−cos

b*x+a*x

trigexpand(%); (%o61) 2*sin

a*x

*sin

b*x

A=b-a; B=b+a; (%o62) A=b−a(%o63) B=b+a ahonnan az integrĂĄl %pi*a/2; (%o64) %pi*a2 Röviden: kill(all); (%o0) done assume(b-a>0,a>0); (%o1) [b>a,a>0] integrate(sin(a*x)*sin(b*x)/x^2,x,0,inf); (%o2) %pi*a2 (4) kill(all); (%o0) done integrate((sin(x))^3/x^3,x,0,inf); (%o2) 3*%pi8 (5) kill(all); (%o0) done assume(a>0); (%o1) [a>0] %pi*%i*residue(exp(%i*z)/(z^2+a^2),z,%i*a); (%o6) %pi*%e−a2*a f:cos(x)/(x^2+a^2); (%o2) cos

x

x2+a2
integrate(f,x,0,inf); (%o3) %pi*%e−a2*a (6) kill(all); (%o0) done assume(a>0); (%o1) [a>0] %pi*residue((exp(%i*z)-1)/(z*(z^2+a^2)),z,%i*a); (%o8) %pi*%e−a*

%ea−1

2*a2
expand(%); (%o9) %pi2*a2−%pi*%e−a2*a2 integrate(sin(x)/(x*(x^2+a^2)),x,0,inf); (%o6) %e−a*

%pi*%ea−%pi

2*a2
(7) kill(all); (%o0) done assume(a>0); (%o1) [a>0] 0;f:exp(-x^2/2)*cos(a*x); (%o2) 0(%o3) %e−x22*cos

a*x

0;integrate(f,x,0,inf); (%o4) 0(%o5) %pi*%e−a222 IntegrĂĄljuk az alĂĄbbi fĂŒggvĂ©nyt egy tĂ©glalap mentĂ©n: 0;f:exp(-z^2/2); (%o23) 0(%o24) %e−z22 0;f0:subst(z=t,f); (%o25) 0(%o26) %e−t22 0;f1:subst(z=t+%i*a,f); (%o27) 0(%o28) %e−

t+%i*a

2
2
0;f1:expand(f1); (%o29) 0(%o30) %e−t22−%i*a*t+a22 0;f2:realpart(f1); (%o31) 0(%o32) %ea22−t22*cos

a*t

0;eq:2*exp(a^2/2)*I=integrate(exp(-t^2/2),t,minf,inf); (%o39) 0(%o40) 2*I*%ea22=2*%pi 0;solve(eq,I); (%o42) 0(%o43) [I=%pi*%e−a222] Feladat Feladat: ÎČ-integrĂĄl Feladat IndukciĂłval Elliptikus integrĂĄlok Ă©s elliptikus fĂŒggvĂ©nyek
PKdlXńB–HmimetypePKdlXQdBV55 5format.txtPKdlXBË{]ú]ú ’content.xmlPK§