Kalkulus II.
J\303\241rai Antal
Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak
1. Line\303\241ris algebra
3. F\303\274ggv\303\251nysorozatok \303\251s f\303\274ggv\303\251nysorok
3.1. Pontonk\303\251nti \303\251s egyenletes konvergencia
restart;
3.1.1. F\303\274ggv\303\251nyterek, f\303\274ggv\303\251nysorozatok.
3.1.2. Cauchy-krit\303\251rium f\303\274ggv\303\251nysorozatokra.
3.1.3. F\303\274ggv\303\251nysorok.
3.1.4. Weierstrass-krit\303\251rium.
*3.1.9. Monoton konvergencia t\303\251tel.
*3.1.10. Domin\303\241lt konvergencia t\303\251tel.
3.1.11. K\303\266vetkezm\303\251ny: hat\303\241r\303\241tmenet \303\251s integr\303\241l felcser\303\251l\303\251se.
3.1.12. K\303\266vetkezm\303\251ny.
3.1.15. F\303\274ggv\303\251nysorozat tagonk\303\251nti differenci\303\241l\303\241sa.
*3.1.17. Weierstrass approxim\303\241ci\303\263s t\303\251tele.
f:=x->sin(Pi*x);
B:=(n,x)->sum(binomial(n,k)*f(k/n)*x^k*(1-x)^(n-k),k=0..n);
plot({f(x),B(2,x),B(10,x),B(30,x)},x=0..1);
3.2. Hatv\303\241nysorok
restart;
3.2.1. Defin\303\255ci\303\263.
3.2.2. Cauchy-Hadamard-t\303\251tel.
3.2.3. Defin\303\255ci\303\263.
3.2.4. K\303\266vetkezm\303\251ny.
3.2.6. Taylor-t\303\251tel.
*3.2.8. Megjegyz\303\251s.
3.2.10. Az exponenci\303\241lis f\303\274ggv\303\251ny, trigonometrikus \303\251s hiperbolikus f\303\274ggv\303\251nyek.
sum(z^n/n!,n=0..infinity); exp(z); exp(z1+z2); expand(%);
plots[complexplot](exp(I*t),t=0..1,x=0..1,y=0..1);
(exp(I*z)+exp(-I*z))/2; convert(%,trig);
(exp(I*z)-exp(-I*z))/(2*I); convert(%,trig);
sum((-1)^n*z^(2*n)/(2*n)!,n=0..infinity);
sum((-1)^n*z^(2*n+1)/(2*n+1)!,n=0..infinity);
exp(I*t); convert(%,trig);
cos(z)^2+sin(z)^2; simplify(%);
cos(z1+z2); expand(%); sin(z1+z2); expand(%);
(exp(z)+exp(-z))/2; convert(%,trigh);
(exp(z)-exp(-z))/2; convert(%,trigh);
sum(z^(2*n)/(2*n)!,n=0..infinity);
sum(z^(2*n+1)/(2*n+1)!,n=0..infinity);
cosh(z)^2-sinh(z)^2; simplify(%);
cosh(z1+z2); expand(%); sinh(z1+z2); expand(%);
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3.3. Fourier-sorok
restart;
3.3.1. \303\201ltal\303\241nos\303\255tott Pitagorasz-t\303\251tel.
3.3.5. Defin\303\255ci\303\263.
3.3.6. Defin\303\255ci\303\263.
3.3.7. Riesz-Fischer-t\303\251tel.
3.3.8. Klasszikus Fourier-sorok.
f:=x->abs(x); l:=1/2; a0:=1/l*int(f(x),x=-l..l);
a:=k->(1/l*int(f(x)*cos(Pi*k*x/l),x=-l..l) assuming(k::posint));
b:=k->(1/l*int(f(x)*sin(Pi*k*x/l),x=-l..l) assuming(k::posint));
eval(a(k)); eval(b(k));
s:=(n,x)->a0/2+sum(a(k)*cos(Pi*k*x/l)+b(k)*sin(Pi*k*x/l),k=1..n);
plot({f(x),s(2,x),s(5,x),s(10,x)},x=-l..l);
f:=x->signum(x); l:=1/2; a0:=1/l*int(f(x),x=-l..l);
a:=k->(1/l*int(f(x)*cos(Pi*k*x/l),x=-l..l) assuming(k::posint));
b:=k->(1/l*int(f(x)*sin(Pi*k*x/l),x=-l..l) assuming(k::posint));
eval(a(k)); eval(b(k));
plot({f(x),s(2,x),s(5,x),s(10,x)},x=-l..l);
3.3.10. Dirichlet-formula.
3.3.11. Lipschitz-krit\303\251rium.
3.3.12. K\303\266vetkezm\303\251ny.
3.3.13. K\303\266vetkezm\303\251ny: Riemann-f\303\251le lokaliz\303\241ci\303\263s t\303\251tel.
3.3.14. Klasszikus ortogon\303\241lis polinomok.
with(orthopoly);
JacobiP(0,1,3/4,x); p0:=simplify(%,'JacobiP');
JacobiP(1,1,3/4,x); p1:=simplify(%,'JacobiP');
JacobiP(2,1,3/4,x); p2:=simplify(%,'JacobiP');
JacobiP(3,1,3/4,x); p3:=simplify(%,'JacobiP');
int((1-x)^1*(1+x)^(3/4)*p1*p2,x=-1..1);
P(0,x); P(1,x); P(2,x); P(3,x);
p0:=P(0,3/4,3/4,x); p1:=P(1,3/4,3/4,x);
p2:=P(2,3/4,3/4,x); p3:=P(3,3/4,3/4,x);
plot({p0,p1,p2,p3},x=-1..1);
LaguerreL(0,0,x); p0:=simplify(%,'LaguerreL');
LaguerreL(1,0,x); p1:=simplify(%,'LaguerreL');
LaguerreL(2,0,x); p2:=simplify(%,'LaguerreL');
LaguerreL(3,0,x); p3:=simplify(%,'LaguerreL');
HermiteH(0,x); p0:=simplify(%,'HermiteH');
HermiteH(1,x); p1:=simplify(%,'HermiteH');
HermiteH(2,x); p2:=simplify(%,'HermiteH');
HermiteH(3,x); p3:=simplify(%,'HermiteH');
plot({p0,p1,p2,p3},x=0..infinity);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
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