Integrali s parametrom
Integrali s parametrom
Integrali s parametrom
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Marko Razpet<br />
<strong>Integrali</strong> s <strong>parametrom</strong><br />
Študijsko gradivo<br />
Do preklica naj pomeni<br />
R = [a, b] × [A, B] ⊂ IR × IR ,<br />
pri čemer so a, b, A, B realna števila in a < b ter A < B. Množica točk R je<br />
v pravokotnem kartezičnem koordinatnem sistemu xy pravokotnik, prikazan<br />
na sliki. Vse funkcije, nastopajoče v pričujočem besedilu, so realne.<br />
y<br />
B<br />
A<br />
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R<br />
a<br />
b<br />
x<br />
..<br />
Na pravokotniku R naj bo definirana funkcija f : R ↦→ IR. Njeno vrednost v<br />
točki (x, y) ∈ R označimo z f(x, y). Če za vsak y ∈ [A, B] obstaja integral<br />
∫ b<br />
a<br />
f(x, y) dx ,<br />
obravnavamo integral s <strong>parametrom</strong> in v njem spremenljivko y imenujemo<br />
parameter. S tem imamo definirano novo funkcijo F : [A, B] ↦→ IR s predpisom:<br />
F (y) =<br />
∫ b<br />
a<br />
1<br />
f(x, y) dx .
V nadaljevanju pomeni vselej zveznost funkcije v krajiščih intervala njeno<br />
zveznost z leve oziroma z desne strani, prav tako pa pomeni vedno odvod<br />
funkcije v krajiščih intervala njen levi oziroma desni odvod.<br />
Osnovni trditvi<br />
1. Če je funkcija f zvezna na pravokotniku R, potem je funkcija F zvezna<br />
na intervalu [A, B].<br />
2.<br />
Če je funkcija f zvezna na pravokotniku R in ima f na R zvezen parcialni<br />
odvod ∂f/∂y, potem je funkcija F na intervalu [A, B] tudi odvedljiva in velja<br />
Leibnizovo pravilo odvajanja pod integralskim znakom:<br />
Posplošitvi<br />
F ′ (y) = d dy<br />
∫ b<br />
a<br />
f(x, y) dx =<br />
∫ b<br />
a<br />
∂f<br />
(x, y) dx .<br />
∂y<br />
3. Če je funkcija f zvezna na pravokotniku R, funkciji α in β pa sta zvezni<br />
na intervalu [A, B], pri čemer za vsak y ∈ [A, B] velja relacija<br />
a ≤ α(y) ≤ β(y) ≤ b ,<br />
potem je funkcija F , definirana na intervalu [A, B] s predpisom<br />
zvezna na [A, B].<br />
F (y) =<br />
∫ β(y)<br />
α(y)<br />
f(x, y) dx ,<br />
2
y<br />
B<br />
A<br />
. .<br />
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. .<br />
x = α(y) .<br />
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x = β(y)<br />
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R<br />
a<br />
b<br />
x<br />
..<br />
4. Če je funkcija f zvezna na pravokotniku R in ima f na R zvezen parcialni<br />
odvod ∂f/∂y, funkciji α in β pa sta zvezni in odvedljivi na intervalu [A, B],<br />
pri čemer za vsak y ∈ [A, B] velja relacija<br />
a ≤ α(y) ≤ β(y) ≤ b ,<br />
potem je funkcija F , definirana na intervalu [A, B] s predpisom<br />
F (y) =<br />
∫ β(y)<br />
α(y)<br />
odvedljiva na intervalu [A, B] in velja:<br />
F ′ (y) = d dy<br />
∫ β(y)<br />
α(y)<br />
f(x, y) dx =<br />
f(x, y) dx<br />
=<br />
∫ β(y)<br />
α(y)<br />
∂f<br />
∂y (x, y) dx + f(β(y), y) β′ (y) − f(α(y), y) α ′ (y) .<br />
5. Če je funkcija f zvezna na pravokotniku R, potem sta funkciji F : [A, B] ↦→<br />
IR in G : [a, b] ↦→ IR, definirani s predpisoma<br />
F (y) =<br />
∫ b<br />
a<br />
f(x, y) dx in G(x) =<br />
∫ B<br />
A<br />
f(x, y) dy ,<br />
3
integrabilni in velja enakost<br />
∫ b<br />
a<br />
G(x) dx =<br />
∫ B<br />
A<br />
F (y) dy<br />
oziroma<br />
∫ b<br />
a<br />
dx<br />
∫ B<br />
A<br />
f(x, y) dy =<br />
∫ B<br />
A<br />
dy<br />
∫ b<br />
a<br />
f(x, y) dx .<br />
Enakomerna konvergenca integralov<br />
V nadaljevanju naj pomeni<br />
R = [a, ∞) × [A, B] ⊂ IR × IR .<br />
Množica točk R je tokrat v pravokotnem kartezičnem koordinatnem sistemu<br />
xy neomejen pravokotnik, prikazan na sliki.<br />
..<br />
. .<br />
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a<br />
A<br />
B<br />
R<br />
Naj bo funkcija f definirana na neomejenem pravokotniku R in I ⊆ [A, B].<br />
Integral<br />
∫ ∞<br />
a<br />
f(x, y) dx<br />
enakomerno konvergira na množici I, če za vsak ε > 0 obstaja X 0 > a tak,<br />
da za vsak y ∈ I in za vsak X ≥ X 0 velja:<br />
∣<br />
∣<br />
∣<br />
∣<br />
∫ ∞<br />
X<br />
f(x, y) dx<br />
∣<br />
∣<br />
∣<br />
∣ < ε .<br />
4
Weierstrassov (zadostni) pogoj<br />
Če obstaja taka nenegativna funkcija ϕ : [a, ∞) ↦→ IR, za katero obstaja<br />
integral<br />
∫ ∞<br />
a<br />
ϕ(x) dx < ∞<br />
in za katero je pri vsakem x ∈ [a, ∞) in vsakem y ∈ I ⊆ IR izpolnjen pogoj<br />
|f(x, y)| ≤ ϕ(x) ,<br />
potem integral<br />
∫ ∞<br />
enakomerno konvergira na množici I.<br />
a<br />
f(x, y) dx<br />
6.<br />
Če je funkcija f zvezna na neomejenem pravokotniku R in če integral<br />
∫ ∞<br />
a<br />
f(x, y) dx<br />
enakomerno konvergira na intervalu [A, B], potem je funkcija F : [A, B] ↦→ IR,<br />
ki je definirana s predpisom<br />
F (y) =<br />
zvezna na intervalu [A, B].<br />
∫ ∞<br />
a<br />
f(x, y) dx ,<br />
7.<br />
Če je funkcija f zvezna na neomejenem pravokotniku R in če integral<br />
∫ ∞<br />
a<br />
f(x, y) dx<br />
enakomerno konvergira na intervalu [A, B], potem sta funkciji F : [A, B] ↦→ IR<br />
in G : [a, ∞) ↦→ IR, definirani s predpisoma<br />
F (y) =<br />
∫ ∞<br />
a<br />
f(x, y) dx in G(x) =<br />
5<br />
∫ B<br />
A<br />
f(x, y) dy ,
integrabilni in velja enakost<br />
∫ ∞<br />
a<br />
G(x) dx =<br />
∫ B<br />
A<br />
F (y) dy<br />
oziroma<br />
∫ ∞<br />
a<br />
dx<br />
∫ B<br />
A<br />
f(x, y) dy =<br />
∫ B<br />
A<br />
dy<br />
∫ ∞<br />
a<br />
f(x, y) dx .<br />
8. Če je funkcija f zvezna na neomejenem pravokotniku R in ima f na R<br />
zvezen parcialni odvod ∂f/∂y, če integral<br />
∫ ∞<br />
a<br />
f(x, y) dx<br />
konvergira za vsak y ∈ [A, B] in če integral<br />
∫ ∞<br />
a<br />
∂f<br />
(x, y) dx<br />
∂y<br />
enakomerno konvergira na intervalu [A, B], potem je funkcija F : [A, B] ↦→ IR,<br />
ki je definirana s predpisom<br />
F (y) =<br />
∫ ∞<br />
a<br />
f(x, y) dx ,<br />
odvedljiva na intervalu [A, B] in velja Leibnizovo pravilo odvajanja pod integralskim<br />
znakom:<br />
F ′ (y) = d dy<br />
∫ ∞<br />
a<br />
f(x, y) dx =<br />
∫ ∞<br />
a<br />
∂f<br />
(x, y) dx .<br />
∂y<br />
Končno naj bo<br />
R = [a, ∞) × [A, ∞) ⊂ IR × IR .<br />
Množica točk R je sedaj v pravokotnem kartezičnem koordinatnem sistemu<br />
xy kvadrant z vrhom v točki (a, A), kot prikazuje slika.<br />
6
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a<br />
A<br />
R<br />
9. Naj bo na kvadrantu R definirana zvezna in pozitivna (negativna) funkcija<br />
f, integrala<br />
∫ ∞<br />
a<br />
f(x, y) dx<br />
in<br />
∫ ∞<br />
A<br />
f(x, y) dy<br />
pa naj enakomerno konvergirata, prvi na poltraku [A, ∞), drugi pa na poltraku<br />
[a, ∞). Če obstajata integrala<br />
∫ ∞<br />
a<br />
G(x) dx<br />
in<br />
∫ ∞<br />
A<br />
F (y) dy<br />
funkcij F : [A, ∞) ↦→ IR in G : [a, ∞) ↦→ IR, ki sta definirani s predpisoma<br />
F (y) =<br />
∫ ∞<br />
a<br />
f(x, y) dx in G(x) =<br />
∫ ∞<br />
A<br />
f(x, y) dy ,<br />
potem velja enakost<br />
∫ ∞<br />
a<br />
G(x) dx =<br />
∫ ∞<br />
A<br />
F (y) dy<br />
oziroma<br />
∫ ∞<br />
a<br />
dx<br />
∫ ∞<br />
A<br />
f(x, y) dy =<br />
∫ ∞<br />
A<br />
dy<br />
∫ ∞<br />
a<br />
f(x, y) dx .<br />
Vir<br />
Ivan Vidav, Višja matematika I, DMFA, Ljubljana, 1994.<br />
7
CIP - Kataložni zapis o publikaciji<br />
Narodna in univerzitetna knjižnica, Ljubljana<br />
517.38(075.8)(076)<br />
RAZPET, Marko<br />
<strong>Integrali</strong> s <strong>parametrom</strong> [Elektronski vir] : študijsko gradivo /<br />
Marko Razpet. - Besedilni podatki. - [Domžale :samozal.], 2006<br />
Način dostopa (URL): http://javor.pef.uni-lj.si/~marko/matematika/i<br />
int_par.pdf. - Opis temelji na verziji z dne 10.02.2006<br />
ISBN 961-6589-24-5<br />
225020160<br />
8