articulo de la integral de cauchy a la integral de riemann omar garcia
DESCRIPTION
INTEGRAL DE CAUCHYTRANSCRIPT
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5/19/2018 Articulo de La Integral de Cauchy a La Integral de Riemann Omar Garcia
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f
[a, b]
lmx0
n
i=1
f(xi)(xixi1)
{x0, x1, x2,...,xn} [a, b] x
[a, b]
f : [a, b]R
f
[a, b]
x [a, b]
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f(x) = lmtx
f(t) f(x+) = lmtx+
f(t)
x = a
f(a+)
x = b
f(b) G[a, b] [a, b]
f : [a, b] R p [a, b]
f(p) = sup {f(x)|x [a, p)} f(p+) = nf{f(x)|x (p, b]}
f(p) f(p) f(p+) q, r [a, b] q < r f(q+) f(r)
f(p) = nf{f(x)|x [a, p)} f(p+) = sup {f(x)|x (p, b]}
f(p+) f(p) f(p) q, r [a, b] q < r f(r) f(q+).
x= a f(a+) x= b f(b)
E[a, b]
C[a, b]
2,1
f G[a, b] >0 E[a, b]
|f(x)(x)|< , x [a, b]
f 0 [a, b] 0
f G[a, b] {n}nN E[a, b] f [a, b]
2,3
f G[a, b] f
f G[a, b] {n(x)}nN E[a, b] n(x) f [a, b]
ba
n(x) dx
nN
{n(x)}nN E[a, b] n(x) f [a, b]
>0
n0 N n > n0 |n(x)n(x)|< , x [a, b]
lmn
ba
n(x) dx = lmn
ba
n(x) dx
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f G[a, b] f [a, b]
lmn ba n(x) dx
{n(x)}nN E[a, b] n(x) f [a, b] f C
ba
f(x) dxC
C
f [a, b]
U(f) ={ (x) E[a, b]|f(x) (x), x [a, b]}
L(f) ={ (x) E[a, b]|(x) f(x), x [a, b]}
f U(f) L(f) (x) L(f) (x) U(f)
(x) f(x)
(x), x [a, b]
U=
ba
(x) dx| (x) U(f)
; L=
ba
(x) dx|(x) L(f)
nf (x)U(f)
U sup (x)L(f)
L
f [a, b]
b
a
f(x) dxR
b
a
f(x) dxR
f R b
a
f(x) dxR=
ba
f(x) dxR
f [a, b]
ba
f(x) dxR
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f : [a, b] R R >0 (x), (x) E[a, b]
(x) f(x) (x), x [a, b] ba
(x) dx ba
(x) dx <
f C f R
f g [a, b]
f G[a, b] g(x) =g(a) +
xa
f(t) dtC x [a, b]
x [a, b) g+(x) =f(x+) x (a, b] g(x) =f(x)
f G[a, b] g f g C[a, b] [a, b] g(x) g(x) =f(x)
f C g: [a, b] R
g(x) =g(a) +
xa
f(t) dtCparacadax [a, b]
f c [a, b] g c
g(c) =f(c)
c= a c= b g(c) g a g
f C
g: [a, b] R
g
(x) =f(x)
x [a, b] b
a
f(x) dxC =g(b)g(a)
5,2
5,3
f
C R
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