The limit passes under the sign of integral
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I have$f:[a,b)times[c,d]rightarrowmathbb{R}, $ (a) $lim_{yto y_0} f(x,y)=l(x)$ uniform with respect to $x$ on any compact included in $[a,b) $ and (b) $int_{a}^{b-0} f(x,y) dx$ is uniform convergent with respect to $y$ on a neighborhood of $y_0$. Then:$int_a^{b-0}l(x)dx$ is convergent and $lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
//I tried to prove it://
Let $epsilon >0$. from (a) i have: $existsdelta_epsilon>0 $ such that $ forall xin[a,b) $ and $forall y in [c,d] $ with $|y-y_0|<delta_epsilon $=>$ |f(x,y)-l(x)|<epsilon$ now if we put it all under the sign of the integral=>$int_a^{b-0} |f(x,y)-l(x)|<int_a^{b-0}epsilon$ and this show:
$lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
but i don't used (b) and i feel this is wrong can you help me with a proof please?
calculus analysis multivariable-calculus
asked Nov 15 at 12:02
Ica Sandu
515
add a comment |
up vote
0
down vote
favorite
I have$f:[a,b)times[c,d]rightarrowmathbb{R}, $ (a) $lim_{yto y_0} f(x,y)=l(x)$ uniform with respect to $x$ on any compact included in $[a,b) $ and (b) $int_{a}^{b-0} f(x,y) dx$ is uniform convergent with respect to $y$ on a neighborhood of $y_0$. Then:$int_a^{b-0}l(x)dx$ is convergent and $lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
//I tried to prove it://
Let $epsilon >0$. from (a) i have: $existsdelta_epsilon>0 $ such that $ forall xin[a,b) $ and $forall y in [c,d] $ with $|y-y_0|<delta_epsilon $=>$ |f(x,y)-l(x)|<epsilon$ now if we put it all under the sign of the integral=>$int_a^{b-0} |f(x,y)-l(x)|<int_a^{b-0}epsilon$ and this show:
$lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
but i don't used (b) and i feel this is wrong can you help me with a proof please?
calculus analysis multivariable-calculus
asked Nov 15 at 12:02
Ica Sandu
515
This is just a part of the proof. To complete you should show "$int_a^{b-0} ell (x ),mathrm dx$ is convergent" as well. You would use (b) to do this.
– xbh
Nov 15 at 13:22
but how to proof second part
– Ica Sandu
Nov 16 at 20:02
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have$f:[a,b)times[c,d]rightarrowmathbb{R}, $ (a) $lim_{yto y_0} f(x,y)=l(x)$ uniform with respect to $x$ on any compact included in $[a,b) $ and (b) $int_{a}^{b-0} f(x,y) dx$ is uniform convergent with respect to $y$ on a neighborhood of $y_0$. Then:$int_a^{b-0}l(x)dx$ is convergent and $lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
//I tried to prove it://
Let $epsilon >0$. from (a) i have: $existsdelta_epsilon>0 $ such that $ forall xin[a,b) $ and $forall y in [c,d] $ with $|y-y_0|<delta_epsilon $=>$ |f(x,y)-l(x)|<epsilon$ now if we put it all under the sign of the integral=>$int_a^{b-0} |f(x,y)-l(x)|<int_a^{b-0}epsilon$ and this show:
$lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
but i don't used (b) and i feel this is wrong can you help me with a proof please?
calculus analysis multivariable-calculus
asked Nov 15 at 12:02
Ica Sandu
515
I have$f:[a,b)times[c,d]rightarrowmathbb{R}, $ (a) $lim_{yto y_0} f(x,y)=l(x)$ uniform with respect to $x$ on any compact included in $[a,b) $ and (b) $int_{a}^{b-0} f(x,y) dx$ is uniform convergent with respect to $y$ on a neighborhood of $y_0$. Then:$int_a^{b-0}l(x)dx$ is convergent and $lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
//I tried to prove it://
Let $epsilon >0$. from (a) i have: $existsdelta_epsilon>0 $ such that $ forall xin[a,b) $ and $forall y in [c,d] $ with $|y-y_0|<delta_epsilon $=>$ |f(x,y)-l(x)|<epsilon$ now if we put it all under the sign of the integral=>$int_a^{b-0} |f(x,y)-l(x)|<int_a^{b-0}epsilon$ and this show:
$lim_{yto y_0}int_a^{b-0}f(x,y)dx= int_a^{b-0}lim{yto y_0}f(x,y)dx $
but i don't used (b) and i feel this is wrong can you help me with a proof please?
calculus analysis multivariable-calculus
calculus analysis multivariable-calculus
asked Nov 15 at 12:02
Ica Sandu
515
asked Nov 15 at 12:02
Ica Sandu
515
asked Nov 15 at 12:02
Ica Sandu
515
asked Nov 15 at 12:02
Ica Sandu
515
asked Nov 15 at 12:02
Ica Sandu
515
515
This is just a part of the proof. To complete you should show "$int_a^{b-0} ell (x ),mathrm dx$ is convergent" as well. You would use (b) to do this.
– xbh
Nov 15 at 13:22
but how to proof second part
– Ica Sandu
Nov 16 at 20:02
add a comment |
This is just a part of the proof. To complete you should show "$int_a^{b-0} ell (x ),mathrm dx$ is convergent" as well. You would use (b) to do this.
– xbh
Nov 15 at 13:22
but how to proof second part
– Ica Sandu
Nov 16 at 20:02
This is just a part of the proof. To complete you should show "$int_a^{b-0} ell (x ),mathrm dx$ is convergent" as well. You would use (b) to do this.
– xbh
Nov 15 at 13:22
This is just a part of the proof. To complete you should show "$int_a^{b-0} ell (x ),mathrm dx$ is convergent" as well. You would use (b) to do this.
– xbh
Nov 15 at 13:22
but how to proof second part
– Ica Sandu
Nov 16 at 20:02
but how to proof second part
– Ica Sandu
Nov 16 at 20:02
add a comment |
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This is just a part of the proof. To complete you should show "$int_a^{b-0} ell (x ),mathrm dx$ is convergent" as well. You would use (b) to do this.
– xbh
Nov 15 at 13:22
but how to proof second part
– Ica Sandu
Nov 16 at 20:02