Find the value of the given limit
2
Find the value of this limit:
$$lim_{xto infty}dfrac{1}{pi}int _0^infty cosleft(dfrac{t^3}{3}+txright) dt$$.
Now as $xto infty implies dfrac{t^3}{3}+txto infty $ for any fixed $t$.
then $cosleft(dfrac{t^3}{3}+txright)$ is oscillating
Now I dont understand how should I compute the limit in this case?
Can someone please help.
calculus integration limits
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
|
show 6 more comments
2
Find the value of this limit:
$$lim_{xto infty}dfrac{1}{pi}int _0^infty cosleft(dfrac{t^3}{3}+txright) dt$$.
Now as $xto infty implies dfrac{t^3}{3}+txto infty $ for any fixed $t$.
then $cosleft(dfrac{t^3}{3}+txright)$ is oscillating
Now I dont understand how should I compute the limit in this case?
Can someone please help.
calculus integration limits
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
1
@KaviRamaMurthy I believe finiteness should follow from the following argument, (though I don't have time to check my work right now): $$begin{align} left|int_0^{infty}cosleft(frac{t^3}{3}+txright)dtright| &= left|int_0^{infty}cos(t^3/3)cos(tx)dt - int_0^{infty}sin(t^3/3)sin(tx)dtright| \& le int_0^{infty}left|cos(t^3/3)cos(tx)right|dt + int_0^{infty}left|sin(t^3/3)sin(tx)right|dt \&le int_0^{infty} |cos(t^3/3)|dt + int_0^{infty}|sin(t^3/3)|dt \&= frac{Gamma(1/3)}{2sqrt{3}} + frac{Gamma(1/3)}{6} end{align}$$
– Brevan Ellefsen
Nov 23 '18 at 8:17
1
@BrevanEllefsen Thank you very much.
– Kavi Rama Murthy
Nov 23 '18 at 8:23
1
Having a look at Airy function may be useful.
– Kemono Chen
Nov 23 '18 at 8:26
1
@KemonoChen;I checked it but it does not give a proof why the integral converges to 0
– Join_PhD
Nov 23 '18 at 8:30
3
Take a look at the Riemann–Lebesgue lemma. (hint: the limit is 0)
– Fabian
Nov 23 '18 at 9:26
|
show 6 more comments
2
2
2
1
Find the value of this limit:
$$lim_{xto infty}dfrac{1}{pi}int _0^infty cosleft(dfrac{t^3}{3}+txright) dt$$.
Now as $xto infty implies dfrac{t^3}{3}+txto infty $ for any fixed $t$.
then $cosleft(dfrac{t^3}{3}+txright)$ is oscillating
Now I dont understand how should I compute the limit in this case?
Can someone please help.
calculus integration limits
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
Find the value of this limit:
$$lim_{xto infty}dfrac{1}{pi}int _0^infty cosleft(dfrac{t^3}{3}+txright) dt$$.
Now as $xto infty implies dfrac{t^3}{3}+txto infty $ for any fixed $t$.
then $cosleft(dfrac{t^3}{3}+txright)$ is oscillating
Now I dont understand how should I compute the limit in this case?
Can someone please help.
calculus integration limits
calculus integration limits
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
edited Nov 23 '18 at 8:06
Rebellos
14.5k31246
14.5k31246
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
asked Nov 23 '18 at 7:56
Join_PhDJoin_PhD
3088
3088
1
@KaviRamaMurthy I believe finiteness should follow from the following argument, (though I don't have time to check my work right now): $$begin{align} left|int_0^{infty}cosleft(frac{t^3}{3}+txright)dtright| &= left|int_0^{infty}cos(t^3/3)cos(tx)dt - int_0^{infty}sin(t^3/3)sin(tx)dtright| \& le int_0^{infty}left|cos(t^3/3)cos(tx)right|dt + int_0^{infty}left|sin(t^3/3)sin(tx)right|dt \&le int_0^{infty} |cos(t^3/3)|dt + int_0^{infty}|sin(t^3/3)|dt \&= frac{Gamma(1/3)}{2sqrt{3}} + frac{Gamma(1/3)}{6} end{align}$$
– Brevan Ellefsen
Nov 23 '18 at 8:17
1
@BrevanEllefsen Thank you very much.
– Kavi Rama Murthy
Nov 23 '18 at 8:23
1
Having a look at Airy function may be useful.
– Kemono Chen
Nov 23 '18 at 8:26
1
@KemonoChen;I checked it but it does not give a proof why the integral converges to 0
– Join_PhD
Nov 23 '18 at 8:30
3
Take a look at the Riemann–Lebesgue lemma. (hint: the limit is 0)
– Fabian
Nov 23 '18 at 9:26
|
show 6 more comments
1
@KaviRamaMurthy I believe finiteness should follow from the following argument, (though I don't have time to check my work right now): $$begin{align} left|int_0^{infty}cosleft(frac{t^3}{3}+txright)dtright| &= left|int_0^{infty}cos(t^3/3)cos(tx)dt - int_0^{infty}sin(t^3/3)sin(tx)dtright| \& le int_0^{infty}left|cos(t^3/3)cos(tx)right|dt + int_0^{infty}left|sin(t^3/3)sin(tx)right|dt \&le int_0^{infty} |cos(t^3/3)|dt + int_0^{infty}|sin(t^3/3)|dt \&= frac{Gamma(1/3)}{2sqrt{3}} + frac{Gamma(1/3)}{6} end{align}$$
– Brevan Ellefsen
Nov 23 '18 at 8:17
1
@BrevanEllefsen Thank you very much.
– Kavi Rama Murthy
Nov 23 '18 at 8:23
1
Having a look at Airy function may be useful.
– Kemono Chen
Nov 23 '18 at 8:26
1
@KemonoChen;I checked it but it does not give a proof why the integral converges to 0
– Join_PhD
Nov 23 '18 at 8:30
3
Take a look at the Riemann–Lebesgue lemma. (hint: the limit is 0)
– Fabian
Nov 23 '18 at 9:26
1
1
@KaviRamaMurthy I believe finiteness should follow from the following argument, (though I don't have time to check my work right now): $$begin{align} left|int_0^{infty}cosleft(frac{t^3}{3}+txright)dtright| &= left|int_0^{infty}cos(t^3/3)cos(tx)dt - int_0^{infty}sin(t^3/3)sin(tx)dtright| \& le int_0^{infty}left|cos(t^3/3)cos(tx)right|dt + int_0^{infty}left|sin(t^3/3)sin(tx)right|dt \&le int_0^{infty} |cos(t^3/3)|dt + int_0^{infty}|sin(t^3/3)|dt \&= frac{Gamma(1/3)}{2sqrt{3}} + frac{Gamma(1/3)}{6} end{align}$$
– Brevan Ellefsen
Nov 23 '18 at 8:17
@KaviRamaMurthy I believe finiteness should follow from the following argument, (though I don't have time to check my work right now): $$begin{align} left|int_0^{infty}cosleft(frac{t^3}{3}+txright)dtright| &= left|int_0^{infty}cos(t^3/3)cos(tx)dt - int_0^{infty}sin(t^3/3)sin(tx)dtright| \& le int_0^{infty}left|cos(t^3/3)cos(tx)right|dt + int_0^{infty}left|sin(t^3/3)sin(tx)right|dt \&le int_0^{infty} |cos(t^3/3)|dt + int_0^{infty}|sin(t^3/3)|dt \&= frac{Gamma(1/3)}{2sqrt{3}} + frac{Gamma(1/3)}{6} end{align}$$
– Brevan Ellefsen
Nov 23 '18 at 8:17
1
1
@BrevanEllefsen Thank you very much.
– Kavi Rama Murthy
Nov 23 '18 at 8:23
@BrevanEllefsen Thank you very much.
– Kavi Rama Murthy
Nov 23 '18 at 8:23
1
1
Having a look at Airy function may be useful.
– Kemono Chen
Nov 23 '18 at 8:26
Having a look at Airy function may be useful.
– Kemono Chen
Nov 23 '18 at 8:26
1
1
@KemonoChen;I checked it but it does not give a proof why the integral converges to 0
– Join_PhD
Nov 23 '18 at 8:30
@KemonoChen;I checked it but it does not give a proof why the integral converges to 0
– Join_PhD
Nov 23 '18 at 8:30
3
3
Take a look at the Riemann–Lebesgue lemma. (hint: the limit is 0)
– Fabian
Nov 23 '18 at 9:26
Take a look at the Riemann–Lebesgue lemma. (hint: the limit is 0)
– Fabian
Nov 23 '18 at 9:26
|
show 6 more comments
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1
@KaviRamaMurthy I believe finiteness should follow from the following argument, (though I don't have time to check my work right now): $$begin{align} left|int_0^{infty}cosleft(frac{t^3}{3}+txright)dtright| &= left|int_0^{infty}cos(t^3/3)cos(tx)dt - int_0^{infty}sin(t^3/3)sin(tx)dtright| \& le int_0^{infty}left|cos(t^3/3)cos(tx)right|dt + int_0^{infty}left|sin(t^3/3)sin(tx)right|dt \&le int_0^{infty} |cos(t^3/3)|dt + int_0^{infty}|sin(t^3/3)|dt \&= frac{Gamma(1/3)}{2sqrt{3}} + frac{Gamma(1/3)}{6} end{align}$$
– Brevan Ellefsen
Nov 23 '18 at 8:17
1
@BrevanEllefsen Thank you very much.
– Kavi Rama Murthy
Nov 23 '18 at 8:23
1
Having a look at Airy function may be useful.
– Kemono Chen
Nov 23 '18 at 8:26
1
@KemonoChen;I checked it but it does not give a proof why the integral converges to 0
– Join_PhD
Nov 23 '18 at 8:30
3
Take a look at the Riemann–Lebesgue lemma. (hint: the limit is 0)
– Fabian
Nov 23 '18 at 9:26