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What should I do if limits of integration are ones where Dirac Delta gets Infinite?
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Does the dirac delta function have a residue?How do I find the Laplace Transform of $ delta(t-2pi)cos(t) $?Dirac delta function - is it okay to use it?Does the definition of distribution and Dirac delta function capture the physicists' idea?Is it true that the integral of $delta(x)/x$ between symmetrical limits is zero?Problem of Integrating Dirac delta function (Singular point at integration limit)Dirac delta of multi-variant function with infinite zerosIs a Finite Integration Interval, for Dirac Delta Functionals, Allowed?We Can Think of the Dirac Delta Function as Being the Limit Point of a Series of Functions That Put Less and Less Mass On All Points Other Than Zero?Integration with Dirac delta function of two-argument function
$begingroup$
I am working on an integral involving Dirac delta function. I am from Physics background. I know that if I integrate from a to b and in between there is a point where argument of Dirac delta function is zero then the answer to this integral is unity. What if at say 'a' the argument of Dirac Delta Function is zero. What should I do in that case? What if I have a Riemann Integrable Function along with Dirac delta function? Please help.
integration dirac-delta
edited Apr 8 at 13:52
Amad
5410
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I am working on an integral involving Dirac delta function. I am from Physics background. I know that if I integrate from a to b and in between there is a point where argument of Dirac delta function is zero then the answer to this integral is unity. What if at say 'a' the argument of Dirac Delta Function is zero. What should I do in that case? What if I have a Riemann Integrable Function along with Dirac delta function? Please help.
integration dirac-delta
edited Apr 8 at 13:52
Amad
5410
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Would you show an example?
$endgroup$
– Botond
Apr 8 at 13:10
$begingroup$
en.wikipedia.org/wiki/… and by definition $int_-infty^infty delta(f(x))g(x)dx = lim_n to inftyint_-infty^infty fracn2 1_f(x)g(x)dx$
$endgroup$
– reuns
Apr 8 at 13:13
1
$begingroup$
@Botond $int_0^inftydelta(sin(x))e^-xdx
$endgroup$
– Sudeep Tiwari
Apr 8 at 13:36
1
$begingroup$
You are no longer doing a Riemann integral, and it depends whether the set you are integrating over contains the endpoint $a$ or not. A Riemann integral is insensitive to finitely many points, but this integral is not.
$endgroup$
– csprun
Apr 8 at 14:19
add a comment |
$begingroup$
I am working on an integral involving Dirac delta function. I am from Physics background. I know that if I integrate from a to b and in between there is a point where argument of Dirac delta function is zero then the answer to this integral is unity. What if at say 'a' the argument of Dirac Delta Function is zero. What should I do in that case? What if I have a Riemann Integrable Function along with Dirac delta function? Please help.
integration dirac-delta
edited Apr 8 at 13:52
Amad
5410
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am working on an integral involving Dirac delta function. I am from Physics background. I know that if I integrate from a to b and in between there is a point where argument of Dirac delta function is zero then the answer to this integral is unity. What if at say 'a' the argument of Dirac Delta Function is zero. What should I do in that case? What if I have a Riemann Integrable Function along with Dirac delta function? Please help.
integration dirac-delta
integration dirac-delta
edited Apr 8 at 13:52
Amad
5410
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 8 at 13:52
Amad
5410
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Apr 8 at 13:52
Amad
5410
edited Apr 8 at 13:52
Amad
5410
edited Apr 8 at 13:52
Amad
5410
5410
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
asked Apr 8 at 13:06
Sudeep TiwariSudeep Tiwari
14
14
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Sudeep Tiwari is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Would you show an example?
$endgroup$
– Botond
Apr 8 at 13:10
$begingroup$
en.wikipedia.org/wiki/… and by definition $int_-infty^infty delta(f(x))g(x)dx = lim_n to inftyint_-infty^infty fracn2 1_f(x)g(x)dx$
$endgroup$
– reuns
Apr 8 at 13:13
1
$begingroup$
@Botond $int_0^inftydelta(sin(x))e^-xdx
$endgroup$
– Sudeep Tiwari
Apr 8 at 13:36
1
$begingroup$
You are no longer doing a Riemann integral, and it depends whether the set you are integrating over contains the endpoint $a$ or not. A Riemann integral is insensitive to finitely many points, but this integral is not.
$endgroup$
– csprun
Apr 8 at 14:19
add a comment |
$begingroup$
Would you show an example?
$endgroup$
– Botond
Apr 8 at 13:10
$begingroup$
en.wikipedia.org/wiki/… and by definition $int_-infty^infty delta(f(x))g(x)dx = lim_n to inftyint_-infty^infty fracn2 1_f(x)g(x)dx$
$endgroup$
– reuns
Apr 8 at 13:13
1
$begingroup$
@Botond $int_0^inftydelta(sin(x))e^-xdx
$endgroup$
– Sudeep Tiwari
Apr 8 at 13:36
1
$begingroup$
You are no longer doing a Riemann integral, and it depends whether the set you are integrating over contains the endpoint $a$ or not. A Riemann integral is insensitive to finitely many points, but this integral is not.
$endgroup$
– csprun
Apr 8 at 14:19
$begingroup$
Would you show an example?
$endgroup$
– Botond
Apr 8 at 13:10
$begingroup$
Would you show an example?
$endgroup$
– Botond
Apr 8 at 13:10
$begingroup$
en.wikipedia.org/wiki/… and by definition $int_-infty^infty delta(f(x))g(x)dx = lim_n to inftyint_-infty^infty fracn2 1_f(x)g(x)dx$
$endgroup$
– reuns
Apr 8 at 13:13
$begingroup$
en.wikipedia.org/wiki/… and by definition $int_-infty^infty delta(f(x))g(x)dx = lim_n to inftyint_-infty^infty fracn2 1_f(x)g(x)dx$
$endgroup$
– reuns
Apr 8 at 13:13
1
1
$begingroup$
@Botond $int_0^inftydelta(sin(x))e^-xdx
$endgroup$
– Sudeep Tiwari
Apr 8 at 13:36
$begingroup$
@Botond $int_0^inftydelta(sin(x))e^-xdx
$endgroup$
– Sudeep Tiwari
Apr 8 at 13:36
1
1
$begingroup$
You are no longer doing a Riemann integral, and it depends whether the set you are integrating over contains the endpoint $a$ or not. A Riemann integral is insensitive to finitely many points, but this integral is not.
$endgroup$
– csprun
Apr 8 at 14:19
$begingroup$
You are no longer doing a Riemann integral, and it depends whether the set you are integrating over contains the endpoint $a$ or not. A Riemann integral is insensitive to finitely many points, but this integral is not.
$endgroup$
– csprun
Apr 8 at 14:19
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Consider a graph of the delta function (distribution) to obtain:
$$int_a^bdelta(x-t),mathrm dx
= cases0&$t<a,text or t>b$\
1&$a<t<b$\
colorred1/2&$colorredt=a,text or t=b$$$
So that
$$int_a^bf(x)delta(x-a),mathrm dx
= tfrac12f(a)$$
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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active
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$begingroup$
Consider a graph of the delta function (distribution) to obtain:
$$int_a^bdelta(x-t),mathrm dx
= cases0&$t<a,text or t>b$\
1&$a<t<b$\
colorred1/2&$colorredt=a,text or t=b$$$
So that
$$int_a^bf(x)delta(x-a),mathrm dx
= tfrac12f(a)$$
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
$endgroup$
add a comment |
$begingroup$
Consider a graph of the delta function (distribution) to obtain:
$$int_a^bdelta(x-t),mathrm dx
= cases0&$t<a,text or t>b$\
1&$a<t<b$\
colorred1/2&$colorredt=a,text or t=b$$$
So that
$$int_a^bf(x)delta(x-a),mathrm dx
= tfrac12f(a)$$
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
$endgroup$
add a comment |
$begingroup$
Consider a graph of the delta function (distribution) to obtain:
$$int_a^bdelta(x-t),mathrm dx
= cases0&$t<a,text or t>b$\
1&$a<t<b$\
colorred1/2&$colorredt=a,text or t=b$$$
So that
$$int_a^bf(x)delta(x-a),mathrm dx
= tfrac12f(a)$$
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
$endgroup$
Consider a graph of the delta function (distribution) to obtain:
$$int_a^bdelta(x-t),mathrm dx
= cases0&$t<a,text or t>b$\
1&$a<t<b$\
colorred1/2&$colorredt=a,text or t=b$$$
So that
$$int_a^bf(x)delta(x-a),mathrm dx
= tfrac12f(a)$$
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
edited Apr 8 at 14:28
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
answered Apr 8 at 14:23
Elements in SpaceElements in Space
1,28211228
1,28211228
add a comment |
add a comment |
Sudeep Tiwari is a new contributor. Be nice, and check out our Code of Conduct.
Sudeep Tiwari is a new contributor. Be nice, and check out our Code of Conduct.
Sudeep Tiwari is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
Would you show an example?
$endgroup$
– Botond
Apr 8 at 13:10
$begingroup$
en.wikipedia.org/wiki/… and by definition $int_-infty^infty delta(f(x))g(x)dx = lim_n to inftyint_-infty^infty fracn2 1_f(x)g(x)dx$
$endgroup$
– reuns
Apr 8 at 13:13
1
$begingroup$
@Botond $int_0^inftydelta(sin(x))e^-xdx
$endgroup$
– Sudeep Tiwari
Apr 8 at 13:36
1
$begingroup$
You are no longer doing a Riemann integral, and it depends whether the set you are integrating over contains the endpoint $a$ or not. A Riemann integral is insensitive to finitely many points, but this integral is not.
$endgroup$
– csprun
Apr 8 at 14:19