Fractional/Integer Based integralsIs there any body of knowledge or study of the fractional calculus on definite integrals?Practical use and applications of improper integralsExamples of pairs of difficult integralsWhat level of math is needed to learn fractional calculus?Is there a rearrangement theorem for conditionally convergent improper integrals?Are there physical interpretations of fractional order integrals and derivatives?Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram AlphaHow get a good approximation of integrals involving the gamma function, exponentials and the fractional part?Request for crazy integralsSpecial Functions defined by Integrals.
Does a druid starting with a bow start with no arrows?
1960's book about a plague that kills all white people
What exploit are these user agents trying to use?
Should I tell management that I intend to leave due to bad software development practices?
intersection of two sorted vectors in C++
What does it mean to describe someone as a butt steak?
Assassin's bullet with mercury
Can I make "comment-region" comment empty lines?
Emailing HOD to enhance faculty application
Watching something be written to a file live with tail
How could indestructible materials be used in power generation?
Is it possible to create light that imparts a greater proportion of its energy as momentum rather than heat?
Why do I get two different answers for this counting problem?
Infinite Abelian subgroup of infinite non Abelian group example
Forgetting the musical notes while performing in concert
Neighboring nodes in the network
What to put in ESTA if staying in US for a few days before going on to Canada
Could gravitational lensing be used to protect a spaceship from a laser?
Reserved de-dupe rules
Today is the Center
How to model explosives?
Why is consensus so controversial in Britain?
UK: Is there precedent for the governments e-petition site changing the direction of a government decision?
How to show the equivalence between the regularized regression and their constraint formulas using KKT
Fractional/Integer Based integrals
Is there any body of knowledge or study of the fractional calculus on definite integrals?Practical use and applications of improper integralsExamples of pairs of difficult integralsWhat level of math is needed to learn fractional calculus?Is there a rearrangement theorem for conditionally convergent improper integrals?Are there physical interpretations of fractional order integrals and derivatives?Mathematical reasoning to get closed-forms or nice definite integrals from these outputs of Wolfram AlphaHow get a good approximation of integrals involving the gamma function, exponentials and the fractional part?Request for crazy integralsSpecial Functions defined by Integrals.
$begingroup$
I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?
integration definite-integrals soft-question indefinite-integrals big-list
$endgroup$
add a comment |
$begingroup$
I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?
integration definite-integrals soft-question indefinite-integrals big-list
$endgroup$
$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55
1
$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51
$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46
$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47
add a comment |
$begingroup$
I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?
integration definite-integrals soft-question indefinite-integrals big-list
$endgroup$
I see a lot of integrals on this page that involve the fractional/integer component of a Real variable $x$. I was wondering what applications these are founded in?
integration definite-integrals soft-question indefinite-integrals big-list
integration definite-integrals soft-question indefinite-integrals big-list
asked Jan 2 at 3:31
user150203
$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55
1
$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51
$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46
$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47
add a comment |
$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55
1
$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51
$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46
$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47
$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55
$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55
1
1
$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51
$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51
$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03
$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46
$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47
$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.
They might appear when working with the Gauss transformation $T(x)=frac1x-lfloorfrac1xrfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac1ln(2)int_Bfracdt1+t$.
answered Mar 22 at 11:49
user647486user647486
887111
$endgroup$
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
add a comment |
$begingroup$
Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
$$int_a^bf([x]),mathrmdx=sum_k=a^b-1f(k).$$
Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059115%2ffractional-integer-based-integrals%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.
They might appear when working with the Gauss transformation $T(x)=frac1x-lfloorfrac1xrfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac1ln(2)int_Bfracdt1+t$.
answered Mar 22 at 11:49
user647486user647486
887111
$endgroup$
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
add a comment |
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.
They might appear when working with the Gauss transformation $T(x)=frac1x-lfloorfrac1xrfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac1ln(2)int_Bfracdt1+t$.
answered Mar 22 at 11:49
user647486user647486
887111
$endgroup$
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
add a comment |
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.
They might appear when working with the Gauss transformation $T(x)=frac1x-lfloorfrac1xrfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac1ln(2)int_Bfracdt1+t$.
answered Mar 22 at 11:49
user647486user647486
887111
$endgroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
In the Wikipedia article for Floor and ceiling functions you can find other applications. Some of the ones listed there are , for example, in analytic number theory, in formulas expressing the Euler constant or the Riemann theta function.
They might appear when working with the Gauss transformation $T(x)=frac1x-lfloorfrac1xrfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac1ln(2)int_Bfracdt1+t$.
answered Mar 22 at 11:49
user647486user647486
887111
edited Mar 22 at 12:04
answered Mar 22 at 11:49
user647486user647486
887111
answered Mar 22 at 11:49
user647486user647486
887111
answered Mar 22 at 11:49
user647486user647486
887111
887111
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
add a comment |
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
$begingroup$
It's forcing me to wait 24 hours. Will award tomorrow this time.
$endgroup$
– user150203
Mar 22 at 11:50
add a comment |
$begingroup$
Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
$$int_a^bf([x]),mathrmdx=sum_k=a^b-1f(k).$$
Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
$endgroup$
add a comment |
$begingroup$
Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
$$int_a^bf([x]),mathrmdx=sum_k=a^b-1f(k).$$
Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
$endgroup$
add a comment |
$begingroup$
Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
$$int_a^bf([x]),mathrmdx=sum_k=a^b-1f(k).$$
Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
$endgroup$
Another very useful property of the floor-, ceiling- and nearest integer function is that they are locally constant. That means that for any integrable function $f$ of a real variable, and integers $a<b$, you have
$$int_a^bf([x]),mathrmdx=sum_k=a^b-1f(k).$$
Here $[x]$ is a placeholder for either the floor-, ceiling or nearest integer function. Also the integral (and sum) need not be bounded; in stead of $a$ and $b$ you can take $-infty$ and $infty$, respectively.
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
answered Mar 29 at 1:03
ServaesServaes
29.8k342101
29.8k342101
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059115%2ffractional-integer-based-integrals%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Measure theory? Real analysis?
$endgroup$
– clathratus
Jan 2 at 4:55
1
$begingroup$
On which page...?
$endgroup$
– Darkrai
Jan 2 at 10:51
$begingroup$
@Digamma - this page... which in the context of the post is MSE. Sorry for any confusion caused.
$endgroup$
– user150203
Jan 6 at 12:03
$begingroup$
$y=x$ is the Sawtooth wave, you compute integrals with it to find the amplitudes of its harmonics, its Fourier coefficients.
$endgroup$
– user647486
Mar 22 at 11:46
$begingroup$
Post it as an answer and I'll award you the points.
$endgroup$
– user150203
Mar 22 at 11:47