Fractional/Integer Based integrals The 2019 Stack Overflow Developer Survey Results Are In ...
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Fractional/Integer Based integrals
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)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.
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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 $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.
answered Mar 22 at 11:49
user647486user647486
917111
$endgroup$
$begingroup$
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
$$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
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
30.5k342101
$endgroup$
add a comment |
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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 $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.
answered Mar 22 at 11:49
user647486user647486
917111
$endgroup$
$begingroup$
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 $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.
answered Mar 22 at 11:49
user647486user647486
917111
$endgroup$
$begingroup$
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 $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.
answered Mar 22 at 11:49
user647486user647486
917111
$endgroup$
$y={x}$ 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 $T(x)=frac{1}{x}-lfloorfrac{1}{x}rfloor$, used in the study of continued fractions and rational approximation, and proving that it is Ergodic with respect to its invariant measure $mu(B)=frac{1}{ln(2)}int_{B}frac{dt}{1+t}$.
answered Mar 22 at 11:49
user647486user647486
917111
edited Mar 22 at 12:04
answered Mar 22 at 11:49
user647486user647486
917111
answered Mar 22 at 11:49
user647486user647486
917111
answered Mar 22 at 11:49
user647486user647486
917111
917111
$begingroup$
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
$$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
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
30.5k342101
$endgroup$
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
$$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
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
30.5k342101
$endgroup$
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
$$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
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
30.5k342101
$endgroup$
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
$$int_a^bf([x]),mathrm{d}x=sum_{k=a}^{b-1}f(k).$$
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
30.5k342101
answered Mar 29 at 1:03
ServaesServaes
30.5k342101
answered Mar 29 at 1:03
ServaesServaes
30.5k342101
answered Mar 29 at 1:03
ServaesServaes
30.5k342101
30.5k342101
add a comment |
add a comment |
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$begingroup$
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