General formula for integrating analytic function with linear term
$begingroup$
Given an analytic function $, f : mathbb{R} rightarrow mathbb{R}$ with no known closed formula for its antiderivative. Assume further that by some clever tricks you managed to calculate the definite integral for a fixed number $c$:
$$A := int_0^c f(t) ,dt ; .$$
Is there a way to calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f^{(n)}$?
Integration by parts yields
$$int_0^c t , f(t) , dt = cA - int_0^c int_0^t , f(tau) , dtau ,dt ; ,$$
which made it seem plausible to me that such a formula exists.
Related Question:
This problem is an abstraction of my more concrete question asked here.
integration definite-integrals analytic-functions
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
$endgroup$
add a comment |
$begingroup$
Given an analytic function $, f : mathbb{R} rightarrow mathbb{R}$ with no known closed formula for its antiderivative. Assume further that by some clever tricks you managed to calculate the definite integral for a fixed number $c$:
$$A := int_0^c f(t) ,dt ; .$$
Is there a way to calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f^{(n)}$?
Integration by parts yields
$$int_0^c t , f(t) , dt = cA - int_0^c int_0^t , f(tau) , dtau ,dt ; ,$$
which made it seem plausible to me that such a formula exists.
Related Question:
This problem is an abstraction of my more concrete question asked here.
integration definite-integrals analytic-functions
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
$endgroup$
$begingroup$
You need to be more explicit about what "calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f'$" means. Your integration-by-parts formula is such a calculation, which uses only $A$ and $f$. Why is this not acceptable? What is it you are after that this does not provide?
$endgroup$
– Paul Sinclair
Jan 5 at 12:02
$begingroup$
My integration-by-parts formula is not acceptable because the antiderivative of $f$ is not known. Hence, this formula is useless as it cannot be evaluated (at least not in an obvious way). Ultimately I just want to be able to explicitly calculate the integral $int_{0}^c tf(t)dt$, only knowing $A$ and how to evaluate $f$, $f^{(n)}$.
$endgroup$
– chickenNinja123
Jan 5 at 12:18
add a comment |
$begingroup$
Given an analytic function $, f : mathbb{R} rightarrow mathbb{R}$ with no known closed formula for its antiderivative. Assume further that by some clever tricks you managed to calculate the definite integral for a fixed number $c$:
$$A := int_0^c f(t) ,dt ; .$$
Is there a way to calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f^{(n)}$?
Integration by parts yields
$$int_0^c t , f(t) , dt = cA - int_0^c int_0^t , f(tau) , dtau ,dt ; ,$$
which made it seem plausible to me that such a formula exists.
Related Question:
This problem is an abstraction of my more concrete question asked here.
integration definite-integrals analytic-functions
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
$endgroup$
Given an analytic function $, f : mathbb{R} rightarrow mathbb{R}$ with no known closed formula for its antiderivative. Assume further that by some clever tricks you managed to calculate the definite integral for a fixed number $c$:
$$A := int_0^c f(t) ,dt ; .$$
Is there a way to calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f^{(n)}$?
Integration by parts yields
$$int_0^c t , f(t) , dt = cA - int_0^c int_0^t , f(tau) , dtau ,dt ; ,$$
which made it seem plausible to me that such a formula exists.
Related Question:
This problem is an abstraction of my more concrete question asked here.
integration definite-integrals analytic-functions
integration definite-integrals analytic-functions
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
edited Jan 5 at 12:22
chickenNinja123
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
asked Jan 4 at 21:35
chickenNinja123chickenNinja123
7713
7713
$begingroup$
You need to be more explicit about what "calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f'$" means. Your integration-by-parts formula is such a calculation, which uses only $A$ and $f$. Why is this not acceptable? What is it you are after that this does not provide?
$endgroup$
– Paul Sinclair
Jan 5 at 12:02
$begingroup$
My integration-by-parts formula is not acceptable because the antiderivative of $f$ is not known. Hence, this formula is useless as it cannot be evaluated (at least not in an obvious way). Ultimately I just want to be able to explicitly calculate the integral $int_{0}^c tf(t)dt$, only knowing $A$ and how to evaluate $f$, $f^{(n)}$.
$endgroup$
– chickenNinja123
Jan 5 at 12:18
add a comment |
$begingroup$
You need to be more explicit about what "calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f'$" means. Your integration-by-parts formula is such a calculation, which uses only $A$ and $f$. Why is this not acceptable? What is it you are after that this does not provide?
$endgroup$
– Paul Sinclair
Jan 5 at 12:02
$begingroup$
My integration-by-parts formula is not acceptable because the antiderivative of $f$ is not known. Hence, this formula is useless as it cannot be evaluated (at least not in an obvious way). Ultimately I just want to be able to explicitly calculate the integral $int_{0}^c tf(t)dt$, only knowing $A$ and how to evaluate $f$, $f^{(n)}$.
$endgroup$
– chickenNinja123
Jan 5 at 12:18
$begingroup$
You need to be more explicit about what "calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f'$" means. Your integration-by-parts formula is such a calculation, which uses only $A$ and $f$. Why is this not acceptable? What is it you are after that this does not provide?
$endgroup$
– Paul Sinclair
Jan 5 at 12:02
$begingroup$
You need to be more explicit about what "calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f'$" means. Your integration-by-parts formula is such a calculation, which uses only $A$ and $f$. Why is this not acceptable? What is it you are after that this does not provide?
$endgroup$
– Paul Sinclair
Jan 5 at 12:02
$begingroup$
My integration-by-parts formula is not acceptable because the antiderivative of $f$ is not known. Hence, this formula is useless as it cannot be evaluated (at least not in an obvious way). Ultimately I just want to be able to explicitly calculate the integral $int_{0}^c tf(t)dt$, only knowing $A$ and how to evaluate $f$, $f^{(n)}$.
$endgroup$
– chickenNinja123
Jan 5 at 12:18
$begingroup$
My integration-by-parts formula is not acceptable because the antiderivative of $f$ is not known. Hence, this formula is useless as it cannot be evaluated (at least not in an obvious way). Ultimately I just want to be able to explicitly calculate the integral $int_{0}^c tf(t)dt$, only knowing $A$ and how to evaluate $f$, $f^{(n)}$.
$endgroup$
– chickenNinja123
Jan 5 at 12:18
add a comment |
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$begingroup$
You need to be more explicit about what "calculate $int_0^c t , f(t) , dt$ in terms of $A$, $f$ and $f'$" means. Your integration-by-parts formula is such a calculation, which uses only $A$ and $f$. Why is this not acceptable? What is it you are after that this does not provide?
$endgroup$
– Paul Sinclair
Jan 5 at 12:02
$begingroup$
My integration-by-parts formula is not acceptable because the antiderivative of $f$ is not known. Hence, this formula is useless as it cannot be evaluated (at least not in an obvious way). Ultimately I just want to be able to explicitly calculate the integral $int_{0}^c tf(t)dt$, only knowing $A$ and how to evaluate $f$, $f^{(n)}$.
$endgroup$
– chickenNinja123
Jan 5 at 12:18