Integral of squared Hypergeometric Function
up vote
0
down vote
favorite
I am trying to integrate the following
$int_{0}^{1} {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
where $minBbb Z$ and $ninBbb Z$ with $m>0$, $ngeq 0$.
(Basically I want to normalise the function). I am getting this as a part of an effort to find some uniform large approximation for a certain associated Legendre polynomial (in case people are wondering whether it is homework).
I know that
$int_{0}^{1} (z(1-z))^m {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
is integrable, under the same conditions, and gives
$frac{n! Gamma (m+1)^2 Gamma (m+n+1) Gamma (2 m+2 n+1)}{2Gamma (m+n+1) Gamma (2 m+n+1) Gamma (2 m+2 n+2)}quad;{_2{F}_1}big(0, 1 + 2 m + 2 n, 2 m + 2 + 2 n, 1big)$.
The above is a table integral from I. S. Gradshteyn and I. M. Ryzhik 8th edition (doesn't appear in the 7th)
I welcome any hints or suggestions.
integration definite-integrals hypergeometric-function
asked Nov 15 at 17:16
ThunderBiggi
1356
add a comment |
up vote
0
down vote
favorite
I am trying to integrate the following
$int_{0}^{1} {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
where $minBbb Z$ and $ninBbb Z$ with $m>0$, $ngeq 0$.
(Basically I want to normalise the function). I am getting this as a part of an effort to find some uniform large approximation for a certain associated Legendre polynomial (in case people are wondering whether it is homework).
I know that
$int_{0}^{1} (z(1-z))^m {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
is integrable, under the same conditions, and gives
$frac{n! Gamma (m+1)^2 Gamma (m+n+1) Gamma (2 m+2 n+1)}{2Gamma (m+n+1) Gamma (2 m+n+1) Gamma (2 m+2 n+2)}quad;{_2{F}_1}big(0, 1 + 2 m + 2 n, 2 m + 2 + 2 n, 1big)$.
The above is a table integral from I. S. Gradshteyn and I. M. Ryzhik 8th edition (doesn't appear in the 7th)
I welcome any hints or suggestions.
integration definite-integrals hypergeometric-function
asked Nov 15 at 17:16
ThunderBiggi
1356
I can help with the Mathjax integers. Which one do you want, $$m,nin Bbb Z$$ Or $$m,ninBbb N$$
– clathratus
Nov 15 at 18:09
@clathratus the first one, but isn't that just a normal $Z$ ?
– ThunderBiggi
Nov 15 at 21:17
2
You can see how someone else created a formula by right-clicking on it and selecting "Show Math As > TeX commands". In this case, you well seeBbb Z. the "Bbb" is just an abbreviation for "mathbb", and you will note that the { } are not needed if you are only applying the operation to a single character (though a space is needed to allow it to recognize where the operation name ends and the target begins). "Bbb Z" is the same as "mathbb{Z}".
– Paul Sinclair
Nov 16 at 0:24
As Paul said, $Bbb Z$ is made by "Bbb Z". Same thing with $Bbb N$: "Bbb N". hell, you could even do $Bbb H$
– clathratus
Nov 16 at 0:38
Thank you people!
– ThunderBiggi
Nov 16 at 9:25
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to integrate the following
$int_{0}^{1} {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
where $minBbb Z$ and $ninBbb Z$ with $m>0$, $ngeq 0$.
(Basically I want to normalise the function). I am getting this as a part of an effort to find some uniform large approximation for a certain associated Legendre polynomial (in case people are wondering whether it is homework).
I know that
$int_{0}^{1} (z(1-z))^m {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
is integrable, under the same conditions, and gives
$frac{n! Gamma (m+1)^2 Gamma (m+n+1) Gamma (2 m+2 n+1)}{2Gamma (m+n+1) Gamma (2 m+n+1) Gamma (2 m+2 n+2)}quad;{_2{F}_1}big(0, 1 + 2 m + 2 n, 2 m + 2 + 2 n, 1big)$.
The above is a table integral from I. S. Gradshteyn and I. M. Ryzhik 8th edition (doesn't appear in the 7th)
I welcome any hints or suggestions.
integration definite-integrals hypergeometric-function
asked Nov 15 at 17:16
ThunderBiggi
1356
I am trying to integrate the following
$int_{0}^{1} {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
where $minBbb Z$ and $ninBbb Z$ with $m>0$, $ngeq 0$.
(Basically I want to normalise the function). I am getting this as a part of an effort to find some uniform large approximation for a certain associated Legendre polynomial (in case people are wondering whether it is homework).
I know that
$int_{0}^{1} (z(1-z))^m {_2{F}_1}big(-n,1+2m+n,1+m,1-zbig)^2 dz$,
is integrable, under the same conditions, and gives
$frac{n! Gamma (m+1)^2 Gamma (m+n+1) Gamma (2 m+2 n+1)}{2Gamma (m+n+1) Gamma (2 m+n+1) Gamma (2 m+2 n+2)}quad;{_2{F}_1}big(0, 1 + 2 m + 2 n, 2 m + 2 + 2 n, 1big)$.
The above is a table integral from I. S. Gradshteyn and I. M. Ryzhik 8th edition (doesn't appear in the 7th)
I welcome any hints or suggestions.
integration definite-integrals hypergeometric-function
integration definite-integrals hypergeometric-function
asked Nov 15 at 17:16
ThunderBiggi
1356
asked Nov 15 at 17:16
ThunderBiggi
1356
edited Nov 16 at 9:24
asked Nov 15 at 17:16
ThunderBiggi
1356
asked Nov 15 at 17:16
ThunderBiggi
1356
asked Nov 15 at 17:16
ThunderBiggi
1356
1356
I can help with the Mathjax integers. Which one do you want, $$m,nin Bbb Z$$ Or $$m,ninBbb N$$
– clathratus
Nov 15 at 18:09
@clathratus the first one, but isn't that just a normal $Z$ ?
– ThunderBiggi
Nov 15 at 21:17
2
You can see how someone else created a formula by right-clicking on it and selecting "Show Math As > TeX commands". In this case, you well seeBbb Z. the "Bbb" is just an abbreviation for "mathbb", and you will note that the { } are not needed if you are only applying the operation to a single character (though a space is needed to allow it to recognize where the operation name ends and the target begins). "Bbb Z" is the same as "mathbb{Z}".
– Paul Sinclair
Nov 16 at 0:24
As Paul said, $Bbb Z$ is made by "Bbb Z". Same thing with $Bbb N$: "Bbb N". hell, you could even do $Bbb H$
– clathratus
Nov 16 at 0:38
Thank you people!
– ThunderBiggi
Nov 16 at 9:25
add a comment |
I can help with the Mathjax integers. Which one do you want, $$m,nin Bbb Z$$ Or $$m,ninBbb N$$
– clathratus
Nov 15 at 18:09
@clathratus the first one, but isn't that just a normal $Z$ ?
– ThunderBiggi
Nov 15 at 21:17
2
You can see how someone else created a formula by right-clicking on it and selecting "Show Math As > TeX commands". In this case, you well seeBbb Z. the "Bbb" is just an abbreviation for "mathbb", and you will note that the { } are not needed if you are only applying the operation to a single character (though a space is needed to allow it to recognize where the operation name ends and the target begins). "Bbb Z" is the same as "mathbb{Z}".
– Paul Sinclair
Nov 16 at 0:24
As Paul said, $Bbb Z$ is made by "Bbb Z". Same thing with $Bbb N$: "Bbb N". hell, you could even do $Bbb H$
– clathratus
Nov 16 at 0:38
Thank you people!
– ThunderBiggi
Nov 16 at 9:25
I can help with the Mathjax integers. Which one do you want, $$m,nin Bbb Z$$ Or $$m,ninBbb N$$
– clathratus
Nov 15 at 18:09
I can help with the Mathjax integers. Which one do you want, $$m,nin Bbb Z$$ Or $$m,ninBbb N$$
– clathratus
Nov 15 at 18:09
@clathratus the first one, but isn't that just a normal $Z$ ?
– ThunderBiggi
Nov 15 at 21:17
@clathratus the first one, but isn't that just a normal $Z$ ?
– ThunderBiggi
Nov 15 at 21:17
2
2
You can see how someone else created a formula by right-clicking on it and selecting "Show Math As > TeX commands". In this case, you well see
Bbb Z. the "Bbb" is just an abbreviation for "mathbb", and you will note that the { } are not needed if you are only applying the operation to a single character (though a space is needed to allow it to recognize where the operation name ends and the target begins). "Bbb Z" is the same as "mathbb{Z}".– Paul Sinclair
Nov 16 at 0:24
You can see how someone else created a formula by right-clicking on it and selecting "Show Math As > TeX commands". In this case, you well see
Bbb Z. the "Bbb" is just an abbreviation for "mathbb", and you will note that the { } are not needed if you are only applying the operation to a single character (though a space is needed to allow it to recognize where the operation name ends and the target begins). "Bbb Z" is the same as "mathbb{Z}".– Paul Sinclair
Nov 16 at 0:24
As Paul said, $Bbb Z$ is made by "Bbb Z". Same thing with $Bbb N$: "Bbb N". hell, you could even do $Bbb H$
– clathratus
Nov 16 at 0:38
As Paul said, $Bbb Z$ is made by "Bbb Z". Same thing with $Bbb N$: "Bbb N". hell, you could even do $Bbb H$
– clathratus
Nov 16 at 0:38
Thank you people!
– ThunderBiggi
Nov 16 at 9:25
Thank you people!
– ThunderBiggi
Nov 16 at 9:25
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f2999981%2fintegral-of-squared-hypergeometric-function%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
I can help with the Mathjax integers. Which one do you want, $$m,nin Bbb Z$$ Or $$m,ninBbb N$$
– clathratus
Nov 15 at 18:09
@clathratus the first one, but isn't that just a normal $Z$ ?
– ThunderBiggi
Nov 15 at 21:17
2
You can see how someone else created a formula by right-clicking on it and selecting "Show Math As > TeX commands". In this case, you well see
Bbb Z. the "Bbb" is just an abbreviation for "mathbb", and you will note that the { } are not needed if you are only applying the operation to a single character (though a space is needed to allow it to recognize where the operation name ends and the target begins). "Bbb Z" is the same as "mathbb{Z}".– Paul Sinclair
Nov 16 at 0:24
As Paul said, $Bbb Z$ is made by "Bbb Z". Same thing with $Bbb N$: "Bbb N". hell, you could even do $Bbb H$
– clathratus
Nov 16 at 0:38
Thank you people!
– ThunderBiggi
Nov 16 at 9:25