Orthogonality of Legendre polynomials with logarithmic functions
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I have to find the value of this integral:
$int_{-1}^1 ln(1-x)*P_3(x),dx$
where $P_3(x)$ is the Legendre polynomial.
I thought I can write $ln(1-x)$ as a summation of Legendre polynomials and then use the orthogonality relation to find the answer. That didn't really work, the closest I got was to:
$ln(1-x)$=$sum_{n=1}^infty P_n(1)*x^n$ and that's not getting me anywhere.
Is this the right track of thought? Or is this question done in an entirely different method?
P.S The answer is $frac {-1}6$
integration orthogonality legendre-polynomials
edited Nov 17 at 13:09
Bernard
116k637108
asked Nov 17 at 13:04
Shikhar Asthana
61
add a comment |
up vote
1
down vote
favorite
I have to find the value of this integral:
$int_{-1}^1 ln(1-x)*P_3(x),dx$
where $P_3(x)$ is the Legendre polynomial.
I thought I can write $ln(1-x)$ as a summation of Legendre polynomials and then use the orthogonality relation to find the answer. That didn't really work, the closest I got was to:
$ln(1-x)$=$sum_{n=1}^infty P_n(1)*x^n$ and that's not getting me anywhere.
Is this the right track of thought? Or is this question done in an entirely different method?
P.S The answer is $frac {-1}6$
integration orthogonality legendre-polynomials
edited Nov 17 at 13:09
Bernard
116k637108
asked Nov 17 at 13:04
Shikhar Asthana
61
Writing a transcendental function as polynomial sounds difficult. How are Legendre polynomials defined?
– mathreadler
Nov 17 at 13:16
Got it! Replace it by Rodrigues' formula and use integration by parts 3 times
– Shikhar Asthana
Nov 17 at 13:21
Yes to consider the differential equation properties of these functions can be useful.
– mathreadler
Nov 17 at 13:33
Thank you for your help !
– Shikhar Asthana
Nov 17 at 14:12
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have to find the value of this integral:
$int_{-1}^1 ln(1-x)*P_3(x),dx$
where $P_3(x)$ is the Legendre polynomial.
I thought I can write $ln(1-x)$ as a summation of Legendre polynomials and then use the orthogonality relation to find the answer. That didn't really work, the closest I got was to:
$ln(1-x)$=$sum_{n=1}^infty P_n(1)*x^n$ and that's not getting me anywhere.
Is this the right track of thought? Or is this question done in an entirely different method?
P.S The answer is $frac {-1}6$
integration orthogonality legendre-polynomials
edited Nov 17 at 13:09
Bernard
116k637108
asked Nov 17 at 13:04
Shikhar Asthana
61
I have to find the value of this integral:
$int_{-1}^1 ln(1-x)*P_3(x),dx$
where $P_3(x)$ is the Legendre polynomial.
I thought I can write $ln(1-x)$ as a summation of Legendre polynomials and then use the orthogonality relation to find the answer. That didn't really work, the closest I got was to:
$ln(1-x)$=$sum_{n=1}^infty P_n(1)*x^n$ and that's not getting me anywhere.
Is this the right track of thought? Or is this question done in an entirely different method?
P.S The answer is $frac {-1}6$
integration orthogonality legendre-polynomials
integration orthogonality legendre-polynomials
edited Nov 17 at 13:09
Bernard
116k637108
asked Nov 17 at 13:04
Shikhar Asthana
61
edited Nov 17 at 13:09
Bernard
116k637108
asked Nov 17 at 13:04
Shikhar Asthana
61
edited Nov 17 at 13:09
Bernard
116k637108
edited Nov 17 at 13:09
Bernard
116k637108
edited Nov 17 at 13:09
Bernard
116k637108
116k637108
asked Nov 17 at 13:04
Shikhar Asthana
61
asked Nov 17 at 13:04
Shikhar Asthana
61
asked Nov 17 at 13:04
Shikhar Asthana
61
61
Writing a transcendental function as polynomial sounds difficult. How are Legendre polynomials defined?
– mathreadler
Nov 17 at 13:16
Got it! Replace it by Rodrigues' formula and use integration by parts 3 times
– Shikhar Asthana
Nov 17 at 13:21
Yes to consider the differential equation properties of these functions can be useful.
– mathreadler
Nov 17 at 13:33
Thank you for your help !
– Shikhar Asthana
Nov 17 at 14:12
add a comment |
Writing a transcendental function as polynomial sounds difficult. How are Legendre polynomials defined?
– mathreadler
Nov 17 at 13:16
Got it! Replace it by Rodrigues' formula and use integration by parts 3 times
– Shikhar Asthana
Nov 17 at 13:21
Yes to consider the differential equation properties of these functions can be useful.
– mathreadler
Nov 17 at 13:33
Thank you for your help !
– Shikhar Asthana
Nov 17 at 14:12
Writing a transcendental function as polynomial sounds difficult. How are Legendre polynomials defined?
– mathreadler
Nov 17 at 13:16
Writing a transcendental function as polynomial sounds difficult. How are Legendre polynomials defined?
– mathreadler
Nov 17 at 13:16
Got it! Replace it by Rodrigues' formula and use integration by parts 3 times
– Shikhar Asthana
Nov 17 at 13:21
Got it! Replace it by Rodrigues' formula and use integration by parts 3 times
– Shikhar Asthana
Nov 17 at 13:21
Yes to consider the differential equation properties of these functions can be useful.
– mathreadler
Nov 17 at 13:33
Yes to consider the differential equation properties of these functions can be useful.
– mathreadler
Nov 17 at 13:33
Thank you for your help !
– Shikhar Asthana
Nov 17 at 14:12
Thank you for your help !
– Shikhar Asthana
Nov 17 at 14:12
add a comment |
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Writing a transcendental function as polynomial sounds difficult. How are Legendre polynomials defined?
– mathreadler
Nov 17 at 13:16
Got it! Replace it by Rodrigues' formula and use integration by parts 3 times
– Shikhar Asthana
Nov 17 at 13:21
Yes to consider the differential equation properties of these functions can be useful.
– mathreadler
Nov 17 at 13:33
Thank you for your help !
– Shikhar Asthana
Nov 17 at 14:12