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 $frac {-1}6$










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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















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 $frac {-1}6$










share|cite|improve this question
























  • 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













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 $frac {-1}6$










share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 17 at 13:09









Bernard

116k637108




116k637108










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


















  • 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















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