Hermite polynomials, prove the solution [closed] Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)limiting behaviour of the heat kernel on the real lineUse the full Fourier Transform in x to solveUsing Fourier Transforms to evaluate $int_-infty^infty x^k space f(x)dx $Hermite functions as eigenvectors of Fourier transformIntegral of $int_-infty^+inftyleft |fracsinxx(1+x^2)right|^2,dx $Evaluating the Fourier transform of the following piecewise functionsConstruct a sequence using the fourier transformWrite the Fourier integral of x^2Fourier representation of complex Dirac functionIntegration of modified bessel function $int_x_1-sigma^x_1+sigmaxexp(-t_1x^2)cdot I_0(t_2x) dx$
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Hermite polynomials, prove the solution [closed]
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)limiting behaviour of the heat kernel on the real lineUse the full Fourier Transform in x to solveUsing Fourier Transforms to evaluate $int_-infty^infty x^k space f(x)dx $Hermite functions as eigenvectors of Fourier transformIntegral of $int_-infty^+inftyleft |fracsinxx(1+x^2)right|^2,dx $Evaluating the Fourier transform of the following piecewise functionsConstruct a sequence using the fourier transformWrite the Fourier integral of x^2Fourier representation of complex Dirac functionIntegration of modified bessel function $int_x_1-sigma^x_1+sigmaxexp(-t_1x^2)cdot I_0(t_2x) dx$
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text The Hermite polynomials, H_n(x) text , satisfy the following: $
beginarrayltext i. <H_N, H_M>=int_-infty^infty e^-x^2 H_n(x) H_m(x) d x=sqrtpi 2^n n ! delta_n, m \ text ii. quad H_n^prime(x)=2 n H_n-1(x) \ text iii. H_n+1(x)=2 x H_n(x)-2 n H_n-1(x) \ text iv. H_n(x)=(-1)^n e^x^2 fracd^nd x^nleft(e^-x^2right)endarray
Using these, show that
beginarrayltext a. quad H_n^prime prime-2 x H_n^prime+2 n H_n=0 . text [Use properties ii. and iii.] endarray
Any help on this would be greatly appreciated!
fourier-analysis bessel-functions
edited Apr 1 at 0:33
MarianD
2,2611618
asked Apr 1 at 0:09
JetJet
82
$endgroup$
closed as off-topic by Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu Apr 1 at 4:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu
add a comment |
$begingroup$
$
text The Hermite polynomials, H_n(x) text , satisfy the following: $
beginarrayltext i. <H_N, H_M>=int_-infty^infty e^-x^2 H_n(x) H_m(x) d x=sqrtpi 2^n n ! delta_n, m \ text ii. quad H_n^prime(x)=2 n H_n-1(x) \ text iii. H_n+1(x)=2 x H_n(x)-2 n H_n-1(x) \ text iv. H_n(x)=(-1)^n e^x^2 fracd^nd x^nleft(e^-x^2right)endarray
Using these, show that
beginarrayltext a. quad H_n^prime prime-2 x H_n^prime+2 n H_n=0 . text [Use properties ii. and iii.] endarray
Any help on this would be greatly appreciated!
fourier-analysis bessel-functions
edited Apr 1 at 0:33
MarianD
2,2611618
asked Apr 1 at 0:09
JetJet
82
$endgroup$
closed as off-topic by Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu Apr 1 at 4:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu
1
$begingroup$
Welcome to Math.SE. Please use MathJax when writing equations, so that they are easier to read. Furthermore, if you show what you were been able to do it may be easier to get an helpful answer.
$endgroup$
– Ertxiem
Apr 1 at 0:12
add a comment |
$begingroup$
$
text The Hermite polynomials, H_n(x) text , satisfy the following: $
beginarrayltext i. <H_N, H_M>=int_-infty^infty e^-x^2 H_n(x) H_m(x) d x=sqrtpi 2^n n ! delta_n, m \ text ii. quad H_n^prime(x)=2 n H_n-1(x) \ text iii. H_n+1(x)=2 x H_n(x)-2 n H_n-1(x) \ text iv. H_n(x)=(-1)^n e^x^2 fracd^nd x^nleft(e^-x^2right)endarray
Using these, show that
beginarrayltext a. quad H_n^prime prime-2 x H_n^prime+2 n H_n=0 . text [Use properties ii. and iii.] endarray
Any help on this would be greatly appreciated!
fourier-analysis bessel-functions
edited Apr 1 at 0:33
MarianD
2,2611618
asked Apr 1 at 0:09
JetJet
82
$endgroup$
$
text The Hermite polynomials, H_n(x) text , satisfy the following: $
beginarrayltext i. <H_N, H_M>=int_-infty^infty e^-x^2 H_n(x) H_m(x) d x=sqrtpi 2^n n ! delta_n, m \ text ii. quad H_n^prime(x)=2 n H_n-1(x) \ text iii. H_n+1(x)=2 x H_n(x)-2 n H_n-1(x) \ text iv. H_n(x)=(-1)^n e^x^2 fracd^nd x^nleft(e^-x^2right)endarray
Using these, show that
beginarrayltext a. quad H_n^prime prime-2 x H_n^prime+2 n H_n=0 . text [Use properties ii. and iii.] endarray
Any help on this would be greatly appreciated!
fourier-analysis bessel-functions
fourier-analysis bessel-functions
edited Apr 1 at 0:33
MarianD
2,2611618
asked Apr 1 at 0:09
JetJet
82
edited Apr 1 at 0:33
MarianD
2,2611618
asked Apr 1 at 0:09
JetJet
82
edited Apr 1 at 0:33
MarianD
2,2611618
edited Apr 1 at 0:33
MarianD
2,2611618
edited Apr 1 at 0:33
MarianD
2,2611618
2,2611618
asked Apr 1 at 0:09
JetJet
82
asked Apr 1 at 0:09
JetJet
82
asked Apr 1 at 0:09
JetJet
82
82
closed as off-topic by Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu Apr 1 at 4:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu
closed as off-topic by Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu Apr 1 at 4:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Morgan Rodgers, John Omielan, Martin Argerami, Lord Shark the Unknown, Tianlalu
1
$begingroup$
Welcome to Math.SE. Please use MathJax when writing equations, so that they are easier to read. Furthermore, if you show what you were been able to do it may be easier to get an helpful answer.
$endgroup$
– Ertxiem
Apr 1 at 0:12
add a comment |
1
$begingroup$
Welcome to Math.SE. Please use MathJax when writing equations, so that they are easier to read. Furthermore, if you show what you were been able to do it may be easier to get an helpful answer.
$endgroup$
– Ertxiem
Apr 1 at 0:12
1
1
$begingroup$
Welcome to Math.SE. Please use MathJax when writing equations, so that they are easier to read. Furthermore, if you show what you were been able to do it may be easier to get an helpful answer.
$endgroup$
– Ertxiem
Apr 1 at 0:12
$begingroup$
Welcome to Math.SE. Please use MathJax when writing equations, so that they are easier to read. Furthermore, if you show what you were been able to do it may be easier to get an helpful answer.
$endgroup$
– Ertxiem
Apr 1 at 0:12
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1 Answer
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$H_n^''-2xH_n^'+2nH_n = (2nH_n-1)^'-4xnH_n-1+2nH_n$ (by ii)
= $4n^2 H_n-2 - 4xnH_n-1+2nH_n $ (by ii again)
= $2n(H_n-2xH_n-1+2nH_n-2) = 0$ (by iii)
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$H_n^''-2xH_n^'+2nH_n = (2nH_n-1)^'-4xnH_n-1+2nH_n$ (by ii)
= $4n^2 H_n-2 - 4xnH_n-1+2nH_n $ (by ii again)
= $2n(H_n-2xH_n-1+2nH_n-2) = 0$ (by iii)
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
$endgroup$
add a comment |
$begingroup$
$H_n^''-2xH_n^'+2nH_n = (2nH_n-1)^'-4xnH_n-1+2nH_n$ (by ii)
= $4n^2 H_n-2 - 4xnH_n-1+2nH_n $ (by ii again)
= $2n(H_n-2xH_n-1+2nH_n-2) = 0$ (by iii)
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
$endgroup$
add a comment |
$begingroup$
$H_n^''-2xH_n^'+2nH_n = (2nH_n-1)^'-4xnH_n-1+2nH_n$ (by ii)
= $4n^2 H_n-2 - 4xnH_n-1+2nH_n $ (by ii again)
= $2n(H_n-2xH_n-1+2nH_n-2) = 0$ (by iii)
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
$endgroup$
$H_n^''-2xH_n^'+2nH_n = (2nH_n-1)^'-4xnH_n-1+2nH_n$ (by ii)
= $4n^2 H_n-2 - 4xnH_n-1+2nH_n $ (by ii again)
= $2n(H_n-2xH_n-1+2nH_n-2) = 0$ (by iii)
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
answered Apr 1 at 0:51
George DewhirstGeorge Dewhirst
90714
90714
add a comment |
add a comment |
1
$begingroup$
Welcome to Math.SE. Please use MathJax when writing equations, so that they are easier to read. Furthermore, if you show what you were been able to do it may be easier to get an helpful answer.
$endgroup$
– Ertxiem
Apr 1 at 0:12