- #1
Gablar16
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Hi This is one of the problems for my take home final exam on differential equations.
I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is
Use the identities in exercise 6 to show that:
[tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}=1[/tex]
and
[tex]\sum^{\infty}_{n=0}(2n+1)J^{2}_{n}J^{2}_{n+1}=\frac{1}{2}x[/tex]
The identities that I already proved in exercise 6 are:
[tex]\frac{1}{2}\frac{d}{dx}(J^{2}_{\alpha}+J^{2}_{\alpha+1})=\frac{\alpha}{x}J^{2}_{\alpha}-\frac{\alpha+1}{x}J^{2}_{\alpha+1}[/tex]
and
[tex]\frac{d}{dx}(xJ^{2}_{\alpha}J^{2}_{\alpha+1})=x(J^{2}_{\alpha}-J^{2}_{\alpha+1})[/tex]
As I mentioned in the introduction I have looked for possible solutions and background information for a while but I'm still stuck.
The best source for information on this problem is http://books.google.com/books?id=d4...SQ&sig=t6mTG7P19IxaJ797Ri-1NojAgWc#PPA361,M1" but I also don't undertstand how hansen changes the from
[tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}[/tex]
to
[tex]\left(e^{\frac{1}{2}z\frac{t-1}{t}}\right)\left(e^{\frac{1}{2}z\frac{-t+1}{t}}\right)[/tex]
I think the problem is that I just don't have enough knowledge of power series to know how to handle this problem. I just don't know how to approach the problem. I though t that maybe:
[tex]2\sum^{\infty}_{n=1}\frac{J^{2}_{0}}{2}+J^{2}_{n}[/tex](is this even right?)
and then try to change it into something that looks like a recurrence relation
other approach I tried was to expand the series and try recurrence relations there. but I got stuck there too.
Thanks very much for any help or links to relevant information.
I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is
Homework Statement
Use the identities in exercise 6 to show that:
[tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}=1[/tex]
and
[tex]\sum^{\infty}_{n=0}(2n+1)J^{2}_{n}J^{2}_{n+1}=\frac{1}{2}x[/tex]
Homework Equations
The identities that I already proved in exercise 6 are:
[tex]\frac{1}{2}\frac{d}{dx}(J^{2}_{\alpha}+J^{2}_{\alpha+1})=\frac{\alpha}{x}J^{2}_{\alpha}-\frac{\alpha+1}{x}J^{2}_{\alpha+1}[/tex]
and
[tex]\frac{d}{dx}(xJ^{2}_{\alpha}J^{2}_{\alpha+1})=x(J^{2}_{\alpha}-J^{2}_{\alpha+1})[/tex]
The Attempt at a Solution
As I mentioned in the introduction I have looked for possible solutions and background information for a while but I'm still stuck.
The best source for information on this problem is http://books.google.com/books?id=d4...SQ&sig=t6mTG7P19IxaJ797Ri-1NojAgWc#PPA361,M1" but I also don't undertstand how hansen changes the from
[tex]J^{2}_{0}+2\sum^{\infty}_{n=1}J^{2}_{n}[/tex]
to
[tex]\left(e^{\frac{1}{2}z\frac{t-1}{t}}\right)\left(e^{\frac{1}{2}z\frac{-t+1}{t}}\right)[/tex]
I think the problem is that I just don't have enough knowledge of power series to know how to handle this problem. I just don't know how to approach the problem. I though t that maybe:
[tex]2\sum^{\infty}_{n=1}\frac{J^{2}_{0}}{2}+J^{2}_{n}[/tex](is this even right?)
and then try to change it into something that looks like a recurrence relation
other approach I tried was to expand the series and try recurrence relations there. but I got stuck there too.
Thanks very much for any help or links to relevant information.
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