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Moolan
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Joined: 05 Aug 2005
Posts: 30
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 Posted: Mon, 15 May 2006 09:16:28 UTC Post subject: Binomial series |
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Hi,
How do you do an expansion of (1+s^-2)^-1/2 about s= inf. I only know how to expand it, if the power in an integer. Help pls. Thanks. |
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Kungsman
Member of the 'S.O.S. Math' Hall of Fame
Joined: 04 Sep 2004
Posts: 3400
Location: Uppsala, Sweden
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| Posted: Mon, 15 May 2006 11:27:37 UTC Post subject: Re: Binomial series |
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| Moolan wrote: | Hi,
How do you do an expansion of (1+s^-2)^-1/2 about s= inf. I only know how to expand it, if the power in an integer. Help pls. Thanks. |
Call the expression  . To expand it at  , expand
at  . This is done just as usual:
where
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Moolan
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Joined: 05 Aug 2005
Posts: 30
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| Posted: Mon, 15 May 2006 14:09:44 UTC Post subject: |
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The s above, does it represent my function ((1+s^-2)^-1/2) or just some arbitary function? If so, how do you get s^-2k?
Could you expand this term a bit more, i dont really see the pattern.
Does it mean that wehn k=0, i get (-1/2)(-3/2)/0!
and k=1, (-1/2)(-3/2)(-5/2)/1! |
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Kungsman
Member of the 'S.O.S. Math' Hall of Fame
Joined: 04 Sep 2004
Posts: 3400
Location: Uppsala, Sweden
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| Posted: Mon, 15 May 2006 14:36:43 UTC Post subject: |
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| Moolan wrote: |
The s above, does it represent my function ((1+s^-2)^-1/2) or just some arbitary function? If so, how do you get s^-2k? |
Of course it isn't some arbitrary function. At the beginning of my post I (we?) agreed to call the given function . This  is of course the very same . And you see, to get back to from  we need to reciprocate back  to  .
| Quote: |
Could you expand this term a bit more, i dont really see the pattern.
Does it mean that wehn k=0, i get (-1/2)(-3/2)/0!
and k=1, (-1/2)(-3/2)(-5/2)/1! |
It's just as usual. For integers  and  (with further restrictions such as  ) we have
For real numbers the middle expression suits badly (what is  ?), so we prefer the last one to generalize the binomial coefficient. And the fact that the binomial theorem generalizes as well is just Taylor's formula.
The first three coefficients would then be
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