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Re: Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !
*To*: mathgroup at smc.vnet.net
*Subject*: [mg70237] Re: Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !
*From*: dimmechan at yahoo.com
*Date*: Sun, 8 Oct 2006 02:05:04 -0400 (EDT)
*References*: <eg845g$nfd$1@smc.vnet.net>
Let me state that your post is quite unreadable.
Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, Assumptions ->
C ≥ 0 && L ≥ 0 && n ≥ 0]
\!\(If[C > 0 && L > 0, L\^n\ \@\(C\ L\)\ \@π\ Gamma[1 + n]\
Hypergeometric2F1Regularized[\(-\(1\/2\)\),
1 + n, 3\/2 + n, \(-\(L\/C\)\)], Integrate[x\^n\
\@\(\(C +
x\)\/\(L - \x\)\), {x, 0, L},
Assumptions -> C ≤ 0 || L ≤ 0]]\)
Are you sure that the input and output match together?
I have my reasons to object this. I got
Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, (Assumptions -> C &
)*#8805; (0 && L & )*#8805; (0 && n & )*#8805; 0]
Integrate::"ilim" : "Invalid integration variable or limit(s) in
\!\(0\) ."
Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, 0]
Anyway, I will try to say a few things.
First of all avoid to use capital letters if you are on introductory
level with Mathematica.
There might be great problems because of possible conflicts with
built-in symbols.
Above you use C. However C is a built in symbol.
Information["C", LongForm -> True]
"C[i] is the default form for the ith parameter or constant generated
in representing the results of various symbolic
computations."*Button[More..., ButtonData :> "C", Active -> True,
ButtonStyle -> "RefGuideLink"]
Attributes[C] = {NHoldAll, Protected}
Here is your integral
Integrate[x^n*Sqrt[(c + x)/(l - x)], {x, 0, l}]
Mathematica 5.2 fails to evaluate the integral (or to state better, I
with 5.2
fail to evaluate the integral; I believe it must be a way to get the
desired result).
In Mathematica 4.0 I took the following result
Integrate[x^n*Sqrt[(c + x)/(l - x)], {x, 0, l}]
(l^(1 + n)*Sqrt[(c + l)/l]*Sqrt[Pi]*Gamma[2 + n]*
Hypergeometric2F1[1 + n, -(1/2), 3/2 + n, -(l/c)])/
(Sqrt[(c + l)/c]*(1 + n)*Gamma[3/2 + n])
which look a little strange on me since I wait a conditional result...
By the way I want to know what version you use.
Supposing now that the last result of 4.0 is correct (you must check it
numerically; always check Integrate with NIntegrate) you can get even
further
FullSimplify[%]
(l^(1 + n)*Sqrt[(c + l)/l]*Sqrt[Pi]*Gamma[1 +
n]*Hypergeometric2F1Regularized[1 + n, -(1/2), 3/2 + n,
-(l/c)])/Sqrt[(c + l)/c]
On this stage, you must understand that Hypergeometric2F1Regularized is
indeed a CLOSED FORM solution proper and approachable to every kind of
calculation (e.g. plotting).
I will be very glad to help you more but this time send your post in
more readable format.
It is quite easy.
After you finish with your inputs/outputs select the cells and press
Shift+Ctrl+I and you will get everything in Input format. Then
Copy/Paste on your message.
Regards
Dimitris
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