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Topic: Re: Series expansion for an integral - involving incomplete elliptic integral
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Dan Luecking

Posts: 26
Registered: 11/12/08
Re: Series expansion for an integral - involving incomplete elliptic integral
Posted: Dec 9, 2011 8:02 AM
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Resent-From: <bergv@illinois.edu>
From: Dan Luecking <LookInSig@uark.edu>
Date: December 8, 2011 4:00:31 PM MST
To: "sci-math-research@moderators.isc.org"
<sci-math-research@moderators.isc.org>
Subject: Re: Series expansion for an integral - involving incomplete
elliptic integral

On Wed, 7 Dec 2011 09:14:30 +0000, victorphy <vbapst@gmail.com> wrote:

> Hi,
>
>
> I have a little question about a series expansion, which may be very
> basic.
>
> I would like to expand the integral of ArgCosh[1 + x^2 + a] between 0
> and y >0, at first order in the small parameter a.


I don't see how this is possible. For such a first order
expansion to exist, one would need at least that

 integral(0 to y) ArgCosh[1 + x^2 + a] - ArgCosh[1 + x^2] dx

be less than a constant times a. By the mean value theorem
(differentiating with respect to a), that would require that

 integral(0 to y) 1/sqrt((1 + x^2 + a)^2 - 1) dx

remain bounded as a tends to 0. But now the monotone
convergence theorem says that this integral tends to:

 integral(0 to y) 1/sqrt((1 + x^2)^2 - 1) dx

And now a little algebra shows that this is equal to

 integral(0 to y) 1/( x * sqrt(x^2 + 2)) dx
= (1/sqrt(y^2+2)) * integral(0 to y) 1/x dx
    = infinity.


Dan
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