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An exact asymptotics challenge - 2

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Vladimir Bondarenko

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Apr 9, 2007, 11:18:42 AM4/9/07
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Hello noble computer algebra warlords,

Hello noble computer algebra warlords,

None of the modern computer algebra systems can calculate
this asymptotics straightforwardly

asympt(int(t^(a-1)/(1-t)^a, t= 0..1/2), a, 2);

Is there a field marshal who can, via a sequence of a CAS
commands, force this fortress to surrender?

Best wishes,

Vladimir Bondarenko

VM and GEMM architect
Co-founder, CEO, Mathematical Director

http://www.cybertester.com/ Cyber Tester, LLC
http://maple.bug-list.org/ Maple Bugs Encyclopaedia
http://www.CAS-testing.org/ CAS Testing

.........................................................

Axel Vogt

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Apr 9, 2007, 11:58:55 AM4/9/07
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Vladimir Bondarenko wrote:
> Hello noble computer algebra warlords,
>
> Hello noble computer algebra warlords,
>
> None of the modern computer algebra systems can calculate
> this asymptotics straightforwardly
>
> asympt(int(t^(a-1)/(1-t)^a, t= 0..1/2), a, 2);
>
> Is there a field marshal who can, via a sequence of a CAS
> commands, force this fortress to surrender?
>
> Best wishes,
>
> Vladimir Bondarenko


I think this is asking for the asymptotics of 2^(-a)*hypergeom([a, a],[a+1],1/2)/a
and I give up instead looking what Temme provides for that.

Mate

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Apr 9, 2007, 12:33:56 PM4/9/07
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On Apr 9, 6:18 pm, "Vladimir Bondarenko" <v...@cybertester.com> wrote:
>
>
> asympt(int(t^(a-1)/(1-t)^a, t= 0..1/2), a, 2);
>
>
>
> .........................................................

1/(2*a)+1/(4*a^2)-1/(8*a^4)+1/(4*a^6)+O(1/(a^7))


Mate


Vladimir Bondarenko

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Apr 9, 2007, 12:44:57 PM4/9/07
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On Apr 9, 9:33 am, "Mate" <mmat...@personal.ro> wrote:

M> 1/(2*a)+1/(4*a^2)-1/(8*a^4)+1/(4*a^6)+O(1/(a^7))

Gee!

What about a harder version

assume(b>0,b<1):
asympt(int(t^(a-1)/(1-t)^a, t= 0..b), a, 2);

?

G. A. Edgar

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Apr 9, 2007, 1:40:22 PM4/9/07
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In article <1176136436.4...@p77g2000hsh.googlegroups.com>,
Mate <mma...@personal.ro> wrote:


(t/(1-t))^a/t;
a
/ t \
|-----|
\1 - t/
--------
t

I1 := Int(%,t=0..1/2);
a
/1/2 / t \
| |-----|
| \1 - t/
I1 := | -------- dt
| t
/0

with(student):
changevar(u=t/(1-t),I1,u);
/ 2 \
/1 a | 2 u u |
| u (1 + u) |1 - ----- + --------|
| | 1 + u 2|
| \ (1 + u) /
| --------------------------------- du
| u
/0

I2 := simplify(%);
/1 (a - 1)
| u
I2 := | -------- du
| 1 + u
/0

value(%);
LerchPhi(-1, 1, a)

convert(%,Psi);
1 /1 1 \ 1 /1 \
- Psi|- + - a| - - Psi|- a|
2 \2 2 / 2 \2 /

asympt(%,a);
1 1 1 /1 \
--- + ---- - ---- + O|--|
2 a 2 4 | 6|
4 a 8 a \a /

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Vladimir Bondarenko

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Apr 9, 2007, 4:17:57 PM4/9/07
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Can anyone crack a harder version

assume(b>0,b<1):
asympt(int(t^(a-1)/(1-t)^a, t= 0..b), a, 2);

?

> > Mate- Hide quoted text -
>
> - Show quoted text -


Mate

unread,
Apr 9, 2007, 5:17:06 PM4/9/07
to
On Apr 9, 11:17 pm, "Vladimir Bondarenko" <v...@cybertester.com>

wrote:
> Can anyone crack a harder version
>
> assume(b>0,b<1):
> asympt(int(t^(a-1)/(1-t)^a, t= 0..b), a, 2);
>
> ?
>
> On Apr 9, 9:44 am, "Vladimir Bondarenko" <v...@cybertester.com> wrote:
>
> > On Apr 9, 9:33 am, "Mate" <mmat...@personal.ro> wrote:
>
> > M> 1/(2*a)+1/(4*a^2)-1/(8*a^4)+1/(4*a^6)+O(1/(a^7))
>
> > Gee!
>
> > What about a harder version
>
> > assume(b>0,b<1):
> > asympt(int(t^(a-1)/(1-t)^a, t= 0..b), a, 2);
>
> > ?
>
> > > On Apr 9, 6:18 pm, "Vladimir Bondarenko" <v...@cybertester.com> wrote:
>
> > > > asympt(int(t^(a-1)/(1-t)^a, t= 0..1/2), a, 2);
>
> > > > .........................................................
>
> > > 1/(2*a)+1/(4*a^2)-1/(8*a^4)+1/(4*a^6)+O(1/(a^7))
>
> > > Mate- Hide quoted text -
>
> > - Show quoted text -

(b/(1-b))^a * ( (1-b)/a + (b-b^2)/a^2 + O(a^3) )

assuming b>0, b <= 1/2.
The case b>1/2 is left to the proposer.

Mate


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