An exact 1-D integration challenge - 44
Vladimir Bondarenko
None of modern computer algebra systems is
able to calculate this integral.
Is there a soul who can show the steps how
to get to the exact value of
int(frac(z)/z^2, z= 1..infinity);
where frac stands for the fractional part?
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
Raymond Manzoni
> Hello computer algebra buffs,
>
> None of modern computer algebra systems is
> able to calculate this integral.
>
> Is there a soul who can show the steps how
> to get to the exact value of
>
> int(frac(z)/z^2, z= 1..infinity);
>
> where frac stands for the fractional part?
>
> Best wishes,
>
Well isn't this simply :
limit_{n->+oo} int_1^n z/z^2 dz - sum_{k=1}^{n-1} k/k^2
that is limit_{n->+oo} log(n) - H_{n-1} = gamma ?
(gamma is the Euler constant : 0.57721566...)
Raymond
Raymond Manzoni
Oops... No it isn't...
The difference is
limit_{n->+oo} int_1^n int(z)/z^2 dz - sum_{k=1}^{n-1} 1/k
....
Raymond Manzoni
In fact the answer is 1-gamma and this is proved in page 110 of
Havil's "Gamma: exploring Euler's constant" (showing that int_1^n
int(z)/z^2 dz = H_n-1)
Sorry
Raymond
Axel Vogt
> Hello computer algebra buffs,
>
> None of modern computer algebra systems is
> able to calculate this integral.
>
> Is there a soul who can show the steps how
> to get to the exact value of
>
> int(frac(z)/z^2, z= 1..infinity);
>
> where frac stands for the fractional part?
1 - Euler's constant ~ 0.422784335098467139393488
Axel Vogt
> Vladimir Bondarenko wrote:
...
>> int(frac(z)/z^2, z= 1..infinity);
>>
>> where frac stands for the fractional part?
>
> 1 - Euler's constant ~ 0.422784335098467139393488
answering through the Maple group i missed Raymond Manzoni's
reply ... this is Sum(ln((n+1)/n)-1/(n+1), n=1..infinity)
which Maple evaluates to the stated value and the sum comes
from obvious slicing
dimitris
In[32]:=
FullSimplify[(Integrate[FractionalPart[z]/z^2, {z, 1, #1}] & ) /@
Range[2, 10]]
Out[32]=
{-(1/2) + Log[2], -(5/6) + Log[3], -(13/12) + Log[4], -(77/60) +
Log[5], -(29/20) + Log[6], -(223/140) + Log[7],
-(481/280) + Log[8], -(4609/2520) + Log[9], -(4861/2520) + Log[10]}
In[44]:=
FullSimplify[(Sum[Log[(n + 1)/n] - 1/(n + 1), {n, 1, #1}] & ) /@
Range[1, 9]]
Out[44]=
{-(1/2) + Log[2], -(5/6) + Log[3], -(13/12) + Log[4], -(77/60) +
Log[5], -(29/20) + Log[6], -(223/140) + Log[7],
-(481/280) + Log[8], -(4609/2520) + Log[9], -(4861/2520) + Log[10]}
Above sum is equal to 1-EulerGamma for n->Infinity as mentioned by
others.
It is interest to note that in Mma 5.2
In[12]:=
Sum[Log[(n+1)/n]-1/(n+1),{n,1,Infinity}]
Infinity::indet: "Indeterminate expression 1-EulerGamma-SymbolicSum`var
$2949 \
$Failed-SymbolicSum`var$2975 $Failed+ComplexInfinity+ComplexInfinity \
encountered. \!\( \*ButtonBox[\(More...\), ButtonData:>\"General::indet
\", \
ButtonStyle->\"RefGuideLinkText\", ButtonFrame->\"None\"]\)
Out[12]=
Indeterminate
/ Vladimir Bondarenko :
Mate
More generally (and harder to guess):
int(frac(z)/(z+a)^2, z= 1..infinity) assuming a>=0;
answer = -ln(1+a)+Psi(2+a)
Mate
David W. Cantrell
Correct. (And it's valid for a > -1.)
More generally still,
letting [x] denote the floor function and psi(x) the digamma function,
a continuous function which is an antiderivative of frac(x)/(x + a)^2 wrt x
on (-a, oo)\Z is
ln(x + a) - psi([x] + a + 1) + ([x] + a)/(x + a).
David W. Cantrell
Mate
Unfortunately such a continuous antiderivative
is hard to see in any CAS (except Derive sometimes).
Mate
David W. Cantrell
I might have said "often" instead of "sometimes".
Derive _does_ produce the continuous antiderivative I gave above.
David
Vladimir Bondarenko
DWC> Derive _does_ produce the continuous antiderivative
DWC> I gave above.
INT(MOD(x)/(x+a)^2,x)
(FLOOR(x)-(x+a)*DIGAMMA(FLOOR(x)+a+1)+(x+a)*LN(x+a)+a)/(x+a)
A beta tester of Derive, 1990-2004 (since 1.62 by Soft
Warehouse to its very last release, 6.1 by Texas Instruments),
I am happy to find that you David and you Mate use it...
(my efforts are not wasted! ;)
It was a capital mistake by Dave Stoutemyer to sell Derive
to TI... I always knew ADR was against this, in his soul,
badly... they would have more fame and money if would have
continued by their own.
Cheers,
Vladimir
http://www.cas-testing.org/index.php?list=3
gwh
Talk about coincidences--- I had just written to Raymond Manzoni
yesterday telling him that I was again studying Julian Havil's book
"Gamma: exploring Euler's constant" and in fact I was studying the
very topic of your post. The discussion begins at the bottom of page
109 and ends on page111. I would imagine that this book is in your own
library, but if not, I would be pleased to send you copies of the
above pages.
Best regards from an engineer, not a mathematician,
Grover Hughes
Vladimir Bondarenko
GH> Talk about coincidences
;)
GH> in fact I was studying the very topic of
GH> your post.
Humm... if this is a coincidence, it would be
an amazing one. Difficult to believe. Rather,
we could be some cogs of some fine mashinery? :-)
GH> I would imagine that this book is in your
GH> own library,
I will buy this book one fine day; just because
of this event!
GH> I would be pleased to send you copies of
GH> the above pages.
I would be delighted! Your words are so much
warm and moving! Most of all I am happy that
you are fond of math.
On a personal note, during my lifetime, I so
listen to the voice of my Lady Intuition much.
For me your message is a very important signal,
a kind of feedback meaning, All is OK.
GH> Best regards from an engineer,
GH> not a mathematician
Best regards from a QA engineer,
a self-taught mathematician ;)
Vladimir Bondarenko
> On Jun 17, 4:40 am, Vladimir Bondarenko <v...@cybertester.com> wrote:
>
>
>
> > Hello computer algebra buffs,
>
> > None of modern computer algebra systems is
> > able to calculate this integral.
>
> > Is there a soul who can show the steps how
> > to get to the exact value of
>
> > int(frac(z)/z^2, z= 1..infinity);
>
> > where frac stands for the fractional part?
>
> > Best wishes,
>
> > Vladimir Bondarenko
>
> > VM and GEMM architect
> > Co-founder, CEO, Mathematical Director
>
> >http://www.cybertester.com/Cyber Tester, LLChttp://maple.bug-list.org/ Maple Bugs Encyclopaediahttp://www.CAS-testing.org/CAS Testing