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From: I.M. Soloveichik on 18 May 2010 11:59
MAPLE can do your integral i) with the answer expressed in terms of Si(W), the integral of f(x) from 0 to W;

for f(x)^2 MAPLE gives the answer (-1+cos(2*W)+2*Si(2*W)*W)/W


> Sin(x)/x function is famous as Sinc function or
> Sinc(x) (Let say it is
> f(x)). But the anti-derivatives of this function
> cannot be expressed
> as elementary functions. I need to compute two
> definite integral of
> this function (i) from -W to W and (ii) from -INF to
> INF. For, f(x) ^2
> and f(x) ^4. I have to compute those values, more
> precisely speaking
> ratio of (i) and (ii).
>
> I shall be very thankful if you kindly suggest me a
> technique to solve
> this problem. I have also tried with different
> integral tables. But,
> the computation became too complicated to solve it.
> If there is any
> appropriate integral exist, can you please suggest me
> that one?
> Looking forward to your responses.
>
> Thanking You.
>
> Yours Sincerely,
> Md. Sahidullah
From: Ray Vickson on 19 May 2010 02:42
On May 18, 6:03 am, Md Sahidullah <sahidulla...(a)gmail.com> wrote:
> Dear Sir,
>
> I need your suggestion to solve the following problem.
>
> Sin(x)/x function is famous as Sinc function or Sinc(x) (Let say it is
> f(x)). But the anti-derivatives of this function cannot be expressed
> as elementary functions. I need to compute two definite integral of
> this function (i) from -W to W and (ii) from -INF to INF. For, f(x) ^2
> and f(x) ^4. I have to compute those values, more precisely speaking
> ratio of (i) and (ii).
>
> I shall be very thankful if you kindly suggest me a technique to solve
> this problem. I have also tried with different integral tables. But,
> the computation became too complicated to solve it. If there is any
> appropriate integral exist, can you please suggest me that one?
> Looking forward to your responses.
>
> Thanking You.
>
> Yours Sincerely,
> Md. Sahidullah

In Maple 9.5:

f:=sin(x)/x:

Ja:=int(f^2,x=-W..W) assuming W>0;

-1 + cos(2 W) + 2 Si(2 W) W
Ja := ---------------------------
W

> limit(Ja,W=infinity);

Pi

> Jb:=int(f^4,x=-W..W) assuming W>0;

2
Jb := -1/12 (3 + cos(4 W) - 2 sin(4 W) W - 8 cos(4 W) W

3 2
- 32 Si(4 W) W - 4 cos(2 W) + 4 sin(2 W) W + 8 cos(2 W) W

3 / 3
+ 16 Si(2 W) W ) / W
/

> limit(Jb,W=infinity);

2 Pi
----
3

Here, Si(x) = int(sin(t)/t, t=0..x) is the so-called Sine integral, a
non-elementary function.

R.G. Vickson
From: mike3 on 19 May 2010 06:59
On May 18, 7:03 am, Md Sahidullah <sahidulla...(a)gmail.com> wrote:
> Dear Sir,
>
> I need your suggestion to solve the following problem.
>
> Sin(x)/x function is famous as Sinc function or Sinc(x) (Let say it is
> f(x)). But the anti-derivatives of this function cannot be expressed
> as elementary functions. I need to compute two definite integral of
> this function (i) from -W to W and (ii) from -INF to INF. For, f(x) ^2
> and f(x) ^4. I have to compute those values, more precisely speaking
> ratio of (i) and (ii).
>
> I shall be very thankful if you kindly suggest me a technique to solve
> this problem. I have also tried with different integral tables. But,
> the computation became too complicated to solve it. If there is any
> appropriate integral exist, can you please suggest me that one?
> Looking forward to your responses.
>
> Thanking You.
>

i) From -w to w cannot be expressed in elementary terms, as you
mention.
There is a function called "sine integral" denoted by Si(x) that
stands
for the integral of the sinc from 0 to x, in this case we could write
the solution as Si(w) - Si(-w) = 2 Si(w) (note that sinc(-x) = sinc(x)
and via a change of variable in the defining integral we see Si(-x) =
-Si(x)), but whether this "helps" the situation any depends on your
point of view, considering it's _defined_ via the integral we wanted
to
"solve". On one hand, you could say it helps us none, since it is
just naming the integral. On the other hand, the integral of 1/x is
used
to define the function called "log", yet you have no trouble accepting
that to be the "solution" to the problem of "integrate 1/x". There's
nothing "wrong" with using new operations -- indeed, you may have run
into a place where it may be useful to use one. But if you want a
solution
in terms of "familiar" operations, the best that can be done are
infinite expansions, like the Taylor series

Si(x) = x - x^3/(3*3!) + x^5/(5*5!) - (x^7)/(7*7!) + ...

(do the termwise integral of the Taylor series for sinc at 0)

and then we have

int_{-w...w} sin(x)/x dx = 2w - (2w^3)/(3*3!) + (2w^5)/(5*5!) - (2w^7)/
(7*7!) + ...

This series converges for all w. The case for f(x)^2 and f(x)^4 can be
expressed as infinite series as well by using termwise integration,
and
also can be expressed using the Si(x) function with integration by
parts.

ii) This is more difficult. Here's some postings about methods to do
this:

http://www.mathkb.com/Uwe/Forum.aspx/math/3178/0-to-infinity-integral-of-sin-x-x-is-pi-2-how-to-calculate-it-out

One involves a complex contour integral, the other involves the use of
Fourier transforms.

For f(x), the integral from -inf to inf is pi, as it is for f(x)^2.
For f(x)^4
it is 2pi/3. Note that for the f(x)^2 and f(x)^4 cases you could also
just
use the Si(x) function to get the antiderivative, then take the
appropriate
limit.
From: Nasser M. Abbasi on 20 May 2010 04:09

"Axel Vogt" <&noreply(a)axelvogt.de> wrote in message
news:85g9ptFj3cU1(a)mid.individual.net...
> Herman Rubin wrote:
>> On 2010-05-18, Tim Norfolk <timsn274(a)aol.com> wrote:
>>> On May 18, 9:03?am, Md Sahidullah <sahidulla...(a)gmail.com> wrote:
>>>> Dear Sir,
>>
>>>> I need your suggestion to solve the following problem.
>>
>>>> Sin(x)/x function is famous as Sinc function or Sinc(x) (Let say it is
>>>> f(x)).
>
> ...
>>
>> I believe it is the Si function. This, and other special
>> functions, have been studied extensively.
>>
>> On the entire real line, the improper Riemann integral of f(x)
>> and the integral of f(x)^2 are both pi.
>


> Yes, a CAS like Maple 'knows' that:
>
> f:= x -> sin(x)/x;
>
> int(f(x), x);
> Si(x)
>
> int(f(x), x= -infinity .. infinity);
>
> Pi
>
> int(f(x)^2, x= -infinity .. infinity);
>
> Pi

hi,
here is a table of int( sinc(x)^n ) for n=1..20,

In[11]:= TableForm[Table[{n, Integrate[Sinc[x]^n, {x, -Infinity,
Infinity}]}, {n, 1, 20}]]

Out[11]
{{1, Pi},
{2, Pi},
{3, (3*Pi)/4},
{4, (2*Pi)/3},
{5, (115*Pi)/192},
{6, (11*Pi)/20},
{7, (5887*Pi)/11520},
{8, (151*Pi)/315},
{9, (259723*Pi)/573440},
{10, (15619*Pi)/36288},
{11, (381773117*Pi)/928972800},
{12, (655177*Pi)/1663200},
{13, (20646903199*Pi)/54499737600},
{14, (27085381*Pi)/74131200},
{15, (467168310097*Pi)/1322526965760},
{16, (2330931341*Pi)/6810804000},
{17, (75920439315929441*Pi)/228532659683328000},
{18, (12157712239*Pi)/37638881280},
{19, (5278968781483042969*Pi)/16783438527143608320},
{20, (37307713155613*Pi)/121645100408832}}]

In[12]:= TableForm[N[%]]
{{1., 3.141592653589793},
{2., 3.141592653589793},
{3., 2.356194490192345},
{4., 2.0943951023931953},
{5., 1.8816830998063867},
{6., 1.7278759594743864},
{7., 1.605430204139159},
{8., 1.5059698117208216},
{9., 1.4228931863286514},
{10., 1.352197300937472},
{11., 1.2910772195970392},
{12., 1.2375536616167626},
{13., 1.1901737928616742},
{14., 1.1478464394112138},
{15., 1.109733539646427},
{16., 1.0751794908365893},
{17., 1.0436630577998458},
{18., 1.014763939724117},
{19., 0.9881389630387916},
{20., 0.9635047953267126}}]

--Nasser


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