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From: Md Sahidullah on 18 May 2010 09:03
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
From: Tim Norfolk on 18 May 2010 09:20
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)). 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

Check out Schaum's 'Advanced Calculus'. This should give you some of
ii). For i), I don't believe that it can be done, except by
approximation.
From: Herman Rubin on 18 May 2010 15:11
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)). 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

> Check out Schaum's 'Advanced Calculus'. This should give you some of
> ii). For i), I don't believe that it can be done, except by
> approximation.

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.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
From: Axel Vogt on 18 May 2010 15:57
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
From: Cary on 18 May 2010 15:59
On Tue, 18 May 2010 06:03:24 -0700 (PDT), Md Sahidullah
<sahidullahmd(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

See <http://mathworld.wolfram.com/SincFunction.html>

Related, see <http://mathworld.wolfram.com/SineIntegral.html>

If you don't need to do the integrals by hand, perhaps you could use
Mathematica at <http://integrals.wolfram.com/>

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