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Integral Calculus

Blazing goIITian

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4 May 2009 13:37:11 IST

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444

what is INTEGRABLE VS NON-INTEGRABLE.

None

first of all sorry for posting this as a separate thread but i want every one in class 11 to know what is non-integrable . one boy ask a qs of the ine. of sinx/x to which most of people have replied that it is non-integrable. ..........but one expert " Anant " gave an impressive answer.

well i ask u what is the integration of

well u will say it to be non-integrable............but thanks to the genius Carl Fredriech Gauss that we now know its integration it is Li(x) where Li(x) is a new function......defined by Gauss.

so non-integrable function are those whose result are a new function which have not been studied till now...........that's a more intelligent answer.

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bladeX -rise of a new sun

Blazing goIITian

Joined: 12 Dec 2007

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4 May 2009 13:38:12 IST

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Logarithmic integral function

From Wikipedia, the free encyclopedia

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In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It occurs in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.

Logarithmic integral

[hide]

[edit] Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers $x\ne 1$ by the definite integral:

${\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \; .$

Here, ln denotes the natural logarithm. The function 1/ln (t) has a singularity at t = 1, and the integral for x > 1 has to be interpreted as a Cauchy principal value:

${\rm li} (x) = \lim_{\varepsilon \to 0} \left( \int_{0}^{1-\varepsilon} \frac{dt}{\ln (t)} + \int_{1+\varepsilon}^{x} \frac{dt}{\ln (t)} \right) \; .$

[edit] Offset logarithmic integral

The offset logarithmic integral or Eulerean logarithmic integral is defined as

${\rm Li}(x) = {\rm li}(x) - {\rm li}(2) \,$

${\rm Li} (x) = \int_{2}^{x} \frac{dt}{\ln t} \,$

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

[edit] Series representation

The function li(x) is related to the exponential integral Ei(x) via the equation

$\hbox{li}(x)=\hbox{Ei}(\ln(x)) , \,\!$

which is valid for $x > 1$ . This identity provides a series representation of li(x) as

${\rm li} (e^{u}) = \hbox{Ei}(u) =\gamma + \ln u + \sum_{n=1}^{\infty} {u^{n}\over n \cdot n!}\quad {\rm for} \; u \ne 0 \; ,$

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant. A more rapidly convergent series due to Ramanujan^{[citation needed]} is

${\rm li} (x) =\gamma+ \ln \ln x+ \sqrt{x} \sum_{n=1}^{\infty}\frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}}\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} .$

[edit] Special values

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 ...; this number is known as the Ramanujan-Soldner constant.

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…

This is $-(\Gamma\left(0,-\ln 2\right) + i\,\pi)$ where $\Gamma\left(a,x\right)$ is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

[edit] Asymptotic expansion

The asymptotic behavior for x → ∞ is

${\rm li} (x) = \mathcal{O} \left( {x\over \ln x} \right) \; .$

where $\mathcal{O}$ refers to big O notation. The full asymptotic expansion is

${\rm li} (x) \sim \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k}$

$\frac{{\rm li} (x)}{x/\ln x} \sim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \cdots.$

Note that, as an asymptotic expansion, this series is not convergent: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral.

[edit] Infinite logarithmic integral

$\int_{-\infty}^\infty \frac{M(t)}{1+t^2}dt$

and discussed in Paul Koosis, The Logarithmic Integral, volumes I and II, Cambridge University Press, second edition, 1998.

[edit] Number theoretic significance

The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value. For example, the prime number theorem states that:

$\pi(x)\sim\hbox{li}(x)\sim\hbox{Li}(x)$

where π(x) denotes the number of primes smaller than or equal to x.

bladeX -rise of a new sun

Blazing goIITian

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4 May 2009 13:38:56 IST

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the above article from WIKIPEDIA contains interesting things about Li(x) read it if u can.

bladeX -rise of a new sun

Blazing goIITian

Joined: 12 Dec 2007

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4 May 2009 13:51:45 IST

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not interesting ............................well i think i m boring lot of chaps..here

Mr.Nasty ® Retired from Action.

Blazing goIITian

Joined: 16 Jan 2009

Posts: 1041

4 May 2009 14:34:46 IST

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No Blade, not a lot.......just some of us.

Nice work though.

Sai Karthik

Cool goIITian

Joined: 11 Feb 2009

Posts: 82

8 Jun 2009 11:40:15 IST

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thanks for the info blade....

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