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Topic: Logarithmic integral

	Logarithmic integral function - Wikipedia, the free encyclopedia
		In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function.
		The offset logarithmic integral or Eulerean logarithmic integral is defined as
		The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value.
	en.wikipedia.org /wiki/Logarithmic_integral_function (307 words)

	MAXIMA - Online Information article about MAXIMA
		When a change of Integrals variables is made, the limits of integration with respect to the new variable must be such that the domain of and Multiple integration is the same as before.
		When the former integral is taken between assigned limits it represents the volume contained between the surface and two planes which cut the axis of x at right angles.
		The latter integral is to be understood as a line integral taken along the curve, and it represents the area of the portion of the curved surface which is contained between two planes at right angles to the axis of x.
	encyclopedia.jrank.org /MAR_MEC/MAXIMA.html (5569 words)

	Short course on asymptotics
		Li(x) is the integral of 1/log y from 2 to x, called the logarithmic integral.
		Integrals involving sin x, cos x, or a complex exponential, such as the integral of (sin y)/y, from 0 to x.
		Since integrals are usually easier to deal with than sums, this can be used to obtain approximations to sums.
	www.math.uiuc.edu /~hildebr/reu02/asymptotics.html (1019 words)

	Logarithmic integral - Wikipedia, the free encyclopedia
		In mathematics, the logarithmic integral can refer to
		and discussed in Paul Koosis, The Logarithmic Integral, volumes I and II, Cambridge University Press, second edition, 1998.
		This disambiguation page lists articles associated with the same title.
	en.wikipedia.org /wiki/Logarithmic_integral (89 words)

	Exponential integral - Wikipedia, the free encyclopedia
		In mathematics, the exponential integral Ei(x) is defined as
		Since 1/t diverges at t = 0, the above integral has to be understood in terms of the Cauchy principal value.
		The exponential integral is closely related to the logarithmic integral function li(x),
	www.sciencedaily.com /encyclopedia/exponential_integral (154 words)

	Abramowitz and Stegun. Subject Index
		Integral of a bivariate normal distribution over a polygon.....
		Cn, Dn, Sn integrals of the squares of Jacobian elliptic functions.....
		n characteristic of the elliptic integral of the third kind.....
	www.math.sfu.ca /~cbm/aands/subj.htm (463 words)

	BRILL
		This book, presenting all necessary references to the spectral theory, will serve as an efficient instrument for the analysis of logarithmic integral equations and will be of value and interest to researchers in the field of mathematical physics, theoretical electromagnetics, integral equations and spectral theory of operators.
		Special attention is given to the analysis of finite-meromorphic operator-valued functions, and explicit formulas for some inverse operators and characteristic numbers are developed, as well as the perturbation technique for the approximate solution of logarithmic integral equations.
		Fundamentals of the theory of infinite-matrix summation operators and operator-valued functions are presented, including applications to the solution of logarithmic integral equations.
	www.brill.nl /product_id10700.htm (593 words)

	Geometry.Net - Pure_And_Applied_Math: Integral Equations
		In their simplest form, integral equations are equations in one variable (say t) that involve an integral over a domain of another variable (s) of the product of a kernel function K(s,t) and another (unknown) function (f(s)).
		The integral equation is then reduced to a linear equation with the values of f at the quadrature points being unknown at the outset.
		Logarithmic integral equations in Electromagnetics.Yu.V. Shestopalov, Yu.G. Smirnov and EV Chernokozhin.
	www.geometry.net /detail/pure_and_applied_math/integral_equations.html (2758 words)

	Logarithmic integral (Site not responding. Last check: 2007-10-14)
		In mathematics the logarithmic integral or integral logarithm li(x) is a non-elementary function defined for all positive real numbers x ≠ 1 by the definite integral :
		The logarithmic integral is mainly important because occurs in estimates of prime number densities especially in the prime number theorem :
		where π(x) denotes the number of primes smaller or equal to x and Li(x) is the offset logarithmic integral function related to li(x) by Li(x) = li(x) - li(2).
	www.freeglossary.com /Logarithmic_integral (408 words)

	Logarithmic integral - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-10-14)
		or integral logarithm li(''x'') is a non-elementary function defined for all positive real_numbers ''x''≠ 1 by the definite integral: : {\rm li} (x) = \int_{0}^{x} \frac{dt}{\ln (t)} \;.
		Here, ln denotes the 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) \;.
		The logarithmic integral is mainly important because it occurs in estimates of prime_number densities, especially in the prime_number_theorem: :π(''x'') ~ Li(''x'') where π(''x'') denotes the number of primes smaller than or equal to ''x'', and Li(''x'') is the offset_logarithmic_integral function, related to li(''x'') by Li(''x'') = li(''x'') - li(2).
	www.indexsuche.com /Logarithmic_integral.html (278 words)

	Other Special Functions
		This is the value of the square root of (4/pi) times the integral from 0 to x of e^(-u^2) with respect to u, at x = r for r>0, and for r<0 it is defined by erf(x)= - erf(- x), while erf(0)=0.
		Given a real number r, calculate the value of the exponential integral, that is, the principal value of the integral from minus infinity to x of e^u / u with respect to u at x = r.
		This integral is defined to be the principal value of the integral from 0 to x of 1 / log(u) with respect to u.
	www.umich.edu /~gpcc/scs/magma/text573.htm (562 words)

	Logarithmic (Site not responding. Last check: 2007-10-14)
		The logarithmic spiral is a spiral whose polar equation is given by, (1).
		The logarithmic spiral was dubbed the "Spira Mirabilis" by Jakob Bernouilli.
		This logarithmic timeline, based on logarithmic scale, is able to show all history on one page in ten lines.
	www.bigletterlist.net /w/l/Logarithmic.htm (371 words)

	Logarithms
		The exponential and logarithmic functions are generally introduced in the form of the "laws of exponents" and their intuitive exensions.
		These rules are introduced for x and y integral, from which they are obvious by definition, then extended to rationals and finally to reals.
		Natural logarithms are sometimes called Naperian to honor the inventor of logarithms, but Napier did not use natural logarithms himself.
	www.du.edu /~etuttle/math/logs.htm (1077 words)

	No Title
		In particular you should know all the standard forms for integrals that were listed there.
		You should be able to identify an integral as an Arcsin, Arctan, general power, or logarithmic integral, as in problems 25-28 in section 8.6 of the book.
		Another trick is to do integration by parts twice, writing the integral in terms of itself, solving for it, and then tacking on a +C.
	www.math.okstate.edu /~myersr/2133/r3/r3.html (963 words)

	IMSL FORTRAN Libraries Numerical Library Modules (Site not responding. Last check: 2007-10-14)
		ALI (DLI) Logarithmic integral, integral from 0 to x of 1/ln(t).
		BSITG (DBSITG) Evaluate the integral of a spline, given its B-spline representation.
		CPSI Logarithmic derivative of the gamma function for a complex argument.
	www2.andrews.edu /~mattingl/courses/cosc125/imsl.alphabetized_functions.html (7021 words)

	Logarithmic integral (Site not responding. Last check: 2007-10-14)
		In mathematics, the logarithmic integral or integrallogarithm li(x) is a non-elementaryfunction defined for all positive real numbers x≠ 1 by the definite integral :
		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 :
		where π(x) denotes the number of primes smaller than or equal to x, and Li(x) is the offset logarithmic integral function, related toli(x) by Li(x) = li(x) - li(2).
	www.therfcc.org /logarithmic-integral-153231.html (199 words)

	SLATEC Common Mathematical Library -- Table of Contents (Site not responding. Last check: 2007-10-14)
		Integrals of Bessel functions BSKIN-S Compute repeated integrals of the K-zero Bessel function.
		Elliptic integrals RC-S Calculate a double precision approximation to DRC-D DRC(X,Y) = Integral from zero to infinity of -1/2 -1 (1/2)(t+X) (t+Y) dt, where X is nonnegative and Y is positive.
		QC25S-S To compute I = Integral of F*W over (BL,BR), with error DQC25S-D estimate, where the weight function W has a singular behaviour of ALGEBRAICO-LOGARITHMIC type at the points A and/or B. (BL,BR) is a part of (A,B).
	www-acs.ucsd.edu /offerings/userhelp/OLD_HTML/slatec.toc,d.html (9389 words)

	No Title
		Three topics were covered after the third exam: partial fractions, tables of integrals, and improper integrals.
		On the final you will not need to do integrals of this type unless they occur in a problem that uses tables of integrals.
		There are many standard forms for integrals that we have not seen in this course.
	www.math.okstate.edu /~myersr/2133/rf/rf.html (828 words)

	logarithmic integral (Site not responding. Last check: 2007-10-14)
		The logarithmic integral gives the integral over a function with the logarithm in the denominator.
		Some authors use the value 2 as starting point of the integral.
		The logarithmic integral li(x) or Li(x) gives an approximation of the pi function.
	www.2dcurves.com /exponential/exponentiali.html (40 words)

	[No title]
		The K-th DQMOMO-D modified Chebyshev moment is defined as the integral over (-1,1) of W(X)*T(K,X), where T(K,X) is the Chebyshev polynomial of degree K. XLEGF-S Compute normalized Legendre polynomials and associated DXLEGF-D Legendre functions.
		Elliptic integrals RC-S Calculate an approximation to DRC-D RC(X,Y) = Integral from zero to infinity of -1/2 -1 (1/2)(t+X) (t+Y) dt, where X is nonnegative and Y is positive.
		D9LGIT Compute the logarithm of Tricomi's incomplete Gamma function with Perron's continued fraction for large X and A.GE.
	www.netlib.org /slatec/toc (13942 words)

	Transcendental Functions
		In this section the exponential and logarithmic functions to the natural base e are described, as well as the conversion to the logarithm with respect to any base.
		For s with positive real part this is the value of the integral from 0 to infinity of u^(x - 1)e^(-u) with respect to u.
		For free real or complex s (not a non-positive integer) return the principal value of the logarithmic derivative Psi(s)=(d log Gamma(s)/ds)=(Gamma'(s)/Gamma(s)), of the gamma function, which allows the expansion Psi(s)= - gamma - (1/s) + s sum_(n=1)^Infinity(1/n(s + n)); here gamma is Euler's gamma.
	www.math.uiuc.edu /Software/magma/text365.html (3486 words)

	Prime number - Wikipedia, the free encyclopedia
		The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics.
		As an example, we consider the Gaussian integers Z[i], that is, complex numbers of the form a + bi with a and b in Z.
		This is an integral domain, and its prime elements are the Gaussian primes.
	en.wikipedia.org /wiki/Prime_number (4552 words)

	Alibris: Logarithms
		A unique work giving a straightforward presentation of the logarithmic integral, a theme which lies athwart much of 20th century analysis.
		Smoley's tables; parallel tables of logarithms and squares, diagrams for solving right triangles, angles and trigonometric functions, common logarithms and natural trigonometric functions and other tables, for engineers, architects and students.
		Six place logarithmic tables, together with a table of natural sines, cosines, tangents, and cotangents.
	www.alibris.com /search/books/subject/Logarithms (264 words)

	Homepage of Youri Shestopalov
		Results are based on the development of the theory of singular integral operators and the spectral theory of integral OFVs with a logarithmic singularity of the kernel [6].
		Other research fields are semilinear Helmholtz and Schrödinger equations with variable coefficients [7] and direct and inverse scattering problems in domains with noncompact boundaries [8] These problems are connected with the modeling of wave propagation in layered structures filled with nonlinear media [9], [10].
		The developed techniques are of universal character and have been recently developed and applied [11] to the analysis of the mathematical problems of elasticity and related items that enable one to model paper layers and surfaces using specifically created mathematical models.
	www.ingvet.kau.se /~youri (538 words)

	Exponential integral - TheBestLinks.com - Mathematics, Logarithmic integral, Euler-Mascheroni gamma constant, Cauchy ...
		Exponential integral - TheBestLinks.com - Mathematics, Logarithmic integral, Euler-Mascheroni gamma constant, Cauchy principal value,...
		Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value.
		The exponential integral is closely related to the logarithmic integral li(x),
	www.thebestlinks.com /Exponential_integral.html (111 words)

	Untitled Document (Site not responding. Last check: 2007-10-14)
		Because the limitations singled out in Bode's integral are so fundamental, it is natural to expect that similar constraints must exist for systems that are not linear or time-invariant.
		For linear time-varying systems, we first present analogues of the logarithmic integral found in Bode's result, based on an abstract version of Szeg{\H o}'s limit theorem provided by Dym and Ta'asan.
		It is shown that this difference is zero for the sensitivity operator of a stable nonlinear system that possesses fading memory from both the input to output and output to input.
	www.ece.jhu.edu /Seminars/Zang%20Defense.htm (358 words)

	SLATEC Keylist Index
		Evaluate the definite integral of a piecewise cubic Hermite function over an interval whose endpoints are data points.
		Calculate a double precision approximation to DRC(X,Y) = Integral from zero to infinity of -1/2 -1 (1/2)(t+X) (t+Y) dt, where X is nonnegative and Y is positive.
		Calculate an approximation to RC(X,Y) = Integral from zero to infinity of -1/2 -1 (1/2)(t+X) (t+Y) dt, where X is nonnegative and Y is positive.
	www.cse.yorku.ca /~roumani/fortran/labs/slatecIndex.htm (11408 words)

	Talk:Logarithmic integral - Wikipedia, the free encyclopedia
		The logarithmic integral function is by far the most common meaning of the term.
		Scholar.google search "logarithmic integral" shows 118 references to the Koosis' book.
		This page was last modified 06:15, 25 August 2006.
	en.wikipedia.org /wiki/Talk:Logarithmic_integral (74 words)

	Books : The Logarithmic Integral (Cambridge Studies in Advanced Mathematics) (Site not responding. Last check: 2007-10-14)
		The Logarithmic Integral (Cambridge Studies in Advanced Mathematics)
		The theme of this unique work, the logarithmic integral, lies athwart much of twentieth century analysis.
		It is a thread connecting many apparently separate parts of the subject, and is a natural point at which to begin a serious study of real and complex analysis.
	www.target.com /gp/detail.html?asin=0521596726 (142 words)

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