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Topic: Sine integral

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List of integrals of trigonometric functions

Sine

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	PlanetMath: sine integral at infinity
		"sine integral at infinity" is owned by pahio.
		See Also: sine integral, sinc function, substitution notation, incomplete gamma function, example of summation by parts
		This is version 11 of sine integral at infinity, born on 2005-05-20, modified 2007-05-24.
	www.planetmath.org /encyclopedia/SineIntegralInInfinity.html (126 words)

	PlanetMath: sine integral
		The function sine integral (in Latin sinus integralis) from
		See Also: sinc function, sine integral at infinity, logarithmic integral, curvature of Nielsen's spiral
		This is version 10 of sine integral, born on 2005-03-04, modified 2007-05-09.
	www.planetmath.org /encyclopedia/SineIntegral.html (96 words)

	PlanetMath: sinc function
		This is a consequence of a comment in the sine integral entry.
		There is no known simple expression for the integral of sinc.
		However, this function is known as the sine integral.
	planetmath.org /encyclopedia/SineCardinal.html (228 words)

	Systematic Integral New Evangelization (SINE)
		SINE helps us to recognize that Christ has a job for each of us and to determine what His will is for us.
		The SINE pastoral plan is not a movement in the Church, but a parish model that shows us how to become an active evangelizing community.
		If you are interested in having the SINE process in your parish, please contact her at (402) 792-3030 or at (402) 786-2705 or by e-mail at carolyn-may@cdolinc.net.
	www.dioceseoflincoln.org /green/Offices_evang_sine.htm (223 words)

	Sine and Cosine
		The integral is 2sqrt(t), and the area is 2, hence the arcsine of 1 is no more than 2.
		We defined cosine as the derivative of sine, and showed this was the square root of 1 minus sine squared.
		We usually let sine be the y coordinate, while cosine is the x coordinate.
	www.mathreference.com /ca-int,sin.html (693 words)

	Elliptic integral - ExampleProblems.com
		In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Guilio Fagnano and Leonhard Euler.
		Additional insight into the theory of the indefinite integral may be gained through the study of the Schwarz-Christoffel mapping.
		Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular; we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.
	www.exampleproblems.com /wiki/index.php?title=Elliptic_integral&printable=yes (620 words)

	Differentiation, Integration, and Inverse Trig Functions
		Integral of Arc Sine, Arc Cosine, Arc Tangent
		Integral leading to Arc Sine, Arc Tangent, Arc Secant
		Differentiation of the cosine of the inverse sine: cubic argument
	www.jtaylor1142001.net /calcjat/Contents/CDIArcTrig.htm (27 words)

	Calculus II (Math 2414) - Integration Techniques - Integrals Involving Trig Functions
		So, in this case we’ve got both sines and cosines in the problem and in this case the exponent on the sine is even while the exponent on the cosine is odd. So, we can use a similar technique in this integral. This time we’ll strip out a cosine and convert the rest to sines.
		So, we still have an integral that can’t be completely done, however notice that we have managed to reduce the integral down to just one term causing problems (a cosine with an even power) rather than two terms causing problems.
		The two integrals in the last two examples will arise on occasion in some of the work that we’ll be doing in later sections and chapters so it wouldn’t be a bad idea to make sure you’ve got them written down somewhere.
	tutorial.math.lamar.edu /AllBrowsers/2414/IntegralsWithTrig.asp (1787 words)

	[No title]
		The symbol SinIntegral represents the sine integral, it is denoted by $Si(z)$.
		The symbol SinhIntegral represents the hyperbolic sine integral, it is denoted by The hyperbolic sine integral is defined by $Shi(z)=\int_{0}^{z} \frac{sinh(t)}{t} dt$.
		The symbol CoshIntegral represents the hyperbolic cosine integral, it is denoted by $Chi(z)$.
	www.orcca.on.ca /~bill/OpenMath/SpecialFns/cd/trigHypInts.ocd (228 words)

	PlanetMath: sine integral
		The function sine integral (in Latin sinus integralis) from
		See Also: sinc function, sine integral at infinity
		This is version 8 of sine integral, born on 2005-03-04, modified 2005-05-20.
	planetmath.org /encyclopedia/SinusIntegralis.html (91 words)

	[No title]
		The sine integral is defined by $Si(z)=\int_{0}^{z} \frac{sint}{t}dt$.
		The hyperbolic sine integral is defined by $Shi(z)=\int_{0}^{z} \frac{sinht}{t} dt$.
		The Clausen integral is defined as $f(\theta)=-\int_{0}^{\theta}ln(2sin\frac{t}{2})dt =\sum_{k=1}^{\infty}\frac{sink\theta}{k^2}$ $(0 \leq \theta \leq \pi)$.
	www.csd.uwo.ca /faculty/watt/home/research/openmath/spfun1.ocd (3083 words)

	[No title]
		Notice that this pattern is for the inverse sine function, so use the formula for that integral.
		Well, the problem with this integral is that sin 3x does not have a direct substitution.
		Notice that the first integral of the above line is of the form of the inverse tangent function.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental12.htm (907 words)

	fund_thm_calc.html
		This integral cannot be evaluated in closed form.
		This function is a common function in physics called the Fresnel sine integral.
		While the integral cannot be evaluated in closed form, this function is so common that its values have been numerically tabulated.
	www.math.uga.edu /calclab2210/fund_thm_calc1.html (490 words)

	Worksheet 29
		Thus the idea is to use the # non-integral terms to approximate Sic(x) and then estimate the integral to determine the # error.
		This means we have to be able to select the non-integral terms from the # expressions B[k], and also the integral term.
		Err[k] is the integral term, that is the error when we approximate Sic(x), and # C[k] is the approximation.
	www.ugrad.math.ubc.ca /coursedoc/m210/worksh29.html (1836 words)

	Numerical Integration
		Estimate integral by M = (b-a) * f(c), where c = (a + b)/2.
		We can estimate the integral of f(x) from a to b using the formula T = (b-a)/2 (f(a) + f(b)).
		Write a program to compute the sine integral Si(x), which is defined as the integral of (sin t) / t from 0 to x.
	www.cs.princeton.edu /introcs/93integration (1079 words)

	Worksheet 27
		This is a convenient test case for the general procedure of # building tables for functions since Maple already knows the values of the Sine integral.
		Each of the integrals # Int(f(t),t=x[n]..x[n+1]) will be estimated by Simpson's rule S[2] and then added up to give # an estimate of Int(f(t),t=0..x) for x=0.0, 0.1, 0.2,...,10.0.
		Another great advantage is that we can # easily compute derivatives of f(t) because we can differentiate with respect to x inside the # integral sign.
	www.ugrad.math.ubc.ca /coursedoc/m210/worksh27.html (456 words)

	The Mathematica Book Online: Advanced Mathematics in Mathematica \| Mathematical Functions
		Thus, for example, integral representations of functions are valid only when the integral exists, but the functions themselves can usually be defined elsewhere by analytic continuation.
		Polylogarithm functions appear in Feynman diagram integrals in elementary particle physics, as well as in algebraic K-theory.
		The Lerch transcendent is related to integrals of the Fermi-Dirac distribution in statistical mechanics by
	documents.wolfram.com /mathematica/book/section-3.2.11 (2167 words)

	mfunlist (Symbolic Math Toolbox)
		Legendre's complete elliptic integrals of the first, second, and third kind.
		Associated complete elliptic integrals of the first, second, and third kind using complementary modulus.
		Legendre's incomplete elliptic integrals of the first, second, and third kind.
	www-rohan.sdsu.edu /doc/matlab/toolbox/symbolic/mfunlist.html (187 words)

	Miscellaneous Functions (Site not responding. Last check: 2007-10-27)
		Some books define the exponential integral of order 1 as the exponential integral:
		The sine and cosine integrals are defined as:
		The Fresnel sine and cosine integrals are defined as:
	www.efunda.com /math/miscellaneousfun/miscellaneousfun.cfm (62 words)

	Sine Integral (Not in Base Package) - LabVIEW 8 Help
		Evaluates the sine integral for any real number x.
		Si(x) is the result of the calculation of the sine integral for the given value of x.
		The following illustration shows the graph of the sine integral in the interval (0,15).
	zone.ni.com /reference/en-XX/help/371361A-01/gmath/sine_integral (96 words)

	Precalc
		Note that in this case we expect the function to be passed as an algebraic rule, such that input and input+epsilon are meaningful as inputs, or members of the function's domain.
		The runtotal function (below) is an inverse operation, analogous to the integral.
		As you will see from the graphs below, the runtotal function, when run against the output of the wiggle function, gets us back to the original curve, albiet displaced vertically by some constant K. This should help students grasp the Fundamental Theorem and the fact that differentiation and integration are inverse operations.
	www.4dsolutions.net /ocn/precalc.html (1028 words)

	Expression Calculator: a multithread calculator
		EllipticE — elliptic integrals of the second kind
		EllipticCE — complimentary elliptic integral of the second kind
		EllipticK — elliptic integrals of the first kind
	excalc.vestris.com /docs (79 words)

	Random numbers, and estimating area by counting darts
		The goal of this exercise is to give you practice in using the for loop, as well as to introduce you to using the Monte Carlo method of evaluatiing the area under a curve.
		The true value of this area is known, since it is equal to the following definite integral:
		// lab_area - develop a function to evaluate the integral of sine(x) // between the limits [0, PI] by evaluating the area under the // curve.
	www.ent.ohiou.edu /~et181/lab6/MonteCarlo.html (695 words)

	[No title]
		for this integral, therefore it is in the form a
		Let u = ln x and a = 1 for this integral, therefore this is in the form of the inverse tangent integral.
		Notice that this integral is in the form of the inverse sine integral, so we will let u = ln x.
	faculty.eicc.edu /bwood/ma155supplemental/supplemental8.htm (289 words)

	Handbook of Mathematical Functions, p. 227
		Exponential Integral and Related Functions WALTER GAUTSCHI AND WILLIAM F. Con tents Mathematical Properties.
		Sine, Cosine and Exponential Integrals for Large Arguments (lO
		Sine and Cosine Integrals for Arguments m (0 5 s 5 10).
	www.convertit.com /Go/Safetyonline/Reference/AMS55.ASP?Res=150&Page=227 (275 words)

	Using the Symbolic Math Toolbox (Symbolic Math Toolbox)
		This allows you to evaluate functions that are not available in standard MATLAB, such as the Fresnel cosine integral.
		For example, suppose you want to evaluate the hyperbolic cosine integral at the points 2+i, 0, and 4.5.
		The hyperbolic cosine integral has a singularity at z = 0.
	www.technion.ac.il /guides/matlab/toolbox/symbolic/ch223.html (182 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		Spherical Bessel Expansions of Sine, Cosine, and Exponential Integrals
		The exponential integral occurs in many physical problems and is relatively cumbersome to evaluate for an intermediate range of its argument.
		It would be helpful to expand that integral as a quadratic series in spherical Bessel functions, but the mathematics literature contains no expansions of this type parallel to the well-known such expansion of the sine integral.
	www.physics.utah.edu /~harris/Abs175.html (80 words)

	Tables of Integrals, Series, and Products
		Integrals that can be reduced to elliptic or pseudo-elliptic integrals
		Sines and cosines of multiple angles and of linear and more complicated functions of the argument
		The hyperbolic sine integral and hyperbolic cosine integral functions
	www.mathtable.com /gr/gr6_toc (1010 words)

	MATLAB Central File Exchange - Fresnel Cosine and Sine Integral Function
		MATLAB Central > File Exchange > Mathematics > General > Fresnel Cosine and Sine Integral Function
		This function calculates the fresnel cosine and sine integrals.
		This function calculates the fresnel cosine and sine integrals.Input:
	www.mathworks.com /matlabcentral/fileexchange/loadFile.do?objectId=9017&objectType=File (311 words)

	Double Integral
		I need to solve a double integral and I have no idea what to change the variables to:
		After putting in these variables, I can't solve the integral.
		If I change what u and v are, then either the Jacobian is zero or I still can't solve the intergral.
	www.physicsforums.com /showthread.php?p=828913 (659 words)

	[No title] (Site not responding. Last check: 2007-10-27)
		From: ron@dorianresearch.com (Ron Levine) Subject: Re: Fresnel Integrals Date: Fri, 13 Aug 1999 00:16:01 GMT Newsgroups: sci.math On Thu, 12 Aug 1999 18:54:05 -0500, "Anthony J. Bruni" wrote: >What is a good reference for Fresnel Integrals?
		For example, the old one that I used in college many years ago, Jenkins and White, Fundamentals of Optics.
		Here is an excerpt from their help page.
	www.math.niu.edu /~rusin/known-math/99/fresnel (150 words)

	Xmgr: transformations (Site not responding. Last check: 2007-10-27)
		igami(a, p) == inverse of complemented incomplete gamma integral
		A set is loaded (if there is one) with the resulting curve.
		Select the form of the results, the item marked cumulative sum will construct a set composed of the current value of the integral at a given X. Sum only reports just the value on the next line.
	www.nd.edu /~ddenting/xmgr_doc/trans.html (2130 words)

	Abramowitz and Stegun, p. 236
		[5.15] J. Todd, Evaluation of the exponential integral for large complex arguments, J. Research NBS 52, [ 5.161 F. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954).
		Circular and hyperbolic functions, exponential, sine and cosine integrals, etc., 3d ed.
		Si(z), Ci(z), Ei(x), [5.20] B. Gourary and M. Lynam, Tables of the auxiliary molecular integrals A,(z) and the auxiliary functions C,(z), The Johns Hopkins Univ.
	www.convertit.com /go/convertit/reference/ams55.asp?Res=200&Page=236 (736 words)

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