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Encyclopedia > Trigonometric integral

In mathematics, the trigonometric integrals are a family of integrals which involve trigonometric functions. A number of the basic trigonometric integrals are discussed at the list of integrals of trigonometric functions. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, an index set is another name for a function domain. ... In calculus, the integral of a function is an extension of the concept of a sum. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other... The following is a list of integrals (antiderivative functions) of trigonometric functions. ...

Contents

Sine integral

The sine integral is given by In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

{rm Si}(x) = int_0^xfrac{sin t}{t},dt
{rm si}(x) = -int_x^inftyfrac{sin t}{t},dt = {rm Si}(x) - frac{1}{2}pi

Note that

j_0(t)=frac{sin t}{t}

is the zeroth spherical Bessel function. In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real or complex number α. The most common and important special case is where α is an integer, n. ...


Cosine integral

The cosine integral: In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

{rm Ci}(x) = gamma + ln x + int_0^xfrac{cos t-1}{t},dt
{rm Cin}(x) = int_0^xfrac{1-cos t}{t},dt
{rm ci}(x) = -int_x^inftyfrac{cos t}{t},dt

Hyperbolic sine integral

The hyperbolic sine integral: In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

{rm Shi}(x) = int_0^xfrac{sinh t}{t},dt = {rm shi}(x)

Hyperbolic cosine integral

The hyperbolic cosine integral: In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

{rm Chi}(x) = gamma+ln x + int_0^xfrac{cosh t-1}{t},dt = {rm chi}(x)

where γ is the Euler-Mascheroni constant. The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ...


Discussion

The spiral formed by graphing si,ci is known as Nielsen's spiral. This article does not cite its references or sources. ... Plot of Nielsens spiral In mathematics, Nielsens spiral, also called the sici spiral is the spiral with parametric equations where ci is the cosine integral and si is the sine integral. ...


Asymptotic Expansion

Large x

{rm Si}(x)=frac{pi}{2} - frac{cos x}{x}left(1-frac{2!}{x^{2}}+...right) - frac{sin x}{x}left(frac{1}{x}-frac{3!}{x^{3}}+...right)
{rm Ci}(x)= frac{sin x}{x}left(1-frac{2!}{x^{2}}+...right) -frac{cos x}{x}left(frac{1}{x}-frac{3!}{x^{3}}+...right)

Small x

{rm Si}(x)= sum_{n=0}^{infty}frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}=x-frac{x^3}{3!cdot3}+frac{x^5}{5!cdot5}-frac{x^7}{7! cdot7}pmcdots
{rm Ci}(x)= gamma+ln x+sum_{n=1}^{infty}frac{(-1)^{n}x^{2n}}{2n(2n)!}=gamma+ln x-frac{x^2}{2!cdot2}+frac{x^4}{4! cdot4}mpcdots

See also

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. ... 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: Here, ln denotes the natural logarithm. ...

References


  Results from FactBites:
 
List of integrals of trigonometric functions - Wikipedia, the free encyclopedia (235 words)
Integrals of trigonometric functions containing both sin and cos
Integrals of trigonometric functions containing both cos and tan
Integrals of trigonometric functions containing both cos and cot
Powers of Trigonometric Functions (270 words)
Integral of a variable to a power: The integral of a variable to a power is the variable to a power increased by one and divided by the new power.
Integral of the sum of differentiable functions: The integral of an algebraic sum of differentiable functions is the same as the algebraic sum of the integrals of these functions taken separately.
Integral of powers of trigonometric functions: The integrals of powers of trigonometric functions will be limited to those which may, by substitution, be written in the form J u" du.
  More results at FactBites »

 

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