Trigonometric integral 

Si(x) (red) and Ci(x) (blue)

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.

Contents

Sine integral

The different sine integral definitions are:

{\rm Si}(x) = \int_0^x\frac{\sin t}{t}\,dt
{\rm si}(x) = -\int_x^\infty\frac{\sin t}{t}\,dt

Si(x) is the primitive of sinx / x which is zero for x = 0; si(x) is the primitive of sinx / x which is zero for x=\infty. We have:

{\rm si}(x) = {\rm Si}(x) - \frac{\pi}{2}

Note that \frac{\sin t}{t} is the sinc function and also the zeroth spherical Bessel function.

Cosine integral

The different cosine integral definitions are:

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

ci(x) is the primitive of cosx / x which is zero for x=\infty. We have:

{\rm ci}(x)={\rm Ci}(x)\,
{\rm Cin}(x)=\gamma+\ln x-{\rm Ci}(x)\,

Hyperbolic sine integral

The hyperbolic sine integral:

{\rm Shi}(x) = \int_0^x\frac{\sinh t}{t}\,dt = {\rm shi}(x).

Hyperbolic cosine integral

The hyperbolic cosine integral:

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

where γ is the Euler-Mascheroni constant.

Nielsen's spiral

Nielsen's spiral

The spiral formed by parametric plot of si,ci is known as Nielsen's spiral. It is also referred to as the Euler Spiral the Cornu Spiral 1, a clothoid, or as a linear curvature Polynomial Spiral. The spiral is also closely related to the Fresnel Integrals. This spiral has applications in vision processing, road and track construction, among other things.

Expansion

Various expansions can be used for evaluation of Trigonometric integrals, depending on the range of the argument.

Asymptotic series (for large argument)

{\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)

These series are divergent, although can be used for estimates and even precise evaluation at ~{\rm Re} (x) \gg 1~.

Convergent series

{\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}\pm\cdots
{\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}\mp\cdots

These series are convergent at any complex ~x~, although for |x|\gg 1 the evaluation is slow and not precise, if at all.

Relation with the exponential integral of imaginary argument

Function {\rm E}_1(z) = \int_1^\infty \frac {\exp(-zt)} {t} {\rm d} t ~~,~~~~({\rm Re}(z) \ge 0) is called exponential integral. It is closely related with Si and Ci:

{\rm E}_1( {\rm i}\!~ x)= -\frac{\pi}{2} +{\rm Si}(x)-{\rm i}\cdot {\rm Ci}(x)~~~~,~~~~~(x>0)

As each involved function is analytic except the cut at negative values of the argument, the area of validity of the relation should be extended to Re(x) > 0. (Out of this range, additional terms which are integer factors of π appear in the expression).

See also

References

Wikibooks
The Wikibook Calculus has a page on the topic of
Trigonometric integrals