Trigonometric integral

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

[edit] 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.

[edit] 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:

ci(x) = Ci(x)
Cin(x) = γ + lnx − Ci(x)

[edit] Hyperbolic sine integral

The hyperbolic sine integral:

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

[edit] 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.

[edit] Nielsen's spiral

Nielsen's spiral
Nielsen's spiral

The spiral formed by parametric plot of si,ci is known as Nielsen's spiral.

[edit] Expansion

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

[edit] 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~.

[edit] 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.

[edit] 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 of funcitons involved 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).

[edit] See also

[edit] References

Wikibooks
Wikibooks Calculus has a page on the topic of