Trigonometric integral

From Wikipedia, the free encyclopedia

Si(x) (blue) and Ci(x) (green) plotted on the same plot.

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.

[edit] Sine integral

Plot of Si(x) for 0 ≤ x ≤ 8π.

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 $sin x / x$ which is zero for $x = 0$ ; $si(x)$ is the primitive of $sin x / x$ 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.

When $x=\infty$ , this is known as the Dirichlet integral.

In signal processing, the oscillations of the Sine integral cause overshoot and ringing artifacts when using the sinc filter, and frequency domain ringing if using a truncated sinc filter as a low-pass filter.

The Gibbs phenomenon is a related phenomenon: thinking of sinc as a low-pass filter, it corresponds to truncating the Fourier series, which causes the Gibbs phenomenon.

[edit] Cosine integral

Plot of Ci(x) for 0 < x ≤ 8π.

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 $cos x / x$ which is zero for $x=\infty$ . We have:

${\rm ci}(x)={\rm Ci}(x)\,$

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

[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

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.

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

${\rm E}_1( {\rm i}\!~ x)= i\left(-\frac{\pi}{2} +{\rm Si}(x)\right)-{\rm Ci}(x) = i~{\rm si}(x) - {\rm ci}(x) \qquad(x>0)$

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).

[edit] See also

[edit] Signal processing

[edit] References

^ [1]

The Wikibook Calculus has a page on the topic of

Trigonometric integrals

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)
- (Section 5.2, Sine and Cosine Integrals)

[0] [1]

[1]

Trigonometric integral

From Wikipedia, the free encyclopedia

Contents

[edit] Sine integral

[edit] Cosine integral

[edit] Hyperbolic sine integral

[edit] Hyperbolic cosine integral

[edit] Nielsen's spiral

[edit] Expansion

[edit] Asymptotic series (for large argument)

[edit] Convergent series

[edit] Relation with the exponential integral of imaginary argument

[edit] See also

[edit] Signal processing

[edit] References

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