Trigonometric integral

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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 sine integral is given by

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

[edit] Cosine integral

The cosine integral:

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

[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] Discussion

The spiral formed by graphing si,ci is known as Nielsen's spiral.

[edit] Asymptotic Expansion

[edit] 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)

[edit] Small x

{\rm Si}(x)= \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}
{\rm Ci}(x)= \gamma+\ln x+\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n}}{2n(2n)!}

[edit] See also

[edit] References

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