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.

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

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

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:

ci(x) = Ci(x)

Cin(x) = γ + ln x - Ci(x)

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

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

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

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

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

${\rm Si}(x)= \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}=x-\frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}-\frac{x^7}{7! \cdot7}\pm\cdots$

${\rm Ci}(x)= \gamma+\ln x+\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n}}{2n(2n)!}=\gamma+\ln x-\frac{x^2}{2!\cdot2}+\frac{x^4}{4! \cdot4}\mp\cdots$

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)

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