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.
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Sine integral
The different sine integral definitions are:
- <math>{\rm Si}(x) = \int_0^x\frac{\sin t}{t}\,dt</math>
- <math>{\rm si}(x) = -\int_x^\infty\frac{\sin t}{t}\,dt</math>
<math>{\rm Si}(x)</math> is the primitive of <math>\sin x/x</math> which is zero for <math>x=0</math>; <math>{\rm si}(x)</math> is the primitive of <math>\sin x/x</math> which is zero for <math>x=\infty</math>. We have:
- <math>{\rm si}(x) = {\rm Si}(x) - \frac{\pi}{2}</math>
Note that
- <math>j_0(t)=\frac{\sin t}{t}</math>
is the sinc function and also the zeroth spherical Bessel function.
Cosine integral
The different cosine integral definitions are:
- <math>{\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt</math>
- <math>{\rm ci}(x) = -\int_x^\infty\frac{\cos t}{t}\,dt</math>
- <math>{\rm Cin}(x) = \int_0^x\frac{1-\cos t}{t}\,dt</math>
<math>{\rm ci}(x)</math> is the primitive of <math>\cos x/x</math> which is zero for <math>x=\infty</math>. We have:
- <math>{\rm ci}(x)={\rm Ci}(x)</math>
- <math>{\rm Cin}(x)=\gamma+\ln x-{\rm Ci}(x)</math>
Hyperbolic sine integral
The hyperbolic sine integral:
- <math>{\rm Shi}(x) = \int_0^x\frac{\sinh t}{t}\,dt = {\rm shi}(x).</math>
Hyperbolic cosine integral
The hyperbolic cosine integral:
- <math>{\rm Chi}(x) = \gamma+\ln x + \int_0^x\frac{\cosh t-1}{t}\,dt = {\rm chi}(x)</math>
where <math>\gamma</math> is the Euler-Mascheroni constant.
Discussion
The spiral formed by graphing si,ci is known as Nielsen's spiral.
Asymptotic expansion
Large x
- <math>{\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)</math>
- <math>{\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)</math>
Small x
- <math>{\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</math>
- <math>{\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</math>
See also
References
- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)
