Trigonometric integral

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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.

Sine integral[edit]

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 ~.

By definition, Si(x) is the antiderivative of sinx/x which is zero for x = 0; and si(x) is the antiderivative of sinx/x which is zero for x = ∞. Their difference is given by the Dirichlet integral,

{\rm Si}(x) - {\rm si}(x) = \int_0^\infty\frac{\sin t}{t}\,dt = \frac{\pi}{2} ~.

Note that sinx/x is the sinc function, and also the zeroth spherical Bessel function.

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.

Related is the Gibbs phenomenon: if the sine integral is considered as the convolution of the sinc function with the heaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.

Cosine integral[edit]

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~,

where γ is the Euler–Mascheroni constant.

ci(x) is the antiderivative of cosx/x (which vanishes at x = ∞). One further has

{\rm ci}(x)={\rm Ci}(x)\,
{\rm Cin}(x)=\gamma+\ln x-{\rm Ci}(x)~.

Hyperbolic sine integral[edit]

The hyperbolic sine integral:

{\rm Shi}(x) = \int_0^x\frac{\sinh t}{t}\,dt = {\rm shi}(x).
{\rm Shi}(x)=\sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)^2(2n)!}=x+\frac{x^3}{3!\cdot3}+\frac{x^5}{5!\cdot5}+\frac{x^7}{7! \cdot7}+\cdots.

Hyperbolic cosine integral[edit]

The hyperbolic cosine integral is

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

Nielsen's spiral[edit]

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, 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 and other areas.

Expansion[edit]

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

Asymptotic series (for large argument)[edit]

{\rm Si}(x)=\frac{\pi}{2} 
                 - \frac{\cos x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right)
                 - \frac{\sin x}{x}\left(\frac{1}{x}-\frac{3!}{x^3}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right)
{\rm Ci}(x)= \frac{\sin x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right)
                   -\frac{\cos x}{x}\left(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right) ~.

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at ~{\Re}(x) ≫ 1.

Convergent series[edit]

{\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 | ≫ 1 the series will converge slowly initially, requiring many terms for high precisions.

Relation with the exponential integral of imaginary argument[edit]

The function

 {\rm E}_1(z) = \int_1^\infty \frac{\exp(-zt)}{t}\,{\rm d} t \qquad({\Re }(z) \ge 0)

is called the exponential integral. It is closely related to 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 respective function is analytic except for the cut at negative values of the argument, the area of validity of the relation should be extended to ~{\Re}(x) > 0. (Outside this range, additional terms which are integer factors of π appear in the expression.)

Cases of imaginary argument of the generalized integro-exponential function are


\int_1^\infty \cos(ax)\frac{\ln x}{x} \, dx =
-\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2a}{2}
+\sum_{n\ge 1}\frac{(-a^2)^n}{(2n)!(2n)^2} ~,

which is the real part of


\int_1^\infty e^{iax}\frac{\ln x}{x} \, dx = -\frac{\pi^2}{24} + \gamma\left(\frac{\gamma}{2}+\ln a\right)+\frac{\ln^2 a}{2}-\frac{\pi}{2}i(\gamma+\ln a) + \sum_{n\ge 1}\frac{(ia)^n}{n!n^2}  ~.

Similarly


\int_1^\infty e^{iax}\frac{\ln x}{x^2}dx
=1+ia[-\frac{\pi^2}{24}+\gamma\left(\frac{\gamma}{2}+\ln a-1\right)+\frac{\ln^2 a}{2}-\ln a+1
-\frac{i\pi}{2}(\gamma+\ln a-1)]+\sum_{n\ge 1}\frac{(ia)^{n+1}}{(n+1)!n^2}~.

Efficient evaluation[edit]

Padé approximants of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae are accurate to better than 10−16 for 0 ≤ x ≤4 ,



\begin{array}{rcl}
{\rm Si}(x) &=& x \cdot \left( 
\frac{
\begin{array}{l}
1 -4.54393409816329991\cdot 10^{-2} \cdot x^2 + 1.15457225751016682\cdot 10^{-3} \cdot x^4 - 1.41018536821330254\cdot 10^{-5} \cdot x^6 \\
~~~ + 9.43280809438713025 \cdot 10^{-8} \cdot x^8 - 3.53201978997168357 \cdot 10^{-10} \cdot x^{10} + 7.08240282274875911 \cdot 10^{-13} \cdot x^{12} \\
~~~ - 6.05338212010422477 \cdot 10^{-16} \cdot x^{14}
\end{array}
}
{
\begin{array}{l}
1 + 1.01162145739225565 \cdot 10^{-2} \cdot x^2 + 4.99175116169755106 \cdot 10^{-5} \cdot x^4 + 1.55654986308745614 \cdot 10^{-7} \cdot x^6 \\
~~~ + 3.28067571055789734 \cdot 10^{-10} \cdot x^8 + 4.5049097575386581 \cdot 10^{-13} \cdot x^{10} + 3.21107051193712168 \cdot 10^{-16} \cdot x^{12}
\end{array}
}
\right)\\
&~&\\
{\rm Ci}(x) &=& \gamma + \ln(x) +\\
&& x^2 \cdot \left(
\frac{
\begin{array}{l}
-0.25 + 7.51851524438898291 \cdot 10^{-3} \cdot x^2 - 1.27528342240267686 \cdot 10^{-4} \cdot x^4 + 1.05297363846239184 \cdot 10^{-6} \cdot x^6 \\
~~~ -4.68889508144848019 \cdot 10^{-9} \cdot x^8 + 1.06480802891189243 \cdot  10^{-11} \cdot x^{10} - 9.93728488857585407 \cdot 10^{-15} \cdot x^{12} \\
\end{array}
}
{
\begin{array}{l}
1 + 1.1592605689110735 \cdot 10^{-2} \cdot x^2 + 6.72126800814254432 \cdot 10^{-5} \cdot x^4 + 2.55533277086129636 \cdot 10^{-7} \cdot x^6 \\
~~~ + 6.97071295760958946 \cdot 10^{-10} \cdot x^8 + 1.38536352772778619 \cdot 10^{-12} \cdot x^{10} + 1.89106054713059759 \cdot 10^{-15} \cdot x^{12} \\
~~~ + 1.39759616731376855 \cdot 10^{-18} \cdot x^{14} \\
\end{array}
}
\right)
\end{array}


For x > 4, instead, one can use the helper functions,


\begin{array}{rcl}
f(x) 
&=& \int_0^\infty \frac{sin(t)}{t+x} dt = \int_0^\infty \frac{e^{-x t}}{t^2 + 1} dt 
~=~ {\rm Ci}(x) \sin(x) + \left(\frac{\pi}{2} - {\rm Si}(x) \right) \cos(x) \\
g(x)
&=& \int_0^\infty \frac{cos(t)}{t+x} dt = \int_0^\infty \frac{t e^{-x t}}{t^2 + 1} dt 
~=~ -{\rm Ci}(x) \cos(x) + \left(\frac{\pi}{2} - {\rm Si}(x) \right) \sin(x) \\
\end{array}  ~,

using which, the trigonometric integrals may be re-expressed as


\begin{array}{rcl}
{\rm Si}(x) &=& \frac{\pi}{2} - f(x) \cos(x) - g(x) \sin(x) \\
{\rm Ci}(x) &=& f(x) \sin(x) - g(x) \cos(x) \\
\end{array} ~.


Chebyshev-Padé expansions of \;\;\frac{1}{\sqrt{y}} \; f\left(\frac{1}{\sqrt{y}} \right) \;\; and \;\;\frac{1}{y} \; g\left(\frac{1}{\sqrt{y}} \right)\;\; in the interval 0..\frac{1}{4^2} give the following approximants, good to better than 10−16 for x \ge 4:



\begin{array}{rcl}
f(x) &=& \dfrac{1}{x} \cdot \left(\frac{
\begin{array}{l}
1 + 7.44437068161936700618 \cdot 10^2 \cdot x^{-2} + 1.96396372895146869801 \cdot 10^5 \cdot x^{-4} + 2.37750310125431834034 \cdot 10^7 \cdot x^{-6} \\
~~~ + 1.43073403821274636888 \cdot 10^9 \cdot x^{-8} + 4.33736238870432522765 \cdot 10^{10} \cdot x^{-10} + 6.40533830574022022911 \cdot 10^{11} \cdot x^{-12} \\
~~~ + 4.20968180571076940208 \cdot 10^{12} \cdot x^{-14} + 1.00795182980368574617 \cdot 10^{13} \cdot x^{-16} + 4.94816688199951963482 \cdot 10^{12} \cdot x^{-18} \\
~~~ - 4.94701168645415959931 \cdot 10^{11} \cdot x^{-20}
\end{array}
}{
\begin{array}{l}
1 + 7.46437068161927678031 \cdot 10^2 \cdot x^{-2} + 1.97865247031583951450 \cdot 10^5 \cdot x^{-4} + 2.41535670165126845144 \cdot 10^7 \cdot x^{-6} \\
~~~ + 1.47478952192985464958 \cdot 10^9 \cdot x^{-8} + 4.58595115847765779830 \cdot 10^{10} \cdot x^{-10} + 7.08501308149515401563 \cdot 10^{11} \cdot x^{-12} \\
~~~ + 5.06084464593475076774 \cdot 10^{12} \cdot x^{-14} + 1.43468549171581016479 \cdot 10^{13} \cdot x^{-16} + 1.11535493509914254097 \cdot 10^{13} \cdot x^{-18}
\end{array}
}
\right) \\
& &\\
g(x) &=& \dfrac{1}{x^2} \cdot \left(\frac{
\begin{array}{l}
1 + 8.1359520115168615 \cdot 10^2 \cdot x^{-2} + 2.35239181626478200 \cdot 10^5 \cdot x^{-4} +3.12557570795778731 \cdot 10^7 \cdot x^{-6} \\
~~~ + 2.06297595146763354 \cdot 10^9 \cdot x^{-8} + 6.83052205423625007 \cdot 10^{10} \cdot x^{-10} + 1.09049528450362786 \cdot 10^{12} \cdot x^{-12} \\
~~~ + 7.57664583257834349 \cdot 10^{12} \cdot x^{-14} + 1.81004487464664575 \cdot 10^{13} \cdot x^{-16} + 6.43291613143049485 \cdot 10^{12} \cdot x^{-18} \\
~~~ - 1.36517137670871689 \cdot 10^{12} \cdot x^{-20}
\end{array}
}{
\begin{array}{l}
1 + 8.19595201151451564 \cdot 10^2 \cdot x^{-2} + 2.40036752835578777 \cdot 10^5 \cdot x^{-4} + 3.26026661647090822 \cdot 10^7 \cdot x^{-6} \\
~~~ + 2.23355543278099360 \cdot 10^9 \cdot x^{-8} + 7.87465017341829930 \cdot 10^{10} \cdot x^{-10} + 1.39866710696414565 \cdot 10^{12} \cdot x^{-12} \\
~~~ + 1.17164723371736605 \cdot 10^{13} \cdot x^{-14} + 4.01839087307656620 \cdot 10^{13} \cdot x^{-16} + 3.99653257887490811 \cdot 10^{13} \cdot x^{-18}
\end{array}
}
\right) \\
\end{array}


Here are text versions of the above suitable for copying into computer code (using x2 = x*x and y = 1/(x*x) where appropriate):

   Si = x*(1. +
           x2*(-4.54393409816329991e-2 +
               x2*(1.15457225751016682e-3 +
                   x2*(-1.41018536821330254e-5 +
                       x2*(9.43280809438713025e-8 +
                           x2*(-3.53201978997168357e-10 +
                               x2*(7.08240282274875911e-13 +
                                   x2*(-6.05338212010422477e-16))))))))
        / (1. + 
           x2*(1.01162145739225565e-2 +
               x2*(4.99175116169755106e-5 + 
                   x2*(1.55654986308745614e-7 +
                       x2*(3.28067571055789734e-10 +
                           x2*(4.5049097575386581e-13 + 
                               x2*(3.21107051193712168e-16)))))))
   
   Ci = 0.577215664901532861 + ln(x) + 
        x2*(-0.25 +
            x2*(7.51851524438898291e-3 +
                x2*(-1.27528342240267686e-4 + 
                    x2*(1.05297363846239184e-6 +
                        x2*(-4.68889508144848019e-9 +
                            x2*(1.06480802891189243e-11 +
                                x2*(-9.93728488857585407e-15)))))))
        / (1. +
           x2*(1.1592605689110735e-2 +
               x2*(6.72126800814254432e-5 + 
                   x2*(2.55533277086129636e-7 +
                       x2*(6.97071295760958946e-10 +
                           x2*(1.38536352772778619e-12 + 
                               x2*(1.89106054713059759e-15 +
                                   x2*(1.39759616731376855e-18))))))))
   
   f = (1. + 
        y*(7.44437068161936700618e2 +
           y*(1.96396372895146869801e5 +
              y*(2.37750310125431834034e7 +
                 y*(1.43073403821274636888e9 +
                    y*(4.33736238870432522765e10 +
                       y*(6.40533830574022022911e11 +
                          y*(4.20968180571076940208e12 +
                             y*(1.00795182980368574617e13 +
                                y*(4.94816688199951963482e12 +
                                   y*(-4.94701168645415959931e11)))))))))))                                                               
        / (x*(1. +
              y*(7.46437068161927678031e2 +
                 y*(1.97865247031583951450e5 +
                    y*(2.41535670165126845144e7 +
                       y*(1.47478952192985464958e9 +
                          y*(4.58595115847765779830e10 +
                             y*(7.08501308149515401563e11 +
                                y*(5.06084464593475076774e12 +
                                   y*(1.43468549171581016479e13 +
                                      y*(1.11535493509914254097e13)))))))))))
   
   g = y*(1. +
          y*(8.1359520115168615e2 +
             y*(2.35239181626478200e5 +
                y*(3.12557570795778731e7 +
                   y*(2.06297595146763354e9 +
                      y*(6.83052205423625007e10 +
                         y*(1.09049528450362786e12 +
                            y*(7.57664583257834349e12 +
                               y*(1.81004487464664575e13 +
                                  y*(6.43291613143049485e12 +
                                     y*(-1.36517137670871689e12)))))))))))
       / (1. +
          y*(8.19595201151451564e2 +
             y*(2.40036752835578777e5 +
                y*(3.26026661647090822e7 +
                   y*(2.23355543278099360e9 +
                      y*(7.87465017341829930e10 +
                         y*(1.39866710696414565e12 +
                            y*(1.17164723371736605e13 +
                               y*(4.01839087307656620e13 +
                                  y*(3.99653257887490811e13))))))))))

See also[edit]

Signal processing[edit]

References[edit]

External links[edit]