Trigonometric integral

In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions.

Sine integral

The different sine integral definitions are

${\displaystyle \operatorname {Si} (x)=\int _{0}^{x}{\frac {\sin t}{t}}\,dt}$
${\displaystyle \operatorname {si} (x)=-\int _{x}^{\infty }{\frac {\sin t}{t}}\,dt.}$

Note that the integrand sin x / x is the sinc function, and also the zeroth spherical Bessel function. Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.

By definition, Si(x) is the antiderivative whose value is zero at zero of sin x / x for x = 0, and si(x) is the antiderivative whose value at x = ∞ is zero of sin x / x. Their difference is given by the Dirichlet integral,

${\displaystyle \operatorname {Si} (x)-\operatorname {si} (x)=\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt={\frac {\pi }{2}}~.}$

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

The different cosine integral definitions are

${\displaystyle \operatorname {Ci} (x)=-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt=\gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt\qquad (|{\rm {Arg}}(x)|<\pi )}$
${\displaystyle \operatorname {Cin} (x)=\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt~,}$

where γ is the Euler–Mascheroni constant. Some texts use ci instead of Ci.

Ci(x) is the antiderivative of cos x / x (which vanishes at ${\displaystyle x\to \infty }$). The two definitions are related by

${\displaystyle \operatorname {Cin} (x)=\gamma +\ln x-\operatorname {Ci} (x)~.}$

Cin is an even entire function.

Hyperbolic sine integral

The hyperbolic sine integral is defined as

${\displaystyle \operatorname {Shi} (x)=\int _{0}^{x}{\frac {\sinh(t)}{t}}\,dt.}$

It is related to the ordinary sine integral by

${\displaystyle \operatorname {Si} (ix)=i\operatorname {Shi} (x).}$

Hyperbolic cosine integral

The hyperbolic cosine integral is

${\displaystyle \operatorname {Chi} (x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt\qquad (|{\rm {Arg}}(x)|<\pi )}$

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.

It has the series expansion

${\displaystyle \operatorname {Chi} (x)=\gamma +\ln(x)+{\frac {x^{2}}{4}}+{\frac {x^{4}}{96}}+{\frac {x^{6}}{4320}}+{\frac {x^{8}}{322560}}+{\frac {x^{10}}{36288000}}+O(x^{12}).}$

Auxiliary functions

Trigonometric integrals can be understood in terms of the so-called "auxiliary functions"

${\displaystyle f(x)\equiv \int _{0}^{\infty }{\frac {\sin(t)}{t+x}}dt=\int _{0}^{\infty }{\frac {e^{-xt}}{t^{2}+1}}dt=\operatorname {Ci} (x)\sin(x)+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos(x)}$
${\displaystyle g(x)\equiv \int _{0}^{\infty }{\frac {\cos(t)}{t+x}}dt=\int _{0}^{\infty }{\frac {te^{-xt}}{t^{2}+1}}dt=-\operatorname {Ci} (x)\cos(x)+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\sin(x)}$.

Using these functions, the trigonometric integrals may be re-expressed as (cf Abramowitz & Stegun, p. 232)

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

Nielsen's spiral

The spiral formed by parametric plot of si , ci is known as Nielsen's spiral.

${\displaystyle x(t)=a\times \operatorname {ci} (t)}$
${\displaystyle y(t)=a\times \operatorname {si} (t)}$

The spiral is closely related to the Fresnel integrals and the Euler spiral. Nielsen's spiral has applications in vision processing, road and track construction and other areas.

Expansion

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

Asymptotic series (for large argument)

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

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at ℜ(x) ≫ 1.

Convergent series

${\displaystyle \operatorname {Si} (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}}=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}\pm \cdots }$
${\displaystyle \operatorname {Ci} (x)=\gamma +\ln x+\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{2n}}{2n(2n)!}}=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}\mp \cdots }$

These series are convergent at any complex x, although for |x| ≫ 1, the series will converge slowly initially, requiring many terms for high precision.

Derivation of Series Expansion

${\displaystyle \sin \,x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+{\frac {x^{9}}{9!}}-{\frac {x^{11}}{11!}}+\,...}$(Maclaurin Series Expansion)

${\displaystyle {\frac {\sin \,x}{x}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+{\frac {x^{8}}{9!}}-{\frac {x^{10}}{11!}}+\,...}$

${\displaystyle \therefore \int {\frac {\sin \,x}{x}}dx=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}+{\frac {x^{9}}{9!\cdot 9}}-{\frac {x^{11}}{11!\cdot 11}}+\,...}$

Relation with the exponential integral of imaginary argument

The function

${\displaystyle \operatorname {E} _{1}(z)=\int _{1}^{\infty }{\frac {\exp(-zt)}{t}}\,dt\qquad (\Re (z)\geq 0)}$

is called the exponential integral. It is closely related to Si and Ci,

${\displaystyle \operatorname {E} _{1}(ix)=i\left(-{\frac {\pi }{2}}+\operatorname {Si} (x)\right)-\operatorname {Ci} (x)=i\operatorname {si} (x)-\operatorname {ci} (x)\qquad (x>0)~.}$

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 (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

${\displaystyle \int _{1}^{\infty }\cos(ax){\frac {\ln x}{x}}\,dx=-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a\right)+{\frac {\ln ^{2}a}{2}}+\sum _{n\geq 1}{\frac {(-a^{2})^{n}}{(2n)!(2n)^{2}}}~,}$

which is the real part of

${\displaystyle \int _{1}^{\infty }e^{iax}{\frac {\ln x}{x}}\,dx=-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a\right)+{\frac {\ln ^{2}a}{2}}-{\frac {\pi }{2}}i(\gamma +\ln a)+\sum _{n\geq 1}{\frac {(ia)^{n}}{n!n^{2}}}~.}$

Similarly

${\displaystyle \int _{1}^{\infty }e^{iax}{\frac {\ln x}{x^{2}}}dx=1+ia[-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a-1\right)+{\frac {\ln ^{2}a}{2}}-\ln a+1-{\frac {i\pi }{2}}(\gamma +\ln a-1)]+\sum _{n\geq 1}{\frac {(ia)^{n+1}}{(n+1)!n^{2}}}~.}$

Efficient evaluation

Padé approximants of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. The following formulae, given by Rowe et al. (2015), are accurate to better than 10−16 for 0 ≤ x ≤ 4,

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

The integrals may be evaluated indirectly via auxiliary functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$, which are defined by

${\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-f(x)\cos(x)-g(x)\sin(x),}$
${\displaystyle \operatorname {Ci} (x)=f(x)\sin(x)-g(x)\cos(x).}$

For ${\displaystyle x\geq 4}$ the Padé rational functions given below approximate ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ with error less than 10−16:

${\displaystyle {\begin{array}{rcl}f(x)&\approx &{\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)&\approx &{\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}}}$