Trigonometric integral

The different sine integral definitions are $${\displaystyle {\begin{aligned}\operatorname {Si} (x)&={\hphantom {-}}\int _{0}^{x}{\frac {\sin t}{t}}\,dt\\\operatorname {si} (x)&=-\int _{x}^{\infty }{\frac {\sin t}{t}}\,dt~.\end{aligned}}}$$

Note that the integrand ${\textstyle {\frac {\sin t}{t}}}$ is the sinc function, and also the zeroth spherical Bessel function. Since ⁠${\displaystyle \operatorname {sinc} }$⁠ is an even entire function (holomorphic over the entire complex plane), ⁠${\displaystyle \operatorname {Si} }$⁠ is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.

By definition, ⁠${\displaystyle \operatorname {Si} (x)}$⁠ is the antiderivative of ⁠${\displaystyle \textstyle {\frac {\sin x}{x}}}$⁠ whose value is zero at ⁠${\displaystyle x=0}$⁠, and ⁠${\displaystyle \operatorname {si} (x)}$⁠ is the antiderivative whose value is zero at ⁠${\displaystyle x=\infty }$⁠. Their difference is given by the Dirichlet integral, $${\displaystyle \operatorname {Si} (x)-\operatorname {si} (x)=\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt={\frac {\pi }{2}}\quad {\text{ or }}\quad \operatorname {Si} (x)={\frac {\pi }{2}}+\operatorname {si} (x)~.}$$

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.

The different cosine integral definitions are $${\displaystyle \operatorname {Cin} (x)\equiv \int _{0}^{x}{\frac {1-\cos t}{t}}\,dt.}$$

⁠${\displaystyle \operatorname {Cin} }$⁠ is an even, entire function. For that reason, some texts define ⁠${\displaystyle \operatorname {Cin} }$⁠ as the primary function, and derive ⁠${\displaystyle \operatorname {Ci} }$⁠ in terms of ⁠${\displaystyle \operatorname {Cin} }$⁠.^[1]

$${\displaystyle {\begin{aligned}\operatorname {Ci} (x)&\equiv ~-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt\\&=\gamma +\ln x-\int _{0}^{x}{\frac {1-\cos t}{t}}\,dt\\[0.4em]&=\gamma +\ln x-\operatorname {Cin} (x),\end{aligned}}}$$ for ⁠${\displaystyle |\operatorname {Arg} (x)|<\pi }$⁠, where ⁠${\displaystyle \gamma \approx 0.57721566490\dots }$⁠ is the Euler–Mascheroni constant. Some texts use ⁠${\displaystyle \operatorname {ci} }$⁠ instead of ⁠${\displaystyle \operatorname {Ci} }$⁠. The restriction on Arg(x) is to avoid a discontinuity (shown as the orange vs. blue area on the left half of the plot above) that arises because of a branch cut in the standard logarithm function (⁠${\displaystyle \ln }$⁠).

⁠${\displaystyle \operatorname {Ci} (x)}$⁠ is the antiderivative of ⁠${\displaystyle \textstyle {\frac {\cos x}{x}}}$⁠ (which vanishes as ⁠${\displaystyle x\to \infty }$⁠). The two definitions are related by $${\displaystyle \operatorname {Ci} (x)=\gamma +\ln x-\operatorname {Cin} (x).}$$

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

The hyperbolic cosine integral is

$${\displaystyle \operatorname {Chi} (x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cosh(t)-1}{t}}\,dt\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,}$$ where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.

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

Trigonometric integrals can be understood in terms of the so-called "auxiliary functions" $${\displaystyle {\begin{aligned}f(x)&\equiv \int _{0}^{\infty }{\frac {\sin t}{t+x}}\,dt\quad =\int _{0}^{\infty }{\frac {e^{-xt}}{t^{2}+1}}\,dt\\[.3em]&={\hphantom {-}}\operatorname {Ci} (x)\sin x-\operatorname {si} (x)\cos x\\&={\hphantom {-}}\operatorname {Ci} (x)\sin x+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos x\,,\\[0.5em]g(x)&\equiv \int _{0}^{\infty }{\frac {\cos t}{t+x}}\,dt\quad =\int _{0}^{\infty }{\frac {te^{-xt}}{t^{2}+1}}\,dt\\[.3em]&=-\operatorname {Ci} (x)\cos x-\operatorname {si} (x)\sin x\\&=-\operatorname {Ci} (x)\cos x+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\sin x\,.\end{aligned}}}$$ Using these functions, the trigonometric integrals may be re-expressed as^[2] $${\displaystyle {\begin{aligned}\operatorname {Si} (x)&={\frac {\pi }{2}}-f(x)\cos x+g(x)\sin x\,,\\\operatorname {Ci} (x)&={\hphantom {{\frac {\pi }{2}}-\mathop {} }}f(x)\sin x-g(x)\cos x\,.\end{aligned}}}$$

The spiral formed by parametric plot of ⁠${\displaystyle \operatorname {si} }$⁠, ⁠${\displaystyle \operatorname {ci} }$⁠ is known as Nielsen's spiral. $${\displaystyle {\begin{aligned}x(t)&=a\times \operatorname {ci} (t)\\y(t)&=a\times \operatorname {si} (t)\end{aligned}}}$$

The spiral is closely related to the Fresnel integrals and the Euler spiral.^[3]Nielsen's spiral has applications in vision processing, road and track construction and other areas.^{[failed verification]}

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

The function $${\displaystyle \operatorname {E} _{1}(z)=\int _{1}^{\infty }{\frac {\exp(-zt)}{t}}\,dt\qquad ~{\text{ for }}~\Re (z)\geq 0}$$ is called the exponential integral. It is closely related to Si and Ci, $${\displaystyle \operatorname {E} _{1}(ix)=i\left(-{\frac {\pi }{2}}+\operatorname {Si} (x)\right)-\operatorname {Ci} (x)=i\operatorname {si} (x)-\operatorname {Ci} (x)\qquad ~{\text{ for }}~x>0~.}$$

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

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

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 1×10⁻¹⁶ for ⁠${\displaystyle 0\leq x\leq 4}$⁠:^[4] $${\displaystyle {\begin{aligned}\operatorname {Si} (x)\approx &\mathop {} x\cdot \left({\begin{smallmatrix}1&-4.54393409816329991\cdot 10^{-2}\cdot x^{2}&+1.15457225751016682\cdot 10^{-3}\cdot x^{4}&-1.41018536821330254\cdot 10^{-5}\cdot x^{6}\\&+9.43280809438713025\cdot 10^{-8}\cdot x^{8}&-3.53201978997168357\cdot 10^{-10}\cdot x^{10}&+7.08240282274875911\cdot 10^{-13}\cdot x^{12}\\&-6.05338212010422477\cdot 10^{-16}\cdot x^{14}\\\hline 1&+1.01162145739225565\cdot 10^{-2}\cdot x^{2}&+4.99175116169755106\cdot 10^{-5}\cdot x^{4}&+1.55654986308745614\cdot 10^{-7}\cdot x^{6}\\&+3.28067571055789734\cdot 10^{-10}\cdot x^{8}&+4.5049097575386581\cdot 10^{-13}\cdot x^{10}&+3.21107051193712168\cdot 10^{-16}\cdot x^{12}\end{smallmatrix}}\right)\\[1em]\operatorname {Ci} (x)\approx &\mathop {} \gamma +\ln(x)+\\&x^{2}\cdot \left({\begin{smallmatrix}-0.25&+7.51851524438898291\cdot 10^{-3}\cdot x^{2}&-1.27528342240267686\cdot 10^{-4}\cdot x^{4}&+1.05297363846239184\cdot 10^{-6}\cdot x^{6}\\&-4.68889508144848019\cdot 10^{-9}\cdot x^{8}&+1.06480802891189243\cdot 10^{-11}\cdot x^{10}&-9.93728488857585407\cdot 10^{-15}\cdot x^{12}\\\hline 1&+1.1592605689110735\cdot 10^{-2}\cdot x^{2}&+6.72126800814254432\cdot 10^{-5}\cdot x^{4}&+2.55533277086129636\cdot 10^{-7}\cdot x^{6}\\&+6.97071295760958946\cdot 10^{-10}\cdot x^{8}&+1.38536352772778619\cdot 10^{-12}\cdot x^{10}&+1.89106054713059759\cdot 10^{-15}\cdot x^{12}\\&+1.39759616731376855\cdot 10^{-18}\cdot x^{14}\end{smallmatrix}}\right)\end{aligned}}}$$

The integrals may be evaluated indirectly via auxiliary functions ⁠${\displaystyle f(x)}$⁠ and ⁠${\displaystyle g(x)}$⁠ (see see § Auxilliary functions above) defined as: $${\displaystyle {\begin{aligned}\operatorname {Si} (x)&={\frac {\pi }{2}}-f(x)\cos x-g(x)\sin x\,,\\\operatorname {Ci} (x)&={\hphantom {{\frac {\pi }{2}}-\mathop {} }}f(x)\sin x-g(x)\cos x\,;\end{aligned}}}$$ or equivalently: $${\displaystyle {\begin{aligned}f(x)&\equiv \left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos x+\operatorname {Ci} (x)\sin x\,,\\g(x)&\equiv \left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\sin x-\operatorname {Ci} (x)\cos x\,.\end{aligned}}}$$

For ${\displaystyle x\geq 4}$ the Padé rational functions given below approximate ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ with error less than 1×10⁻¹⁶:^[4]

$${\displaystyle {\begin{aligned}f(x)\approx &\mathop {} {\dfrac {1}{x}}\cdot \left({\begin{smallmatrix}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}\\\hline 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{smallmatrix}}\right)\\[1em]g(x)\approx &\mathop {} {\dfrac {1}{x^{2}}}\cdot \left({\begin{smallmatrix}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}\\\hline 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{smallmatrix}}\right)\end{aligned}}}$$

• Exponential integral
• Logarithmic integral
• Tanc function
• Tanhc function
• Sinhc function
• Coshc function