[[File:Plot of the hyperbolic sine integral function Shi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the hyperbolic sine integral function Shi(z) in the complex plane from −2 − 2i to 2 + 2i|thumb|Plot of the hyperbolic sine integral function Shi(z) in the complex plane from −2 − 2i to 2 + 2i ]] [[Image:sine cosine integral.svg|thumb| Si(x) (blue) and Ci(x) (green) shown on the same plot.]] In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.

Sine integral

[[Image:Sine integral.svg|thumb|Plot of Si(x) for 0 ≤ x ≤ 8π .]] [[File:Plot of the cosine integral function Ci(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the cosine integral function Ci(z) in the complex plane from −2 − 2i to 2 + 2i|thumb|Plot of the cosine integral function Ci(z) in the complex plane from −2 − 2i to 2 + 2i ]] The different sine integral definitions are Note that the integrand 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 of sin x / x whose value is zero at x = 0 , and si(x) is the antiderivative whose value is zero at x = ∞ . Their difference is given by the Dirichlet integral, 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

[[Image:Cosine integral.svg|thumb|Plot of Ci(x) for 0 < x ≤ 8π ]] The different cosine integral definitions are where γ ≈ 0.57721566 ... is the Euler–Mascheroni constant. Some texts use ci instead of Ci . Ci(x) is the antiderivative of cos x / x (which vanishes as ). The two definitions are related by Cin is an even, entire function. For that reason, some texts treat Cin as the primary function, and derive Ci in terms of Cin .

Hyperbolic sine integral

The hyperbolic sine integral is defined as It is related to the ordinary sine integral by

Hyperbolic cosine integral

The hyperbolic cosine integral is [[File:Plot of the hyperbolic cosine integral function Chi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the hyperbolic cosine integral function Chi(z) in the complex plane from −2 − 2i to 2 + 2i|thumb|Plot of the hyperbolic cosine integral function Chi(z) in the complex plane from −2 − 2i to 2 + 2i ]] where \gamma is the Euler–Mascheroni constant. It has the series expansion

Auxiliary functions

Trigonometric integrals can be understood in terms of the so-called "auxiliary functions" Using these functions, the trigonometric integrals may be re-expressed as (cf. Abramowitz & Stegun, p. 232)

Nielsen's spiral

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

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

Convergent series

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

Derivation of series expansion

From the Maclaurin series expansion of sine:

Relation with the exponential integral of imaginary argument

The function is called the exponential integral. It is closely related to Si and Ci , 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 which is the real part of Similarly

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 , The integrals may be evaluated indirectly via auxiliary functions f(x) and g(x), which are defined by For x \ge 4 the Padé rational functions given below approximate f(x) and g(x) with error less than 10−16: