Plot of the hyperbolic sine integral function Shi(z) in the complex plane from −2 − 2i to 2 + 2i
Special function defined by an integral
Si(x) (blue) and Ci(x) (green) shown on the same plot.Sine integral in the complex plane, plotted with a variant of domain coloring.Cosine integral in the complex plane. Note the branch cut along the negative real axis.
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,
Cin is an even, entire function. For that reason, some texts define Cin as the primary function, and derive Ci in terms of Cin.
for where γ ≈ 0.57721566490... is the Euler–Mascheroni constant. Some texts use ci instead of 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 (ln).
Ci(x) is the antiderivative of cos x/x (which vanishes as ). The two definitions are related by
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
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.[1]
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 |x| ≫ 1, the series will converge slowly initially, requiring many terms for high precision.
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),[2] are accurate to better than 10−16 for 0 ≤ x ≤ 4,
The integrals may be evaluated indirectly via auxiliary functions and , which are defined by
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