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
Special function defined by an integral
Si(x) (blue) and Ci(x) (green) plotted on the same plot.Integral sine in the complex plane, plotted with a variant of domain coloring.Integral cosine in the complex plane. Note the branch cut along the negative real axis.
In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.
Sine integral
Plot of Si(x) for 0 ≤ x ≤ 8π.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
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.
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
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.
Derivation of series expansion
From the Maclaurin series expansion of sine:
Relation with the exponential integral of imaginary argument
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
^Gray (1993). Modern Differential Geometry of Curves and Surfaces. Boca Raton. p. 119.{{cite book}}: CS1 maint: location missing publisher (link)
^ abRowe, B.; et al. (2015). "GALSIM: The modular galaxy image simulation toolkit". Astronomy and Computing. 10: 121. arXiv:1407.7676. Bibcode:2015A&C....10..121R. doi:10.1016/j.ascom.2015.02.002. S2CID 62709903.
Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 231. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
Further reading
Mathar, R.J. (2009). "Numerical evaluation of the oscillatory integral over exp(iπx)·x1/x between 1 and ∞". Appendix B. arXiv:0912.3844 [math.CA].
Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). "Section 6.8.2 – Cosine and Sine Integrals". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
Sloughter, Dan. "Sine Integral Taylor series proof" (PDF). Difference Equations to Differential Equations.
Temme, N.M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.