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.
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
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,
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
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.
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
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,
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