# Trigonometric integral

For simple integrals of trigonometric functions, see List of integrals of trigonometric functions. Trigonometric integral_sentence_0

In mathematics, the trigonometric integrals are a family of integrals involving trigonometric functions. Trigonometric integral_sentence_1

## Sine integral Trigonometric integral_section_0

The different sine integral definitions are Trigonometric integral_sentence_2

Note that the integrand ​⁄ x is the sinc function, and also the zeroth spherical Bessel function. Trigonometric integral_sentence_3

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. Trigonometric integral_sentence_4

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 = ∞. Trigonometric integral_sentence_5

Their difference is given by the Dirichlet integral, Trigonometric integral_sentence_6

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. Trigonometric integral_sentence_7

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. Trigonometric integral_sentence_8

## Cosine integral Trigonometric integral_section_1

The different cosine integral definitions are Trigonometric integral_sentence_9

where γ ≈ 0.57721566 ... is the Euler–Mascheroni constant. Trigonometric integral_sentence_10

Some texts use ci instead of Ci. Trigonometric integral_sentence_11

Cin is an even, entire function. Trigonometric integral_sentence_12

For that reason, some texts treat Cin as the primary function, and derive Ci in terms of Cin. Trigonometric integral_sentence_13

## Hyperbolic sine integral Trigonometric integral_section_2

The hyperbolic sine integral is defined as Trigonometric integral_sentence_14

It is related to the ordinary sine integral by Trigonometric integral_sentence_15

## Hyperbolic cosine integral Trigonometric integral_section_3

The hyperbolic cosine integral is Trigonometric integral_sentence_16

It has the series expansion Trigonometric integral_sentence_17

## Auxiliary functions Trigonometric integral_section_4

Trigonometric integrals can be understood in terms of the so-called "auxiliary functions" Trigonometric integral_sentence_18

Using these functions, the trigonometric integrals may be re-expressed as (cf. Trigonometric integral_sentence_19

Abramowitz & Stegun, ) Trigonometric integral_sentence_20

## Nielsen's spiral Trigonometric integral_section_5

The spiral formed by parametric plot of si , ci is known as Nielsen's spiral. Trigonometric integral_sentence_21

The spiral is closely related to the Fresnel integrals and the Euler spiral. Trigonometric integral_sentence_22

Nielsen's spiral has applications in vision processing, road and track construction and other areas. Trigonometric integral_sentence_23

## Expansion Trigonometric integral_section_6

Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument. Trigonometric integral_sentence_24

### Asymptotic series (for large argument) Trigonometric integral_section_7

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at ℜ(x) ≫ 1. Trigonometric integral_sentence_25

### Convergent series Trigonometric integral_section_8

These series are convergent at any complex x, although for |x| ≫ 1, the series will converge slowly initially, requiring many terms for high precision. Trigonometric integral_sentence_26

## Relation with the exponential integral of imaginary argument Trigonometric integral_section_10

The function Trigonometric integral_sentence_27

is called the exponential integral. Trigonometric integral_sentence_28

It is closely related to Si and Ci, Trigonometric integral_sentence_29

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.) Trigonometric integral_sentence_30

Cases of imaginary argument of the generalized integro-exponential function are Trigonometric integral_sentence_31

which is the real part of Trigonometric integral_sentence_32

Similarly Trigonometric integral_sentence_33

## Efficient evaluation Trigonometric integral_section_11

Padé approximants of the convergent Taylor series provide an efficient way to evaluate the functions for small arguments. Trigonometric integral_sentence_34

The following formulae, given by Rowe et al. Trigonometric integral_sentence_35

(2015), are accurate to better than 10 for 0 ≤ x ≤ 4, Trigonometric integral_sentence_36