Trigonometric integral

 ( Trigonometry ) 

In mathematics, trigonometric integrals are a indexed family of nonelementary integrals involving trigonometric functions.

---

Sine integral The different sine integral definitions are $$\backslash operatorname\{Si\}(x)\; =\; \backslash int\_0^x\backslash frac\{\backslash sin\; t\}\{t\}\backslash ,dt$$ $$\backslash operatorname\{si\}(x)\; =\; -\backslash int\_x^\backslash infty\backslash frac\{\backslash sin\; t\}\{t\}\backslash ,dt~.$$

Note that the integrand $\backslash frac\{\backslash sin(t)\}\{t\}$ is the sinc function, and also the zeroth . Since is an even function entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints.

By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral, $$\backslash operatorname\{Si\}(x)\; -\; \backslash operatorname\{si\}(x)\; =\; \backslash int\_0^\backslash infty\backslash frac\{\backslash sin\; t\}\{t\}\backslash ,dt\; =\; \backslash frac\{\backslash pi\}\{2\}\; \backslash quad\; \backslash text\{\; or\; \}\; \backslash quad\; \backslash operatorname\{Si\}(x)\; =\; \backslash frac\{\backslash pi\}\{2\}\; +\; \backslash operatorname\{si\}(x)\; ~.$$

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 The different cosine integral definitions are $$\backslash operatorname\{Cin\}(x)\; ~\backslash equiv~\; \backslash int\_0^x\; \backslash frac\{\backslash \; 1\; -\; \backslash cos\; t\backslash \; \}\{\; t\; \}\backslash \; \backslash operatorname\{d\}\; t\; ~.$$

is an even, [[entire function]]. For that reason, some texts define  as the primary function, and derive  in terms of
     

$$\backslash operatorname\{Ci\}(x)\; ~~\backslash equiv~\; -\backslash int\_x^\backslash infty\; \backslash frac\{\backslash \; \backslash cos\; t\backslash \; \}\{\; t\; \}\backslash \; \backslash operatorname\{d\}\; t\; ~$$ $$~~\; \backslash qquad\; ~=~~\; \backslash gamma\; ~+~\; \backslash ln\; x\; ~-~\; \backslash int\_0^x\; \backslash frac\{\backslash \; 1\; -\; \backslash cos\; t\backslash \; \}\{\; t\; \}\backslash \; \backslash operatorname\{d\}\; t\; ~$$

$$~~\; \backslash qquad\; ~=~~\; \backslash gamma\; ~+~\; \backslash ln\; x\; ~-~\; \backslash operatorname\{Cin\}\; x\; ~$$ for where is the Euler–Mascheroni constant. Some texts use instead of . The restriction on 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 ().

is the antiderivative of  (which vanishes as $\backslash \; x\; \backslash to\; \backslash infty\backslash$). The two definitions are related by
     

$$\backslash operatorname\{Ci\}(x)\; =\; \backslash gamma\; +\; \backslash ln\; x\; -\; \backslash operatorname\{Cin\}(x)\; ~.$$

---

Hyperbolic sine integral The hyperbolic sine integral is defined as $$\backslash operatorname\{Shi\}(x)\; =\backslash int\_0^x\; \backslash frac\; \{\backslash sinh\; (t)\}\{t\}\backslash ,dt.$$

It is related to the ordinary sine integral by $$\backslash operatorname\{Si\}(ix)\; =\; i\backslash operatorname\{Shi\}(x).$$

---

Hyperbolic cosine integral The hyperbolic cosine integral is where $\backslash gamma$ is the Euler–Mascheroni constant.

It has the series expansion $$\backslash operatorname\{Chi\}(x)\; =\; \backslash gamma\; +\; \backslash ln(x)\; +\; \backslash frac\; \{x^2\}\{4\}\; +\; \backslash frac\; \{x^4\}\{96\}\; +\; \backslash frac\; \{x^6\}\{4320\}\; +\; \backslash frac\; \{x^8\}\{322560\}\; +\; \backslash frac\{x^\{10\}\}\{36288000\}\; +\; O(x^\{12\}).$$

---

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

&=& \operatorname{Ci}(x) \sin(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \cos(x)~,  \\
     

g(x) &\equiv& \int_0^\infty \frac{\cos(t)}{t+x} \,dt &=& \int_0^\infty \frac{t e^{-x t}}{t^2 + 1} \,dt

&=& -\operatorname{Ci}(x) \cos(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \sin(x)~.
     

\end{array} 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 is known as Nielsen's spiral. $$x(t)\; =\; a\; \backslash times\; \backslash operatorname\{ci\}(t)$$ $$y(t)\; =\; a\; \backslash times\; \backslash operatorname\{si\}(t)$$

The spiral is closely related to the 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) $$\backslash operatorname\{Si\}(x)\; \backslash sim\; \backslash frac\{\backslash pi\}\{2\}$$

- \frac{\cos x}{x}\left(1-\frac{2!}{x^2}+\frac{4!}{x^4}-\frac{6!}{x^6}\cdots\right)
- \frac{\sin x}{x}\left(\frac{1}{x}-\frac{3!}{x^3}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right)
     

$$\backslash operatorname\{Ci\}(x)\; \backslash sim\; \backslash frac\{\backslash sin\; x\}\{x\}\backslash left(1-\backslash frac\{2!\}\{x^2\}+\backslash frac\{4!\}\{x^4\}-\backslash frac\{6!\}\{x^6\}\backslash cdots\backslash right)$$

- \frac{\cos x}{x}\left(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^5}-\frac{7!}{x^7}\cdots\right) ~.
     

These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at .

---

Convergent series $$\backslash operatorname\{Si\}(x)=\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^\{n\}x^\{2n+1\}\}\{(2n+1)(2n+1)!\}=x-\backslash frac\{x^3\}\{3!\backslash cdot3\}+\backslash frac\{x^5\}\{5!\backslash cdot5\}-\backslash frac\{x^7\}\{7!\; \backslash cdot7\}\backslash pm\backslash cdots$$ $$\backslash operatorname\{Ci\}(x)=\; \backslash gamma+\backslash ln\; x+\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{(-1)^\{n\}x^\{2n\}\}\{2n(2n)!\}=\backslash gamma+\backslash ln\; x-\backslash frac\{x^2\}\{2!\backslash cdot2\}\; +\; \backslash frac\{x^4\}\{4!\; \backslash cdot4\}\backslash mp\backslash cdots$$

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

---

Derivation of series expansion From the Taylor series expansion of sine: $$\backslash sin\backslash ,x\; =\; x\; -\; \backslash frac\{x^3\}\{3!\}+\backslash frac\{x^5\}\{5!\}-\; \backslash frac\{x^7\}\{7!\}+\backslash frac\{x^9\}\{9!\}-\backslash frac\{x^\{11\}\}\{11!\}\; +\; \backslash cdots$$ $$\backslash frac\{\backslash sin\backslash ,x\}\{x\}\; =\; 1\; -\; \backslash frac\{x^2\}\{3!\}+\backslash frac\{x^4\}\{5!\}-\; \backslash frac\{x^6\}\{7!\}+\backslash frac\{x^8\}\{9!\}-\backslash frac\{x^\{10\}\}\{11!\}+\backslash cdots$$ $$\backslash therefore\backslash int\; \backslash frac\{\backslash sin\backslash ,x\}\{x\}dx\; =\; x\; -\; \backslash frac\{x^3\}\{3!\backslash cdot3\}+\backslash frac\{x^5\}\{5!\backslash cdot5\}-\; \backslash frac\{x^7\}\{7!\backslash cdot7\}+\backslash frac\{x^9\}\{9!\backslash cdot9\}-\backslash frac\{x^\{11\}\}\{11!\backslash cdot11\}+\backslash cdots$$

---

Relation with the exponential integral of imaginary argument The function $$\backslash operatorname\{E\}\_1(z)\; =\; \backslash int\_1^\backslash infty\; \backslash frac\{\backslash exp(-zt)\}\{t\}\backslash ,dt\; \backslash qquad~\backslash text\{\; for\; \}~\; \backslash Re(z)\; \backslash ge\; 0$$ is called the exponential integral. It is closely related to and , $$\backslash operatorname\{E\}\_1(i\; x)\; =\; i\backslash left(-\backslash frac\{\backslash pi\}\{2\}\; +\; \backslash operatorname\{Si\}(x)\backslash right)-\backslash operatorname\{Ci\}(x)\; =\; i\; \backslash operatorname\{si\}(x)\; -\; \backslash operatorname\{Ci\}(x)\; \backslash qquad\; ~\backslash text\{\; for\; \}~\; x\; >\; 0\; ~.$$

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 appear in the expression.)

Cases of imaginary argument of the generalized integro-exponential function are $$\backslash int\_1^\backslash infty\; \backslash cos(ax)\backslash frac\{\backslash ln\; x\}\{x\}\; \backslash ,\; dx\; =\; -\backslash frac\{\backslash pi^2\}\{24\}+\backslash gamma\backslash left(\backslash frac\{\backslash gamma\}\{2\}+\backslash ln\; a\backslash right)+\backslash frac\{\backslash ln^2a\}\{2\}\; +\backslash sum\_\{n\backslash ge\; 1\}\; \backslash frac\{(-a^2)^n\}\{(2n)!(2n)^2\}\; ~,$$ which is the real part of $$\backslash int\_1^\backslash infty\; e^\{iax\}\backslash frac\{\backslash ln\; x\}\{x\}\backslash ,dx\; =\; -\backslash frac\{\backslash pi^2\}\{24\}\; +\; \backslash gamma\backslash left(\backslash frac\{\backslash gamma\}\{2\}+\backslash ln\; a\backslash right)+\backslash frac\{\backslash ln^2\; a\}\{2\}\; -\backslash frac\{\backslash pi\}\{2\}i\backslash left(\backslash gamma+\backslash ln\; a\backslash right)\; +\; \backslash sum\_\{n\backslash ge\; 1\}\backslash frac\{(ia)^n\}\{n!n^2\}\; ~.$$

Similarly $$\backslash int\_1^\backslash infty\; e^\{iax\}\backslash frac\{\backslash ln\; x\}\{x^2\}\backslash ,dx\; =$$

1 + ia\left[ -\frac{\pi^2}{24} + \gamma \left( \frac{\gamma}{2} + \ln a - 1 \right) + \frac{\ln^2 a}{2} - \ln a + 1 \right]
+ \frac{\pi a}{2} \Bigl( \gamma+\ln a - 1 \Bigr)
+ \sum_{n\ge 1}\frac{(ia)^{n+1}}{(n+1)!n^2}~.
     

---

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 for ,

The integrals may be evaluated indirectly via auxiliary functions $f(x)$ and $g(x)$, which are defined by

$$\backslash operatorname\{Si\}(x)=\backslash frac\{\backslash pi\}\{2\}-f(x)\backslash cos(x)-g(x)\backslash sin(x)$$		$$\backslash operatorname\{Ci\}(x)=f(x)\backslash sin(x)-g(x)\backslash cos(x)$$
or equivalently
$$f(x)\; \backslash equiv\; \backslash left\backslash frac\{\backslash pi\}\{2\}\; \backslash cos(x)\; +\; \backslash operatorname\{Ci\}(x)\; \backslash sin(x)$$		$$g(x)\; \backslash equiv\; \backslash left\backslash frac\{\backslash pi\}\{2\}\; \backslash sin(x)\; -\; \backslash operatorname\{Ci\}(x)\; \backslash cos(x)$$

For $x\; \backslash ge\; 4$ the Padé rational functions given below approximate $f(x)$ and $g(x)$ with error less than 10⁻¹⁶:

---

See also

• Logarithmic integral
• Tanc function
• Tanhc function
• Sinhc function
• Coshc function

---

Further reading

• (2025). 9780521880688, Cambridge University Press. ISBN 9780521880688

---

External links

• http://mathworld.wolfram.com/SineIntegral.html

Post Comment