Trigonometric Integral

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

Sine integral

The different sine integral definitions are
$\operatorname(x) = \int_0^x\frac\,dt$
$\operatorname(x) = -\int_x^\infty\frac\,dt~.$

Note that the integrand is the sinc function, and also the zeroth spherical Bessel function.
Since is an even 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,
$\operatorname(x) - \operatorname(x) = \int_0^\infty\frac\,dt = \frac \quad \text \quad \operatorname(x) = \frac + \operatorname(x) ~.$

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
$\operatorname(x) = \int_0^x \frac\,dt~,$
$\operatorname(x) = -\int_x^\infty \frac\,dt = \gamma + \ln x - \int_0^x \frac\,dt \qquad ~\text ~\left, \operatorname(x)\ < \pi~,$
where is the Euler–Mascheroni constant. Some texts use instead of .

is the antiderivative of (which vanishes as $x \to \infty$). The two definitions are related by
$\operatorname(x) = \gamma + \ln x - \operatorname(x)~.$

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

Hyperbolic sine integral

The hyperbolic sine integral is defined as
$\operatorname(x) =\int_0^x \frac \,dt.$

It is related to the ordinary sine integral by
$\operatorname(ix) = i\operatorname(x).$

Hyperbolic cosine integral

The hyperbolic cosine integral is

$\operatorname(x) = \gamma+\ln x + \int_0^x\frac\,dt \qquad ~ \text ~ \left, \operatorname(x) \ < \pi~,$
where $\gamma$ is the Euler–Mascheroni constant.

It has the series expansion
$\operatorname(x) = \gamma + \ln(x) + \frac + \frac + \frac + \frac + \frac + O(x^).$

Auxiliary functions

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

\begin
f(x) &\equiv&\displaystyle \int_0^\infty \frac \,dt &=&\displaystyle \int_0^\infty \frac \,dt
&=& \quad \operatorname(x) \sin(x) + \left frac - \operatorname(x) \right\cos(x)~, \qquad \text \\
g(x) &\equiv&\displaystyle \int_0^\infty \frac \,dt &=&\displaystyle \int_0^\infty \frac \,dt
&=& -\operatorname(x) \cos(x) + \left frac - \operatorname(x) \right\sin(x)~.
\end

Using these functions, the trigonometric integrals may be re-expressed as
(cf. Abramowitz & Stegun
p. 232

$\begin \frac - \operatorname(x) = -\operatorname(x) &=& f(x) \cos(x) + g(x) \sin(x)~, \qquad \text \\ \operatorname(x) &=& f(x) \sin(x) - g(x) \cos(x)~. \\ \end$

Nielsen's spiral

The spiral

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