In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.
Sine integral
The different sine integral definitions are
Si ( x ) = − ∫ 0 x sin t t d t si ( x ) = − ∫ x ∞ sin t t d t . {\displaystyle {\begin{aligned}\operatorname {Si} (x)&={\hphantom {-}}\int _{0}^{x}{\frac {\sin t}{t}}\,dt\\\operatorname {si} (x)&=-\int _{x}^{\infty }{\frac {\sin t}{t}}\,dt~.\end{aligned}}}
Note that the integrand sin t t {\textstyle {\frac {\sin t}{t}}} is the sinc function, and also the zeroth spherical Bessel function. Since sinc {\displaystyle \operatorname {sinc} } is an even entire function (holomorphic over the entire complex plane), Si {\displaystyle \operatorname {Si} } is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, Si ( x ) {\displaystyle \operatorname {Si} (x)} is the antiderivative of sin x x {\displaystyle \textstyle {\frac {\sin x}{x}}} whose value is zero at x = 0 {\displaystyle x=0} , and si ( x ) {\displaystyle \operatorname {si} (x)} is the antiderivative whose value is zero at x = ∞ {\displaystyle x=\infty } . Their difference is given by the Dirichlet integral,
Si ( x ) − si ( x ) = ∫ 0 ∞ sin t t d t = π 2 or Si ( x ) = π 2 + si ( x ) . {\displaystyle \operatorname {Si} (x)-\operatorname {si} (x)=\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt={\frac {\pi }{2}}\quad {\text{ or }}\quad \operatorname {Si} (x)={\frac {\pi }{2}}+\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
Cin ( x ) ≡ ∫ 0 x 1 − cos t t d t . {\displaystyle \operatorname {Cin} (x)\equiv \int _{0}^{x}{\frac {1-\cos t}{t}}\,dt.}
Cin {\displaystyle \operatorname {Cin} } is an even, entire function. For that reason, some texts define Cin {\displaystyle \operatorname {Cin} } as the primary function, and derive Ci {\displaystyle \operatorname {Ci} } in terms of Cin {\displaystyle \operatorname {Cin} } .
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