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Trigonometric integral
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<h2 id="sine-integral">Sine integral</h2> <p>The different <a href="/facts/Sine/gQ6LXK8z">sine</a> integral definitions are Si ( x ) = ∫ 0 x sin t t d t {\displaystyle \operatorname {Si} (x)=\int _{0}^{x}{\frac {\sin t}{t}}\,dt} si ( x ) = − ∫ x ∞ sin t t d t . {\displaystyle \operatorname {si} (x)=-\int _{x}^{\infty }{\frac {\sin t}{t}}\,dt~.} </p><p>Note that the integrand sin ( t ) t {\displaystyle {\frac {\sin(t)}{t}}} is the <a href="/facts/Sinc_function/lSvDtHy0">sinc function</a>, and also the zeroth <a href="/facts/Bessel_function/hSyvFPqC">spherical Bessel function</a>. Since sinc is an <a href="/facts/Even_function/tGHWJqAa">even</a> <a href="/facts/Entire_function/XNluKzu7">entire function</a> (<a href="/facts/Holomorphic/kVzthtEf">holomorphic</a> over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along <a href="/facts/Cauchy%2527s_integral_theorem/NIgLv1VB">any path</a> connecting the endpoints. </p><p>By definition, Si(<i>x</i>) is the <a href="/facts/Antiderivative/5qDsudIv">antiderivative</a> of sin <i>x</i> / <i>x</i> whose value is zero at <i>x</i> = 0, and si(<i>x</i>) is the antiderivative whose value is zero at <i>x</i> = ∞. Their difference is given by the <a href="/facts/Dirichlet_integral/fFo9lHcb">Dirichlet integral</a>, 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)~.} </p><p>In <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, the oscillations of the sine integral cause <a href="/facts/Overshoot_(signal)/l450BHuI">overshoot</a> and <a href="/facts/Ringing_artifacts/JC2W86cf">ringing artifacts</a> when using the <a href="/facts/Sinc_filter/1H6q0F5t">sinc filter</a>, and <a href="/facts/Frequency_domain/pyTVrUSq">frequency domain</a> ringing if using a truncated sinc filter as a <a href="/facts/Low-pass_filter/QFsUjw7h">low-pass filter</a>. </p><p>Related is the <a href="/facts/Gibbs_phenomenon/JCXxWvcD">Gibbs phenomenon</a>: If the sine integral is considered as the <a href="/facts/Convolution/PVPrdz9J">convolution</a> of the sinc function with the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>, this corresponds to truncating the <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a>, which is the cause of the Gibbs phenomenon. </p> <h2 id="cosine-integral">Cosine integral</h2> <p>The different <a href="/facts/Cosine/czej7ur0">cosine</a> 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}}\ \operatorname {d} t~.} </p><p>Cin is an <a href="/facts/Even_and_odd_functions/tGHWJqAa">even</a>, <a href="/facts/Entire_function/XNluKzu7">entire function</a>. For that reason, some texts define Cin as the primary function, and derive Ci in terms of Cin . </p><p> Ci ( x ) ≡ − ∫ x ∞ cos t t d t {\displaystyle \operatorname {Ci} (x)~~\equiv ~-\int _{x}^{\infty }{\frac {\ \cos t\ }{t}}\ \operatorname {d} t~} = γ + ln x − ∫ 0 x 1 − cos t t d t {\displaystyle ~~\qquad ~=~~\gamma ~+~\ln x~-~\int _{0}^{x}{\frac {\ 1-\cos t\ }{t}}\ \operatorname {d} t~} </p><p> = γ + ln x − Cin x {\displaystyle ~~\qquad ~=~~\gamma ~+~\ln x~-~\operatorname {Cin} x~} for | Arg ( x ) | < π , {\displaystyle ~{\Bigl |}\ \operatorname {Arg} (x)\ {\Bigr |}<\pi \ ,} where <i>γ</i> ≈ 0.57721566490 ... is the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>. Some texts use ci instead of Ci . The restriction on Arg(x) 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 <a href="/facts/Branch_cut/lJ7Y6Iht">branch cut</a> in the standard <a href="/facts/Natural_logarithm/PnreKEzI">logarithm function</a> ( ln ). </p><p>Ci(<i>x</i>) is the antiderivative of cos <i>x</i> / <i>x</i> (which vanishes as x → ∞ {\displaystyle \ x\to \infty \ } ). The two definitions are related by Ci ( x ) = γ + ln x − Cin ( x ) . {\displaystyle \operatorname {Ci} (x)=\gamma +\ln x-\operatorname {Cin} (x)~.} </p> <h2 id="hyperbolic-sine-integral">Hyperbolic sine integral</h2> <p>The <a href="/facts/Hyperbolic_sine/Y4Cb5S35">hyperbolic sine</a> integral is defined as Shi ( x ) = ∫ 0 x sinh ( t ) t d t . {\displaystyle \operatorname {Shi} (x)=\int _{0}^{x}{\frac {\sinh(t)}{t}}\,dt.} </p><p>It is related to the ordinary sine integral by Si ( i x ) = i Shi ( x ) . {\displaystyle \operatorname {Si} (ix)=i\operatorname {Shi} (x).} </p> <h2 id="hyperbolic-cosine-integral">Hyperbolic cosine integral</h2> <p>The <a href="/facts/Hyperbolic_cosine/Y4Cb5S35">hyperbolic cosine</a> integral is </p> <p> Chi ( x ) = γ + ln x + ∫ 0 x cosh t − 1 t d t for | Arg ( x ) | < π , {\displaystyle \operatorname {Chi} (x)=\gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,} where γ {\displaystyle \gamma } is the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>. </p><p>It has the series expansion Chi ( x ) = γ + ln ( x ) + x 2 4 + x 4 96 + x 6 4320 + x 8 322560 + x 10 36288000 + O ( x 12 ) . {\displaystyle \operatorname {Chi} (x)=\gamma +\ln(x)+{\frac {x^{2}}{4}}+{\frac {x^{4}}{96}}+{\frac {x^{6}}{4320}}+{\frac {x^{8}}{322560}}+{\frac {x^{10}}{36288000}}+O(x^{12}).} </p> <h2 id="auxiliary-functions">Auxiliary functions</h2> <p>Trigonometric integrals can be understood in terms of the so-called "<a href="/facts/Auxiliary_function/QXX80Rnk">auxiliary functions</a>" f ( x ) ≡ ∫ 0 ∞ sin ( t ) t + x d t = ∫ 0 ∞ e − x t t 2 + 1 d t = Ci ( x ) sin ( x ) + [ π 2 − Si ( x ) ] cos ( x ) , g ( x ) ≡ ∫ 0 ∞ cos ( t ) t + x d t = ∫ 0 ∞ t e − x t t 2 + 1 d t = − Ci ( x ) cos ( x ) + [ π 2 − Si ( x ) ] sin ( x ) . {\displaystyle {\begin{array}{rcl}f(x)&\equiv &\int _{0}^{\infty }{\frac {\sin(t)}{t+x}}\,dt&=&\int _{0}^{\infty }{\frac {e^{-xt}}{t^{2}+1}}\,dt&=&\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 {te^{-xt}}{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, <a href="http://people.math.sfu.ca/~cbm/aands/page_232.htm">p. 232</a>) π 2 − Si ( x ) = − si ( x ) = f ( x ) cos ( x ) + g ( x ) sin ( x ) , and Ci ( x ) = f ( x ) sin ( x ) − g ( x ) cos ( x ) . {\displaystyle {\begin{array}{rcl}{\frac {\pi }{2}}-\operatorname {Si} (x)=-\operatorname {si} (x)&=&f(x)\cos(x)+g(x)\sin(x)~,\qquad {\text{ and }}\\\operatorname {Ci} (x)&=&f(x)\sin(x)-g(x)\cos(x)~.\\\end{array}}} </p> <h2 id="nielsens-spiral">Nielsen's spiral</h2> <p>The <a href="/facts/Spiral/4R331aba">spiral</a> formed by parametric plot of si, ci is known as Nielsen's spiral. x ( t ) = a × ci ( t ) {\displaystyle x(t)=a\times \operatorname {ci} (t)} y ( t ) = a × si ( t ) {\displaystyle y(t)=a\times \operatorname {si} (t)} </p><p>The spiral is closely related to the <a href="/facts/Fresnel_integral/5W7z760j">Fresnel integrals</a> and the <a href="/facts/Euler_spiral/umcvnb2J">Euler spiral</a>. Nielsen's spiral has applications in vision processing, road and track construction and other areas.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="expansion">Expansion</h2> <p>Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument. </p> <h3>Asymptotic series (for large argument)</h3> <p> Si ( x ) ∼ π 2 − cos x x ( 1 − 2 ! x 2 + 4 ! x 4 − 6 ! x 6 ⋯ ) − sin x x ( 1 x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 ⋯ ) {\displaystyle \operatorname {Si} (x)\sim {\frac {\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)} Ci ( x ) ∼ sin x x ( 1 − 2 ! x 2 + 4 ! x 4 − 6 ! x 6 ⋯ ) − cos x x ( 1 x − 3 ! x 3 + 5 ! x 5 − 7 ! x 7 ⋯ ) . {\displaystyle \operatorname {Ci} (x)\sim {\frac {\sin x}{x}}\left(1-{\frac {2!}{x^{2}}}+{\frac {4!}{x^{4}}}-{\frac {6!}{x^{6}}}\cdots \right)-{\frac {\cos x}{x}}\left({\frac {1}{x}}-{\frac {3!}{x^{3}}}+{\frac {5!}{x^{5}}}-{\frac {7!}{x^{7}}}\cdots \right)~.} </p><p>These series are <a href="/facts/Asymptotic_series/4W79BNve">asymptotic</a> and divergent, although can be used for estimates and even precise evaluation at ℜ(<i>x</i>) ≫ 1. </p> <h3>Convergent series</h3> <p> Si ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ( 2 n + 1 ) ! = x − x 3 3 ! ⋅ 3 + x 5 5 ! ⋅ 5 − x 7 7 ! ⋅ 7 ± ⋯ {\displaystyle \operatorname {Si} (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)(2n+1)!}}=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}\pm \cdots } Ci ( x ) = γ + ln x + ∑ n = 1 ∞ ( − 1 ) n x 2 n 2 n ( 2 n ) ! = γ + ln x − x 2 2 ! ⋅ 2 + x 4 4 ! ⋅ 4 ∓ ⋯ {\displaystyle \operatorname {Ci} (x)=\gamma +\ln x+\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{2n}}{2n(2n)!}}=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}\mp \cdots } </p><p>These series are convergent at any complex x, although for |<i>x</i>| ≫ 1, the series will converge slowly initially, requiring many terms for high precision. </p> <h3>Derivation of series expansion</h3> <p>From the Maclaurin series expansion of sine: sin x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + x 9 9 ! − x 11 11 ! + ⋯ {\displaystyle \sin \,x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+{\frac {x^{9}}{9!}}-{\frac {x^{11}}{11!}}+\cdots } sin x x = 1 − x 2 3 ! + x 4 5 ! − x 6 7 ! + x 8 9 ! − x 10 11 ! + ⋯ {\displaystyle {\frac {\sin \,x}{x}}=1-{\frac {x^{2}}{3!}}+{\frac {x^{4}}{5!}}-{\frac {x^{6}}{7!}}+{\frac {x^{8}}{9!}}-{\frac {x^{10}}{11!}}+\cdots } ∴ ∫ sin x x d x = x − x 3 3 ! ⋅ 3 + x 5 5 ! ⋅ 5 − x 7 7 ! ⋅ 7 + x 9 9 ! ⋅ 9 − x 11 11 ! ⋅ 11 + ⋯ {\displaystyle \therefore \int {\frac {\sin \,x}{x}}dx=x-{\frac {x^{3}}{3!\cdot 3}}+{\frac {x^{5}}{5!\cdot 5}}-{\frac {x^{7}}{7!\cdot 7}}+{\frac {x^{9}}{9!\cdot 9}}-{\frac {x^{11}}{11!\cdot 11}}+\cdots } </p> <h2 id="relation-with-the-exponential-integral-of-imaginary-argument">Relation with the exponential integral of imaginary argument</h2> <p>The function E 1 ( z ) = ∫ 1 ∞ exp ( − z t ) t d t for ℜ ( z ) ≥ 0 {\displaystyle \operatorname {E} _{1}(z)=\int _{1}^{\infty }{\frac {\exp(-zt)}{t}}\,dt\qquad ~{\text{ for }}~\Re (z)\geq 0} is called the <a href="/facts/Exponential_integral/JF6aMcmL">exponential integral</a>. It is closely related to Si and Ci, E 1 ( i x ) = i ( − π 2 + Si ( x ) ) − Ci ( x ) = i si ( x ) − ci ( x ) for x > 0 . {\displaystyle \operatorname {E} _{1}(ix)=i\left(-{\frac {\pi }{2}}+\operatorname {Si} (x)\right)-\operatorname {Ci} (x)=i\operatorname {si} (x)-\operatorname {ci} (x)\qquad ~{\text{ for }}~x>0~.} </p><p>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 <i>π</i> appear in the expression.) </p><p>Cases of imaginary argument of the generalized integro-exponential function are ∫ 1 ∞ cos ( a x ) ln x x d x = − π 2 24 + γ ( γ 2 + ln a ) + ln 2 a 2 + ∑ n ≥ 1 ( − a 2 ) n ( 2 n ) ! ( 2 n ) 2 , {\displaystyle \int _{1}^{\infty }\cos(ax){\frac {\ln x}{x}}\,dx=-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a\right)+{\frac {\ln ^{2}a}{2}}+\sum _{n\geq 1}{\frac {(-a^{2})^{n}}{(2n)!(2n)^{2}}}~,} which is the real part of ∫ 1 ∞ e i a x ln x x d x = − π 2 24 + γ ( γ 2 + ln a ) + ln 2 a 2 − π 2 i ( γ + ln a ) + ∑ n ≥ 1 ( i a ) n n ! n 2 . {\displaystyle \int _{1}^{\infty }e^{iax}{\frac {\ln x}{x}}\,dx=-{\frac {\pi ^{2}}{24}}+\gamma \left({\frac {\gamma }{2}}+\ln a\right)+{\frac {\ln ^{2}a}{2}}-{\frac {\pi }{2}}i\left(\gamma +\ln a\right)+\sum _{n\geq 1}{\frac {(ia)^{n}}{n!n^{2}}}~.} </p><p>Similarly ∫ 1 ∞ e i a x ln x x 2 d x = 1 + i a [ − π 2 24 + γ ( γ 2 + ln a − 1 ) + ln 2 a 2 − ln a + 1 ] + π a 2 ( γ + ln a − 1 ) + ∑ n ≥ 1 ( i a ) n + 1 ( n + 1 ) ! n 2 . {\displaystyle \int _{1}^{\infty }e^{iax}{\frac {\ln x}{x^{2}}}\,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\geq 1}{\frac {(ia)^{n+1}}{(n+1)!n^{2}}}~.} </p> <h2 id="efficient-evaluation">Efficient evaluation</h2> <p><a href="/facts/Pad%25C3%25A9_approximant/cR5BjAb8">Padé approximants</a> 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),<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> are accurate to better than 10−16 for 0 ≤ <i>x</i> ≤ 4, Si ( x ) ≈ x ⋅ ( 1 − 4.54393409816329991 ⋅ 10 − 2 ⋅ x 2 + 1.15457225751016682 ⋅ 10 − 3 ⋅ x 4 − 1.41018536821330254 ⋅ 10 − 5 ⋅ x 6 + 9.43280809438713025 ⋅ 10 − 8 ⋅ x 8 − 3.53201978997168357 ⋅ 10 − 10 ⋅ x 10 + 7.08240282274875911 ⋅ 10 − 13 ⋅ x 12 − 6.05338212010422477 ⋅ 10 − 16 ⋅ x 14 1 + 1.01162145739225565 ⋅ 10 − 2 ⋅ x 2 + 4.99175116169755106 ⋅ 10 − 5 ⋅ x 4 + 1.55654986308745614 ⋅ 10 − 7 ⋅ x 6 + 3.28067571055789734 ⋅ 10 − 10 ⋅ x 8 + 4.5049097575386581 ⋅ 10 − 13 ⋅ x 10 + 3.21107051193712168 ⋅ 10 − 16 ⋅ x 12 ) Ci ( x ) ≈ γ + ln ( x ) + x 2 ⋅ ( − 0.25 + 7.51851524438898291 ⋅ 10 − 3 ⋅ x 2 − 1.27528342240267686 ⋅ 10 − 4 ⋅ x 4 + 1.05297363846239184 ⋅ 10 − 6 ⋅ x 6 − 4.68889508144848019 ⋅ 10 − 9 ⋅ x 8 + 1.06480802891189243 ⋅ 10 − 11 ⋅ x 10 − 9.93728488857585407 ⋅ 10 − 15 ⋅ x 12 1 + 1.1592605689110735 ⋅ 10 − 2 ⋅ x 2 + 6.72126800814254432 ⋅ 10 − 5 ⋅ x 4 + 2.55533277086129636 ⋅ 10 − 7 ⋅ x 6 + 6.97071295760958946 ⋅ 10 − 10 ⋅ x 8 + 1.38536352772778619 ⋅ 10 − 12 ⋅ x 10 + 1.89106054713059759 ⋅ 10 − 15 ⋅ x 12 + 1.39759616731376855 ⋅ 10 − 18 ⋅ x 14 ) {\displaystyle {\begin{array}{rcl}\operatorname {Si} (x)&\approx &x\cdot \left({\frac {\begin{array}{l}1-4.54393409816329991\cdot 10^{-2}\cdot x^{2}+1.15457225751016682\cdot 10^{-3}\cdot x^{4}-1.41018536821330254\cdot 10^{-5}\cdot x^{6}\\~~~+9.43280809438713025\cdot 10^{-8}\cdot x^{8}-3.53201978997168357\cdot 10^{-10}\cdot x^{10}+7.08240282274875911\cdot 10^{-13}\cdot x^{12}\\~~~-6.05338212010422477\cdot 10^{-16}\cdot x^{14}\end{array}}{\begin{array}{l}1+1.01162145739225565\cdot 10^{-2}\cdot x^{2}+4.99175116169755106\cdot 10^{-5}\cdot x^{4}+1.55654986308745614\cdot 10^{-7}\cdot x^{6}\\~~~+3.28067571055789734\cdot 10^{-10}\cdot x^{8}+4.5049097575386581\cdot 10^{-13}\cdot x^{10}+3.21107051193712168\cdot 10^{-16}\cdot x^{12}\end{array}}}\right)\\&~&\\\operatorname {Ci} (x)&\approx &\gamma +\ln(x)+\\&&x^{2}\cdot \left({\frac {\begin{array}{l}-0.25+7.51851524438898291\cdot 10^{-3}\cdot x^{2}-1.27528342240267686\cdot 10^{-4}\cdot x^{4}+1.05297363846239184\cdot 10^{-6}\cdot x^{6}\\~~~-4.68889508144848019\cdot 10^{-9}\cdot x^{8}+1.06480802891189243\cdot 10^{-11}\cdot x^{10}-9.93728488857585407\cdot 10^{-15}\cdot x^{12}\\\end{array}}{\begin{array}{l}1+1.1592605689110735\cdot 10^{-2}\cdot x^{2}+6.72126800814254432\cdot 10^{-5}\cdot x^{4}+2.55533277086129636\cdot 10^{-7}\cdot x^{6}\\~~~+6.97071295760958946\cdot 10^{-10}\cdot x^{8}+1.38536352772778619\cdot 10^{-12}\cdot x^{10}+1.89106054713059759\cdot 10^{-15}\cdot x^{12}\\~~~+1.39759616731376855\cdot 10^{-18}\cdot x^{14}\\\end{array}}}\right)\end{array}}} </p><p>The integrals may be evaluated indirectly via <a href="/facts/Auxiliary_function/QXX80Rnk">auxiliary functions</a> f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} , which are defined by </p> <table><tbody><tr><td> Si ( x ) = π 2 − f ( x ) cos ( x ) − g ( x ) sin ( x ) {\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-f(x)\cos(x)-g(x)\sin(x)} </td><td></td><td> Ci ( x ) = f ( x ) sin ( x ) − g ( x ) cos ( x ) {\displaystyle \operatorname {Ci} (x)=f(x)\sin(x)-g(x)\cos(x)} </td></tr><tr><td colspan="3"><i>or equivalently</i></td></tr><tr><td> f ( x ) ≡ [ π 2 − Si ( x ) ] cos ( x ) + Ci ( x ) sin ( x ) {\displaystyle f(x)\equiv \left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos(x)+\operatorname {Ci} (x)\sin(x)} </td><td></td><td> g ( x ) ≡ [ π 2 − Si ( x ) ] sin ( x ) − Ci ( x ) cos ( x ) {\displaystyle g(x)\equiv \left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\sin(x)-\operatorname {Ci} (x)\cos(x)} </td></tr></tbody></table> <p>For x ≥ 4 {\displaystyle x\geq 4} the <a href="/facts/Pad%25C3%25A9_approximant/cR5BjAb8">Padé rational functions</a> given below approximate f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} with error less than 10−16:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p> f ( x ) ≈ 1 x ⋅ ( 1 + 7.44437068161936700618 ⋅ 10 2 ⋅ x − 2 + 1.96396372895146869801 ⋅ 10 5 ⋅ x − 4 + 2.37750310125431834034 ⋅ 10 7 ⋅ x − 6 + 1.43073403821274636888 ⋅ 10 9 ⋅ x − 8 + 4.33736238870432522765 ⋅ 10 10 ⋅ x − 10 + 6.40533830574022022911 ⋅ 10 11 ⋅ x − 12 + 4.20968180571076940208 ⋅ 10 12 ⋅ x − 14 + 1.00795182980368574617 ⋅ 10 13 ⋅ x − 16 + 4.94816688199951963482 ⋅ 10 12 ⋅ x − 18 − 4.94701168645415959931 ⋅ 10 11 ⋅ x − 20 1 + 7.46437068161927678031 ⋅ 10 2 ⋅ x − 2 + 1.97865247031583951450 ⋅ 10 5 ⋅ x − 4 + 2.41535670165126845144 ⋅ 10 7 ⋅ x − 6 + 1.47478952192985464958 ⋅ 10 9 ⋅ x − 8 + 4.58595115847765779830 ⋅ 10 10 ⋅ x − 10 + 7.08501308149515401563 ⋅ 10 11 ⋅ x − 12 + 5.06084464593475076774 ⋅ 10 12 ⋅ x − 14 + 1.43468549171581016479 ⋅ 10 13 ⋅ x − 16 + 1.11535493509914254097 ⋅ 10 13 ⋅ x − 18 ) g ( x ) ≈ 1 x 2 ⋅ ( 1 + 8.1359520115168615 ⋅ 10 2 ⋅ x − 2 + 2.35239181626478200 ⋅ 10 5 ⋅ x − 4 + 3.12557570795778731 ⋅ 10 7 ⋅ x − 6 + 2.06297595146763354 ⋅ 10 9 ⋅ x − 8 + 6.83052205423625007 ⋅ 10 10 ⋅ x − 10 + 1.09049528450362786 ⋅ 10 12 ⋅ x − 12 + 7.57664583257834349 ⋅ 10 12 ⋅ x − 14 + 1.81004487464664575 ⋅ 10 13 ⋅ x − 16 + 6.43291613143049485 ⋅ 10 12 ⋅ x − 18 − 1.36517137670871689 ⋅ 10 12 ⋅ x − 20 1 + 8.19595201151451564 ⋅ 10 2 ⋅ x − 2 + 2.40036752835578777 ⋅ 10 5 ⋅ x − 4 + 3.26026661647090822 ⋅ 10 7 ⋅ x − 6 + 2.23355543278099360 ⋅ 10 9 ⋅ x − 8 + 7.87465017341829930 ⋅ 10 10 ⋅ x − 10 + 1.39866710696414565 ⋅ 10 12 ⋅ x − 12 + 1.17164723371736605 ⋅ 10 13 ⋅ x − 14 + 4.01839087307656620 ⋅ 10 13 ⋅ x − 16 + 3.99653257887490811 ⋅ 10 13 ⋅ x − 18 ) {\displaystyle {\begin{array}{rcl}f(x)&\approx &{\dfrac {1}{x}}\cdot \left({\frac {\begin{array}{l}1+7.44437068161936700618\cdot 10^{2}\cdot x^{-2}+1.96396372895146869801\cdot 10^{5}\cdot x^{-4}+2.37750310125431834034\cdot 10^{7}\cdot x^{-6}\\~~~+1.43073403821274636888\cdot 10^{9}\cdot x^{-8}+4.33736238870432522765\cdot 10^{10}\cdot x^{-10}+6.40533830574022022911\cdot 10^{11}\cdot x^{-12}\\~~~+4.20968180571076940208\cdot 10^{12}\cdot x^{-14}+1.00795182980368574617\cdot 10^{13}\cdot x^{-16}+4.94816688199951963482\cdot 10^{12}\cdot x^{-18}\\~~~-4.94701168645415959931\cdot 10^{11}\cdot x^{-20}\end{array}}{\begin{array}{l}1+7.46437068161927678031\cdot 10^{2}\cdot x^{-2}+1.97865247031583951450\cdot 10^{5}\cdot x^{-4}+2.41535670165126845144\cdot 10^{7}\cdot x^{-6}\\~~~+1.47478952192985464958\cdot 10^{9}\cdot x^{-8}+4.58595115847765779830\cdot 10^{10}\cdot x^{-10}+7.08501308149515401563\cdot 10^{11}\cdot x^{-12}\\~~~+5.06084464593475076774\cdot 10^{12}\cdot x^{-14}+1.43468549171581016479\cdot 10^{13}\cdot x^{-16}+1.11535493509914254097\cdot 10^{13}\cdot x^{-18}\end{array}}}\right)\\&&\\g(x)&\approx &{\dfrac {1}{x^{2}}}\cdot \left({\frac {\begin{array}{l}1+8.1359520115168615\cdot 10^{2}\cdot x^{-2}+2.35239181626478200\cdot 10^{5}\cdot x^{-4}+3.12557570795778731\cdot 10^{7}\cdot x^{-6}\\~~~+2.06297595146763354\cdot 10^{9}\cdot x^{-8}+6.83052205423625007\cdot 10^{10}\cdot x^{-10}+1.09049528450362786\cdot 10^{12}\cdot x^{-12}\\~~~+7.57664583257834349\cdot 10^{12}\cdot x^{-14}+1.81004487464664575\cdot 10^{13}\cdot x^{-16}+6.43291613143049485\cdot 10^{12}\cdot x^{-18}\\~~~-1.36517137670871689\cdot 10^{12}\cdot x^{-20}\end{array}}{\begin{array}{l}1+8.19595201151451564\cdot 10^{2}\cdot x^{-2}+2.40036752835578777\cdot 10^{5}\cdot x^{-4}+3.26026661647090822\cdot 10^{7}\cdot x^{-6}\\~~~+2.23355543278099360\cdot 10^{9}\cdot x^{-8}+7.87465017341829930\cdot 10^{10}\cdot x^{-10}+1.39866710696414565\cdot 10^{12}\cdot x^{-12}\\~~~+1.17164723371736605\cdot 10^{13}\cdot x^{-14}+4.01839087307656620\cdot 10^{13}\cdot x^{-16}+3.99653257887490811\cdot 10^{13}\cdot x^{-18}\end{array}}}\right)\\\end{array}}} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Logarithmic_integral/P3xwqRlA">Logarithmic integral</a></li> <li><a href="/facts/Tanc_function/fxuJJL1L">Tanc function</a></li> <li><a href="/facts/Tanhc_function/l4KEjVp6">Tanhc function</a></li> <li><a href="/facts/Sinhc_function/9En7Yfna">Sinhc function</a></li> <li><a href="/facts/Coshc_function/XVyLvY75">Coshc function</a></li></ul> <ul><li><a href="/facts/Milton_Abramowitz/6yRyez7P">Abramowitz, Milton</a>; <a href="/facts/Irene_Stegun/FjSS1LDR">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a href="http://www.math.ubc.ca/~cbm/aands/page_231.htm">"Chapter 5"</a>. <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 231. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-61272-0. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://www.loc.gov/item/65012253">65-12253</a>.</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Mathar, R.J. (2009). "Numerical evaluation of the oscillatory integral over exp(<i>iπx</i>)·<i>x</i>1/<i>x</i> between 1 and ∞". Appendix B. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0912.3844">0912.3844</a> [<a href="https://arxiv.org/archive/math.CA">math.CA</a>].</li> <li>Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). <a href="http://apps.nrbook.com/empanel/index.html#pg=300">"Section 6.8.2 – Cosine and Sine Integrals"</a>. <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.). New York: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-88068-8.</li> <li>Sloughter, Dan. <a href="http://de2de.synechism.org/c5/sec58.pdf">"Sine Integral Taylor series proof"</a> (PDF). <i>Difference Equations to Differential Equations</i>.</li> <li>Temme, N.M. (2010), <a href="http://dlmf.nist.gov/6">"Exponential, Logarithmic, Sine, and Cosine Integrals"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/SineIntegral.html">http://mathworld.wolfram.com/SineIntegral.html</a></li> <li><a href="https://www.encyclopediaofmath.org/index.php?title=Integral_sine">"Integral sine"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="https://www.encyclopediaofmath.org/index.php?title=Integral_cosine">"Integral cosine"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Gray (1993). Modern Differential Geometry of Curves and Surfaces. Boca Raton. p. 119.{{cite book}}: CS1 maint: location missing publisher (link) <a href="/wiki/Template:Cite_book" target="_blank">/wiki/Template:Cite_book</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Rowe, B.; et al. (2015). "GALSIM: The modular galaxy image simulation toolkit". Astronomy and Computing. 10: 121. arXiv:1407.7676. Bibcode:2015A&C....10..121R. doi:10.1016/j.ascom.2015.02.002. S2CID 62709903. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Rowe, B.; et al. (2015). "GALSIM: The modular galaxy image simulation toolkit". Astronomy and Computing. 10: 121. arXiv:1407.7676. Bibcode:2015A&C....10..121R. doi:10.1016/j.ascom.2015.02.002. S2CID 62709903. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>