Si(x) (blue) and Ci(x) (green) plotted on the same plot. In mathematics , the trigonometric integrals are a family of integrals involving trigonometric functions .
Sine integral Plot of Si(x ) for 0 ≤ x ≤ 8 π . The different sine 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~.}
Note that the integrand sin x x sinc function , and also the zeroth spherical Bessel function .
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.
By definition, Si(x ) is the antiderivative of sin x / x whose value is zero at x = 0si(x ) is the antiderivative whose value is zero at x = ∞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 Plot of Ci(x ) for 0 < x ≤ 8π . The different cosine integral definitions are
Cin
(
x
)
=
∫
0
x
1
−
cos
t
t
d
t
,
{\displaystyle \operatorname {Cin} (x)=\int _{0}^{x}{\frac {1-\cos t}{t}}\operatorname {d} t~,}
Ci
(
x
)
=
−
∫
x
∞
cos
t
t
d
t
=
γ
+
ln
x
−
∫
0
x
1
−
cos
t
t
d
t
for
|
Arg
(
x
)
|
<
π
,
{\displaystyle \operatorname {Ci} (x)=-\int _{x}^{\infty }{\frac {\cos t}{t}}\operatorname {d} t=\gamma +\ln x-\int _{0}^{x}{\frac {1-\cos t}{t}}\operatorname {d} t\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,}
where γ ≈ 0.57721566 ... is the Euler–Mascheroni constant . Some texts use ci instead of Ci .
Ci(x ) is the antiderivative of cos x / x (which vanishes as
x
→
∞
{\displaystyle x\to \infty }
Ci
(
x
)
=
γ
+
ln
x
−
Cin
(
x
)
.
{\displaystyle \operatorname {Ci} (x)=\gamma +\ln x-\operatorname {Cin} (x)~.}
Cin is an even , entire function . For that reason, some texts treat Cin as the primary function, and derive Ci in terms of Cin .
Hyperbolic sine integral The hyperbolic sine 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.}
It is related to the ordinary sine integral by
Si
(
i
x
)
=
i
Shi
(
x
)
.
{\displaystyle \operatorname {Si} (ix)=i\operatorname {Shi} (x).}
Hyperbolic cosine integral The hyperbolic cosine integral is
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}}\operatorname {d} t\qquad ~{\text{ for }}~\left|\operatorname {Arg} (x)\right|<\pi ~,}
where
γ
{\displaystyle \gamma }
Euler–Mascheroni constant .
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}).}
Auxiliary functions Trigonometric integrals can be understood in terms of the so-called "auxiliary functions"
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
)
,
and
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}}\mathrm {d} t&=&\int _{0}^{\infty }{\frac {e^{-xt}}{t^{2}+1}}\mathrm {d} t&=&\quad \operatorname {Ci} (x)\sin(x)+\left[{\frac {\pi }{2}}-\operatorname {Si} (x)\right]\cos(x)~,\qquad {\text{ and }}\\g(x)&\equiv &\int _{0}^{\infty }{\frac {\cos(t)}{t+x}}\mathrm {d} t&=&\int _{0}^{\infty }{\frac {te^{-xt}}{t^{2}+1}}\mathrm {d} t&=&-\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 )
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}}}
Nielsen's spiral Nielsen's spiral. The spiral 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)}
The spiral is closely related to the Fresnel integrals and the Euler spiral . Nielsen's spiral has applications in vision processing, road and track construction and other areas.[citation needed
Expansion Various expansions can be used for evaluation of trigonometric integrals, depending on the range of the argument.
Asymptotic series (for large argument)
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)~.}
These series are asymptotic and divergent, although can be used for estimates and even precise evaluation at ℜ(x ) ≫ 1 .
Convergent series
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 }
These series are convergent at any complex x , although for |x , the series will converge slowly initially, requiring many terms for high precision.
Derivation of Series Expansion
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!}}+\,...}
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!}}+\,...}
∴
∫
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}}+\,...}
Relation with the exponential integral of imaginary argument 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 exponential integral . 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~.}
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
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}}\,\operatorname {d} x=-{\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}}}~.}
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}}}\,\operatorname {d} x=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}}}~.}
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),[1] 10−16 for 0 ≤ x ≤ 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}}}
The integrals may be evaluated indirectly via auxiliary functions
f
(
x
)
{\displaystyle f(x)}
g
(
x
)
{\displaystyle g(x)}
x
≥
4
{\displaystyle x\geq 4}
Padé rational functions given below approximate
f
(
x
)
{\displaystyle f(x)}
g
(
x
)
{\displaystyle g(x)}
−16 :[1]
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}}}
See also References Further reading Mathar, R.J. (2009). "Numerical evaluation of the oscillatory integral over exp(iπ x )·x 1/x between 1 and ∞". Appendix B. arXiv :0912.3844 math.CA ]. Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. (2007). "Section 6.8.2 – Cosine and Sine Integrals" . Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8 Sloughter, Dan. "Sine Integral Taylor series proof" (PDF) . Difference Equations to Differential Equations . Temme, N.M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 External links
![{\displaystyle
\begin{array}{rcl}
f(x) &\equiv& \int_0^\infty \frac{\sin(t)}{t+x} \mathrm{d}t &=& \int_0^\infty \frac{e^{-x t}}{t^2 + 1} \mathrm{d}t
&=& \quad \operatorname{Ci}(x) \sin(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \cos(x)~, \qquad \text{ and } \\
g(x) &\equiv& \int_0^\infty \frac{\cos(t)}{t+x} \mathrm{d}t &=& \int_0^\infty \frac{t e^{-x t}}{t^2 + 1} \mathrm{d}t
&=& -\operatorname{Ci}(x) \cos(x) + \left[\frac{\pi}{2} - \operatorname{Si}(x) \right] \sin(x)~.
\end{array}
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba629ad1268d23aebd6a03f14f6720e91fc3183)
![{\displaystyle
\int_1^\infty e^{iax}\frac{\ln x}{x^2}\, \operatorname{d}x =
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}~.
}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb531475b072416d7add59348d55361c50ba523c)
![{\displaystyle f(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \cos(x) + \operatorname{Ci}(x) \sin(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a843910ab6cb92c362e68ac401c28c1e7cda148)
![{\displaystyle g(x) \equiv \left[\frac{\pi}{2} - \operatorname{Si}(x)\right] \sin(x) - \operatorname{Ci}(x) \cos(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f74128afc0519376e13432f0e9f5b0bf6627de7)
