Special function related to the dilogarithm
The inverse tangent integral is a special function , defined by:
Ti
2
(
x
)
=
∫
0
x
arctan
t
t
d
t
{\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt}
Equivalently, it can be defined by a power series , or in terms of the dilogarithm , a closely related special function.
Definition [ ]
The inverse tangent integral is defined by:
Ti
2
(
x
)
=
∫
0
x
arctan
t
t
d
t
{\displaystyle \operatorname {Ti} _{2}(x)=\int _{0}^{x}{\frac {\arctan t}{t}}\,dt}
The arctangent is taken to be the principal branch ; that is, −π /2 < arctan(t ) < π /2 for all real t .[1]
Its power series representation is
Ti
2
(
x
)
=
x
−
x
3
3
2
+
x
5
5
2
−
x
7
7
2
+
⋯
{\displaystyle \operatorname {Ti} _{2}(x)=x-{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}-{\frac {x^{7}}{7^{2}}}+\cdots }
which is absolutely convergent for
|
x
|
≤
1.
{\displaystyle |x|\leq 1.}
[1]
The inverse tangent integral is closely related to the dilogarithm
Li
2
(
z
)
=
∑
n
=
1
∞
z
n
n
2
{\textstyle \operatorname {Li} _{2}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{2}}}}
and can be expressed simply in terms of it:
Ti
2
(
z
)
=
1
2
i
(
Li
2
(
i
z
)
−
Li
2
(
−
i
z
)
)
{\displaystyle \operatorname {Ti} _{2}(z)={\frac {1}{2i}}\left(\operatorname {Li} _{2}(iz)-\operatorname {Li} _{2}(-iz)\right)}
That is,
Ti
2
(
x
)
=
Im
(
Li
2
(
i
x
)
)
{\displaystyle \operatorname {Ti} _{2}(x)=\operatorname {Im} (\operatorname {Li} _{2}(ix))}
for all real x .[1]
Properties [ ]
The inverse tangent integral is an odd function :[1]
Ti
2
(
−
x
)
=
−
Ti
2
(
x
)
{\displaystyle \operatorname {Ti} _{2}(-x)=-\operatorname {Ti} _{2}(x)}
The values of Ti2 (x ) and Ti2 (1/x ) are related by the identity
Ti
2
(
x
)
−
Ti
2
(
1
x
)
=
π
2
log
x
{\displaystyle \operatorname {Ti} _{2}(x)-\operatorname {Ti} _{2}\left({\frac {1}{x}}\right)={\frac {\pi }{2}}\log x}
valid for all x > 0 (or, more generally, for Re(x ) > 0).
This can be proven by differentiating and using the identity
arctan
(
t
)
+
arctan
(
1
/
t
)
=
π
/
2
{\displaystyle \arctan(t)+\arctan(1/t)=\pi /2}
.[2] [3]
The special value Ti2 (1) is Catalan's constant
1
−
1
3
2
+
1
5
2
−
1
7
2
+
⋯
≈
0.915966
{\textstyle 1-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+\cdots \approx 0.915966}
.[3]
Generalizations [ ]
Similar to the polylogarithm
Li
n
(
z
)
=
∑
k
=
1
∞
z
k
k
n
{\textstyle \operatorname {Li} _{n}(z)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{n}}}}
, the function
Ti
n
(
x
)
=
x
−
x
3
3
n
+
x
5
5
n
−
x
7
7
n
+
⋯
{\displaystyle \operatorname {Ti} _{n}(x)=x-{\frac {x^{3}}{3^{n}}}+{\frac {x^{5}}{5^{n}}}-{\frac {x^{7}}{7^{n}}}+\cdots }
is defined analogously. This satisfies the recurrence relation:[4]
Ti
n
(
x
)
=
∫
0
x
Ti
n
−
1
(
t
)
t
d
t
{\displaystyle \operatorname {Ti} _{n}(x)=\int _{0}^{x}{\frac {\operatorname {Ti} _{n-1}(t)}{t}}\,dt}
Relation to other special functions [ ]
The inverse tangent integral is related to the Legendre chi function
χ
2
(
x
)
=
x
+
x
3
3
2
+
x
5
5
2
+
⋯
{\textstyle \chi _{2}(x)=x+{\frac {x^{3}}{3^{2}}}+{\frac {x^{5}}{5^{2}}}+\cdots }
by:[1]
Ti
2
(
x
)
=
−
i
χ
2
(
i
x
)
{\displaystyle \operatorname {Ti} _{2}(x)=-i\chi _{2}(ix)}
Note that
χ
2
(
x
)
{\displaystyle \chi _{2}(x)}
can be expressed as
∫
0
x
artanh
t
t
d
t
{\textstyle \int _{0}^{x}{\frac {\operatorname {artanh} t}{t}}\,dt}
, similar to the inverse tangent integral but with the inverse hyperbolic tangent instead.
The inverse tangent integral can also be written in terms of the Lerch transcendent
Φ
(
z
,
s
,
a
)
=
∑
n
=
0
∞
z
n
(
n
+
a
)
s
:
{\textstyle \Phi (z,s,a)=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+a)^{s}}}:}
[5]
Ti
2
(
x
)
=
1
4
x
Φ
(
−
x
2
,
2
,
1
/
2
)
{\displaystyle \operatorname {Ti} _{2}(x)={\frac {1}{4}}x\Phi (-x^{2},2,1/2)}
History [ ]
The notation Ti2 and Tin is due to Lewin. (1809)[6] studied the function, using the notation
C
n
(
x
)
{\displaystyle {\overset {n}{\operatorname {C} }}(x)}
. The function was also studied by Ramanujan .[2]
References [ ]
^ a b c d e Lewin 1981 , pp. 38–39, Section 2.1
^ a b Ramanujan, S. (1915). "On the integral
∫
0
x
tan
−
1
t
t
d
t
{\displaystyle \int _{0}^{x}{\frac {\tan ^{-1}t}{t}}\,dt}
". Journal of the Indian Mathematical Society . 7 : 93–96. Appears in: Hardy, G. H. ; Seshu Aiyar, P. V.; Wilson, B. M. , eds. (1927). Collected Papers of Srinivasa Ramanujan . pp. 40–43.
^ a b Lewin 1981 , pp. 39–40, Section 2.2
^ Lewin 1981 , p. 190, Section 7.1.2
^ Weisstein, Eric W. "Inverse Tangent Integral" . MathWorld .
^ Spence, William (1809). An essay on the theory of the various orders of logarithmic transcendents; with an inquiry into their applications to the integral calculus and the summation of series . London.
Lewin, L. (1958). Dilogarithms and Associated Functions . London: Macdonald. MR 0105524 . Zbl 0083.35904 .
Lewin, L. (1981). Polylogarithms and Associated Functions . New York: North-Holland. ISBN 978-0-444-00550-2 .