Inverse tangent integral: Difference between revisions
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{{Short description|Special function related to the dilogarithm}} |
{{Short description|Special function related to the dilogarithm}} |
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The '''inverse tangent integral''' is a [[special function]], defined by: |
The '''inverse tangent integral''' is a [[special function]], defined by: |
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:<math>\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} dt</math> |
:<math>\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt</math> |
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Equivalently, it can be defined by a [[power series]], or in terms of the [[dilogarithm]], a closely related special function. |
Equivalently, it can be defined by a [[power series]], or in terms of the [[dilogarithm]], a closely related special function. |
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==Definition== |
==Definition== |
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The inverse tangent integral is defined by: |
The inverse tangent integral is defined by: |
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:<math>\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} dt</math> |
:<math>\operatorname{Ti}_2(x) = \int_0^x \frac{\arctan t}{t} \, dt</math> |
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The [[arctangent]] is taken to be the [[principal branch]]; that is, − |
The [[arctangent]] is taken to be the [[principal branch]]; that is, −{{pi}}/2 < arctan(''t'') < {{pi}}/2 for all real ''t''.<ref name="lewin-1981-2-1">{{harvnb|Lewin|1981|pp=38–39|loc=Section 2.1}}</ref> |
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Its [[power series]] representation is |
Its [[power series]] representation is |
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:<math>\operatorname{Ti}_2(x) - \operatorname{Ti}_2 \left(\frac{1}{x} \right) = \frac{\pi}{2} \log x</math> |
:<math>\operatorname{Ti}_2(x) - \operatorname{Ti}_2 \left(\frac{1}{x} \right) = \frac{\pi}{2} \log x</math> |
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valid for all ''x'' > 0 (or, more generally, for Re(''x'') > 0). |
valid for all ''x'' > 0 (or, more generally, for Re(''x'') > 0). |
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This can be proven by |
This can be proven by differentiating and using the identity <math>\arctan(t) + \arctan(1/t) = \pi/2</math>.<ref name="ramanujan" /><ref name="lewin-1981-2-2" /> |
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The special value Ti<sub>2</sub>(1) is [[Catalan's constant]] <math display="inline">1 - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \approx 0.915966</math>.<ref name="lewin-1981-2-2">{{harvnb|Lewin|1981|pp=39–40|loc=Section 2.2}}</ref> |
The special value Ti<sub>2</sub>(1) is [[Catalan's constant]] <math display="inline">1 - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \approx 0.915966</math>.<ref name="lewin-1981-2-2">{{harvnb|Lewin|1981|pp=39–40|loc=Section 2.2}}</ref> |
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==Generalizations== |
==Generalizations== |
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Similar to the [[polylogarithm]] <math display="inline">\operatorname{Li}_n(z) = \sum_{ |
Similar to the [[polylogarithm]] <math display="inline">\operatorname{Li}_n(z) = \sum_{k=1}^\infty \frac{z^k}{k^n}</math>, the function |
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:<math>\operatorname{Ti} |
:<math>\operatorname{Ti}_{n}(x) = \sum\limits_{k=0}^{\infty}\frac{(-1)^{k}x^{2k+1}}{\left(2k+1\right)^{n}}=x - \frac{x^3}{3^n} + \frac{x^5}{5^n} - \frac{x^7}{7^n} + \cdots</math> |
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is defined analogously. This satisfies the recurrence relation:<ref>{{harvnb|Lewin|1981|p=190|loc=Section 7.1.2}}</ref> |
is defined analogously. This satisfies the recurrence relation:<ref>{{harvnb|Lewin|1981|p=190|loc=Section 7.1.2}}</ref> |
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:<math>\operatorname{Ti} |
:<math>\operatorname{Ti}_{n}(x) = \int_0^x \frac{\operatorname{Ti}_{n-1}(t)}{t} \, dt</math> |
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By this series representation it can be seen that the special values <math>\operatorname{Ti}_{n}(1)=\beta(n)</math>, where <math>\beta(s)</math> represents the [[Dirichlet beta function]]. |
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==Relation to other special functions== |
==Relation to other special functions== |
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The inverse tangent integral is related to the [[Legendre chi function]] <math display="inline">\chi_2(x) = x + \frac{x^3}{3^2} + \frac{x^5}{5^2} + \cdots</math> by:<ref name="lewin-1981-2-1" /> |
The inverse tangent integral is related to the [[Legendre chi function]] <math display="inline">\chi_2(x) = x + \frac{x^3}{3^2} + \frac{x^5}{5^2} + \cdots</math> by:<ref name="lewin-1981-2-1" /> |
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:<math>\operatorname{Ti}_2(x) = -i \chi_2(ix)</math> |
:<math>\operatorname{Ti}_2(x) = -i \chi_2(ix)</math> |
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Note that <math>\chi_2(x)</math> can be expressed as <math display="inline">\int_0^x \frac{\operatorname{artanh} t}{t} dt</math>, similar to the inverse tangent integral but with the [[inverse hyperbolic tangent]] instead. |
Note that <math>\chi_2(x)</math> can be expressed as <math display="inline">\int_0^x \frac{\operatorname{artanh} t}{t} \, dt</math>, similar to the inverse tangent integral but with the [[inverse hyperbolic tangent]] instead. |
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The inverse tangent integral can also be written in terms of the [[Lerch transcendent]] <math display="inline">\Phi(z,s,a) = \sum_{n=0}^\infty \frac{z^n}{(n+a)^s}:</math><ref>{{MathWorld |id=InverseTangentIntegral |title=Inverse Tangent Integral}}</ref> |
The inverse tangent integral can also be written in terms of the [[Lerch transcendent]] <math display="inline">\Phi(z,s,a) = \sum_{n=0}^\infty \frac{z^n}{(n+a)^s}:</math><ref>{{MathWorld |id=InverseTangentIntegral |title=Inverse Tangent Integral}}</ref> |
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==History== |
==History== |
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The notation Ti<sub>2</sub> and Ti<sub>''n''</sub> is due to Lewin. [[William Spence (mathematician)|Spence]] (1809)<ref>{{Cite book | last= Spence | first= William | date= 1809 | title= 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 | location= London | url= https://babel.hathitrust.org/cgi/pt?id=uc1.c3118082}}</ref> studied the function, using the notation <math>\overset{n}{\operatorname{C}}(x)</math>. The function was also studied by [[Ramanujan]].<ref name="ramanujan">{{Cite journal | first= S. | last= Ramanujan | author-link= Srinivasa Ramanujan | journal= Journal of the Indian Mathematical Society | volume= 7 | date= 1915 | pages= 93–96 | title= On the integral <math>\int_0^x \frac{\tan^{-1} t}{t} dt</math>}} Appears in: {{Cite book | title= Collected Papers of Srinivasa Ramanujan | editor-first1= G. H. | editor-last1= Hardy |editor-link= G. H. Hardy | editor-first2= P. V. | editor-last2=Seshu Aiyar | editor-first3= B. M. | editor-last3= Wilson | editor-link3= Bertram Martin Wilson | date = 1927 | pages= 40–43 }}</ref> |
The notation Ti<sub>2</sub> and Ti<sub>''n''</sub> is due to Lewin. [[William Spence (mathematician)|Spence]] (1809)<ref>{{Cite book | last= Spence | first= William | date= 1809 | title= 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 | location= London | url= https://babel.hathitrust.org/cgi/pt?id=uc1.c3118082}}</ref> studied the function, using the notation <math>\overset{n}{\operatorname{C}}(x)</math>. The function was also studied by [[Ramanujan]].<ref name="ramanujan">{{Cite journal | first= S. | last= Ramanujan | author-link= Srinivasa Ramanujan | journal= Journal of the Indian Mathematical Society | volume= 7 | date= 1915 | pages= 93–96 | title= On the integral <math>\int_0^x \frac{\tan^{-1} t}{t} \, dt</math>}} Appears in: {{Cite book | title= Collected Papers of Srinivasa Ramanujan | editor-first1= G. H. | editor-last1= Hardy |editor-link= G. H. Hardy | editor-first2= P. V. | editor-last2=Seshu Aiyar | editor-first3= B. M. | editor-last3= Wilson | editor-link3= Bertram Martin Wilson | date = 1927 | pages= 40–43 }}</ref> |
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==References== |
==References== |
Latest revision as of 19:39, 12 February 2024
The inverse tangent integral is a special function, defined by:
Equivalently, it can be defined by a power series, or in terms of the dilogarithm, a closely related special function.
Definition[edit]
The inverse tangent integral is defined by:
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
which is absolutely convergent for [1]
The inverse tangent integral is closely related to the dilogarithm and can be expressed simply in terms of it:
That is,
for all real x.[1]
Properties[edit]
The inverse tangent integral is an odd function:[1]
The values of Ti2(x) and Ti2(1/x) are related by the identity
valid for all x > 0 (or, more generally, for Re(x) > 0). This can be proven by differentiating and using the identity .[2][3]
The special value Ti2(1) is Catalan's constant .[3]
Generalizations[edit]
Similar to the polylogarithm , the function
is defined analogously. This satisfies the recurrence relation:[4]
By this series representation it can be seen that the special values , where represents the Dirichlet beta function.
Relation to other special functions[edit]
The inverse tangent integral is related to the Legendre chi function by:[1]
Note that can be expressed as , 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 [5]
History[edit]
The notation Ti2 and Tin is due to Lewin. Spence (1809)[6] studied the function, using the notation . The function was also studied by Ramanujan.[2]
References[edit]
- ^ a b c d e Lewin 1981, pp. 38–39, Section 2.1
- ^ a b Ramanujan, S. (1915). "On the integral ". 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.