Transcendental single-variable function
In mathematics, the Clausen function, introduced by Thomas Clausen (1832), is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.
The Clausen function of order 2 – often referred to as "the" Clausen function, despite being but one of a class of many – is given by the integral:
formula_1
In the range formula_2 the sine function inside the absolute value sign remains strictly positive, so the absolute value signs may be omitted. The Clausen function also has the Fourier series representation:
formula_3
The Clausen functions, as a class of functions, feature extensively in many areas of modern mathematical research, particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications with regard to the summation of hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the polygamma function, and Dirichlet L-series.
Basic properties.
The Clausen function (of order 2) has simple zeros at all (integer) multiples of formula_4 since if formula_5 is an integer, then formula_6
formula_7
It has maxima at formula_8
formula_9
and minima at formula_10
formula_11
The following properties are immediate consequences of the series definition:
formula_12
formula_13
See .
General definition.
More generally, one defines the two generalized Clausen functions:
formula_14
formula_15
which are valid for complex "z" with Re "z" ;1. The definition may be extended to all of the complex plane through analytic continuation.
When "z" is replaced with a non-negative integer, the standard Clausen functions are defined by the following Fourier series:
formula_16
formula_17
formula_18
formula_19
N.B. The SL-type Clausen functions have the alternative notation formula_20 and are sometimes referred to as the Glaisher–Clausen functions (after James Whitbread Lee Glaisher, hence the GL-notation).
Relation to the Bernoulli polynomials.
The SL-type Clausen function are polynomials in formula_21, and are closely related to the Bernoulli polynomials. This connection is apparent from the Fourier series representations of the Bernoulli polynomials:
formula_22
formula_23
Setting formula_24 in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:
formula_25
formula_26
where the Bernoulli polynomials formula_27 are defined in terms of the Bernoulli numbers formula_28 by the relation:
formula_29
Explicit evaluations derived from the above include:
formula_30
formula_31
formula_32
formula_33
Duplication formula.
For formula_34, the duplication formula can be proven directly from the integral definition (see also for the result – although no proof is given):
formula_35
Denoting Catalan's constant by formula_36, immediate consequences of the duplication formula include the relations:
formula_37
formula_38
For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace formula_39 with the dummy variable formula_40, and integrate over the interval formula_41 Applying the same process repeatedly yields:
formula_42
formula_43
formula_44
formula_45
And more generally, upon induction on formula_46
formula_47
Use of the generalized duplication formula allows for an extension of the result for the Clausen function of order 2, involving Catalan's constant. For formula_48
formula_49
Where formula_50 is the Dirichlet beta function.
Proof of the duplication formula.
From the integral definition,
formula_51
Apply the duplication formula for the sine function, formula_52 to obtain
formula_53
Apply the substitution formula_54 on both integrals:
formula_55
On that last integral, set formula_56, and use the trigonometric identity formula_57 to show that:
 formula_58
 formula_59
Therefore,
 formula_60
Derivatives of general-order Clausen functions.
Direct differentiation of the Fourier series expansions for the Clausen functions give:
formula_61
formula_62
formula_63
formula_64
By appealing to the First Fundamental Theorem Of Calculus, we also have:
formula_65
Relation to the inverse tangent integral.
The inverse tangent integral is defined on the interval formula_66 by
formula_67
It has the following closed form in terms of the Clausen function:
formula_68
Proof of the inverse tangent integral relation.
From the integral definition of the inverse tangent integral, we have
formula_69
Performing an integration by parts
formula_70
formula_71
Apply the substitution formula_72 to obtain
formula_73
For that last integral, apply the transform :formula_74 to get
formula_75
Finally, as with the proof of the Duplication formula, the substitution formula_76 reduces that last integral to
formula_77
Thus
formula_78
Relation to the Barnes' G-function.
For real formula_66, the Clausen function of second order can be expressed in terms of the Barnes G-function and (Euler) Gamma function:
formula_80
Or equivalently
formula_81
See .
Relation to the polylogarithm.
The Clausen functions represent the real and imaginary parts of the polylogarithm, on the unit circle:
formula_82
formula_83
This is easily seen by appealing to the series definition of the polylogarithm.
formula_84
By Euler's theorem,
formula_85
and by de Moivre's Theorem (De Moivre's formula)
formula_86
Hence
formula_87
formula_88
Relation to the polygamma function.
The Clausen functions are intimately connected to the polygamma function. Indeed, it is possible to express Clausen functions as linear combinations of sine functions and polygamma functions. One such relation is shown here, and proven below:
formula_89
An immediate corollary is this equivalent formula in terms of the Hurwitz zeta function:
formula_90
Relation to the generalized logsine integral.
The generalized logsine integral is defined by:
formula_91
In this generalized notation, the Clausen function can be expressed in the form:
formula_92
Kummer's relation.
Ernst Kummer and Rogers give the relation
formula_93
valid for formula_94.
Relation to the Lobachevsky function.
The Lobachevsky function Λ or Л is essentially the same function with a change of variable:
formula_95
though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function
formula_96
Relation to Dirichlet L-functions.
For rational values of formula_97 (that is, for formula_98 for some integers "p" and "q"), the function formula_99 can be understood to represent a periodic orbit of an element in the cyclic group, and thus formula_100 can be expressed as a simple sum involving the Hurwitz zeta function. This allows relations between certain Dirichlet L-functions to be easily computed.
Series acceleration.
A series acceleration for the Clausen function is given by
formula_101
which holds for formula_102. Here, formula_103 is the Riemann zeta function. A more rapidly convergent form is given by
formula_104
Convergence is aided by the fact that formula_105 approaches zero rapidly for large values of "n". Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series .
Special values.
Recall the Barnes G-function, the Catalan's constant "K" and the Gieseking constant "V". Some special values include
formula_106
formula_107
formula_108
formula_109
formula_110
formula_111
formula_112
formula_113
In general, from the Barnes G-function reflection formula,
formula_114
Equivalently, using Euler's reflection formula for the gamma function, then,
formula_115
Generalized special values.
Some special values for higher order Clausen functions include
formula_116
formula_117
formula_118
formula_119
formula_120
where formula_121 is the Dirichlet beta function, formula_122 is the Dirichlet eta function (also called the alternating zeta function), and formula_123 is the Riemann zeta function.
Integrals of the direct function.
The following integrals are easily proven from the series representations of the Clausen function:
formula_124
formula_125
formula_126
formula_127
Fourier-analytic methods can be used to find the first moments of the square of the function formula_128 on the interval formula_129:
formula_130
formula_131
formula_132
Here formula_133 denotes the multiple zeta function.
Integral evaluations involving the direct function.
A large number of trigonometric and logarithmo-trigonometric integrals can be evaluated in terms of the Clausen function, and various common mathematical constants like formula_134 (Catalan's constant), formula_135, and the special cases of the zeta function, formula_136 and formula_137.
The examples listed below follow directly from the integral representation of the Clausen function, and the proofs require little more than basic trigonometry, integration by parts, and occasional term-by-term integration of the Fourier series definitions of the Clausen functions.
formula_138
formula_139
formula_140
formula_141
formula_142
formula_143
formula_144
References.
 ;templatestyles src="Reflist/styles.css" / ;