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Graph of the Clausen function Cl₂(θ)

In mathematics, the Clausen function - introduced by Template:Harvs - 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 special functions. It is intimately connected with the Polylogarithm, Inverse tangent integral, Polygamma function, Riemann Zeta function, Eta function, and Dirichlet beta function.

The Clausen function of order 2 - often referred to at the Clausen function, despite being but one of a class of many - is given by the integral:

{Cl}_{2} (φ) = - \int_{0}^{φ} \log | 2 \sin \frac{x}{2} | d x :

In the range : $0 < φ < 2 π$ 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:

{Cl}_{2} (φ) = \sum_{k = 1}^{\infty} \frac{\sin k φ}{k^{2}} = \sin φ + \frac{\sin 2 φ}{2^{2}} + \frac{\sin 3 φ}{3^{2}} + \frac{\sin 4 φ}{4^{2}} + \dots

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, Central Binomial sums, 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 : $π,$ since if : $k \in ℤ$ is an integer, : $\sin k π = 0$

{Cl}_{2} (m π) = 0, m = 0, \pm 1, \pm 2, \pm 3, \dots

It has maxima at : $θ = \frac{π}{3} + 2 m π [m \in ℤ]$

{Cl}_{2} (\frac{π}{3} + 2 m π) = 1.01494160 \dots

and minima at : $θ = - \frac{π}{3} + 2 m π [m \in ℤ]$

{Cl}_{2} (- \frac{π}{3} + 2 m π) = - 1.01494160 \dots

The following properties are immediate consequences of the series definition:

{Cl}_{2} (θ + 2 m π) = {Cl}_{2} (θ)

{Cl}_{2} (- θ) = - {Cl}_{2} (θ)

(Ref: See Lu and Perez, 1992, below for these results - although no proofs are given).

General definition

More generally, one defines the two generalized Clausen functions:

S_{z} (θ) = \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{z}}

C_{z} (θ) = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{z}}

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:

{Cl}_{2 m + 2} (θ) = \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m + 2}}

{Cl}_{2 m + 1} (θ) = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m + 1}}

{Sl}_{2 m + 2} (θ) = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m + 2}}

{Sl}_{2 m + 1} (θ) = \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m + 1}}

N.B. The SL-type Clausen functions have the alternative notation : ${Gl}_{m} (θ)$ 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 $θ$ , and are closely related to the Bernoulli polynomials. This connection is apparent from the Fourier series representations of the Bernoulli Polynomials:

B_{2 n - 1} (x) = \frac{2 (- 1)^{n} (2 n - 1)!}{(2 π)^{2 n - 1}} \sum_{k = 1}^{\infty} \frac{\sin 2 π k x}{k^{2 n - 1}}

B_{2 n} (x) = \frac{2 (- 1)^{n - 1} (2 n)!}{(2 π)^{2 n}} \sum_{k = 1}^{\infty} \frac{\cos 2 π k x}{k^{2 n}}

Setting $x = θ / 2 π$ in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:

{Sl}_{2 m} (θ) = \frac{(- 1)^{m - 1} (2 π)^{2 m}}{2 (2 m)!} B_{2 m} (\frac{θ}{2 π})

{Sl}_{2 m - 1} (θ) = \frac{(- 1)^{m} (2 π)^{2 m - 1}}{2 (2 m - 1)!} B_{2 m - 1} (\frac{θ}{2 π})

Where the Bernoulli polynomials $B_{n} (x)$ are defined in terms of the Bernoulli numbers $B_{n} \equiv B_{n} (0)$ by the relation:

B_{n} (x) = \sum_{j = 0}^{n} (\binom{n}{j}) B_{j} x^{n - j}

Explicit evaluations derived from the above include:

{Sl}_{1} (θ) = \frac{π}{2} - \frac{θ}{2}

{Sl}_{2} (θ) = \frac{π^{2}}{6} - \frac{π θ}{2} + \frac{θ^{2}}{4}

{Sl}_{3} (θ) = \frac{π^{2} θ}{6} - \frac{π θ^{2}}{4} + \frac{θ^{3}}{12}

{Sl}_{4} (θ) = \frac{π^{4}}{90} - \frac{π^{2} θ^{2}}{12} + \frac{π θ^{3}}{12} - \frac{θ^{4}}{48}

Duplication formula

For : $0 < θ < π$ , the duplication formula can be proven directly from the Integral definition (see also Lu and Perez, 1992, below for the result - although no proof is given):

{Cl}_{2} (2 θ) = 2 {Cl}_{2} (θ) - 2 {Cl}_{2} (π - θ)

Immediate consequences of the duplication formula, along with use of the special value : ${Cl}_{2} (\frac{π}{2}) = G$ , include the relations:

{Cl}_{2} (\frac{π}{4}) - {Cl}_{2} (\frac{3 π}{4}) = \frac{G}{2}

2 {Cl}_{2} (\frac{π}{3}) = 3 {Cl}_{2} (\frac{2 π}{3})

For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace $θ$ with the dummy variable $x$ , and integrate over the interval $[0, θ] .$ Applying the same process repeatedly yields:

{Cl}_{3} (2 θ) = 4 {Cl}_{3} (θ) + 4 {Cl}_{3} (π - θ)

{Cl}_{4} (2 θ) = 8 {Cl}_{4} (θ) - 8 {Cl}_{4} (π - θ)

{Cl}_{5} (2 θ) = 16 {Cl}_{5} (θ) + 16 {Cl}_{5} (π - θ)

{Cl}_{6} (2 θ) = 32 {Cl}_{6} (θ) - 32 {Cl}_{6} (π - θ)

And more generally, upon induction on $m, m \geq 1$

{Cl}_{m + 1} (2 θ) = 2^{m} [{Cl}_{m + 1} (θ) + (- 1)^{m} {Cl}_{m + 1} (π - θ)]

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 $m \in ℤ \geq 1$

{Cl}_{2 m} (\frac{π}{2}) = 2^{2 m - 1} [{Cl}_{2 m} (\frac{π}{4}) - {Cl}_{2 m} (\frac{3 π}{4})] = β (2 m)

Where $β (x)$ is the Dirichlet beta function.

Proof of the Duplication formula

From the integral definition,

{Cl}_{2} (2 θ) = - \int_{0}^{2 θ} \log | 2 \sin \frac{x}{2} | d x

Apply the duplication formula for the Sine function, : $\sin 2 x = 2 \sin \frac{x}{2} \cos \frac{x}{2}$ to obtain

- \int_{0}^{2 θ} \log | (2 \sin \frac{x}{4}) (2 \cos \frac{x}{4}) | d x =

- \int_{0}^{2 θ} \log | 2 \sin \frac{x}{4} | d x - \int_{0}^{2 θ} \log | 2 \cos \frac{x}{4} | d x =

Apply the substitution $x = 2 y, d x = 2 d y$ on both integrals

$\Rightarrow$

- 2 \int_{0}^{θ} \log | 2 \sin \frac{x}{2} | d x - 2 \int_{0}^{θ} \log | 2 \cos \frac{x}{2} | d x =

2 {Cl}_{2} (θ) - 2 \int_{0}^{θ} \log | 2 \cos \frac{x}{2} | d x

On that last integral, set $y = π - x, x = π - y, d x = - d y$ , and use the trigonometric identity $\cos (x - y) = \cos x \cos y - \sin x \sin y$ to show that:

$\cos (\frac{π - y}{2}) = \sin \frac{y}{2} \Rightarrow$

${Cl}_{2} (2 θ) = 2 {Cl}_{2} (θ) - 2 \int_{0}^{θ} \log | 2 \cos \frac{x}{2} | d x =$

$2 {Cl}_{2} (θ) + 2 \int_{π}^{π - θ} \log | 2 \sin \frac{y}{2} | d y =$

${Cl}_{2} (θ) - 2 {Cl}_{2} (π - θ) + 2 {Cl}_{2} (π)$

${Cl}_{2} (π) = 0$

Therefore

${Cl}_{2} (2 θ) = 2 {Cl}_{2} (θ) - 2 {Cl}_{2} (π - θ) . ◻$

Derivatives of general order Clausen functions

Direct differentiation of the Fourier series expansions for the Clausen functions give:

\frac{d}{d θ} {Cl}_{2 m + 2} (θ) = \frac{d}{d θ} \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m + 2}} = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m + 1}} = {Cl}_{2 m + 1} (θ)

\frac{d}{d θ} {Cl}_{2 m + 1} (θ) = \frac{d}{d θ} \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m + 1}} = - \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m}} = - {Cl}_{2 m} (θ)

\frac{d}{d θ} {Sl}_{2 m + 2} (θ) = \frac{d}{d θ} \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m + 2}} = - \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m + 1}} = - {Sl}_{2 m + 1} (θ)

\frac{d}{d θ} {Sl}_{2 m + 1} (θ) = \frac{d}{d θ} \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m + 1}} = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m}} = {Sl}_{2 m} (θ)

By appealing to the First Fundamental Theorem Of Calculus, we also have:

\frac{d}{d θ} {Cl}_{2} (θ) = \frac{d}{d θ} [- \int_{0}^{θ} \log | 2 \sin \frac{x}{2} | d x] = - \log | 2 \sin \frac{θ}{2} | = {Cl}_{1} (θ)

Relation to the Inverse Tangent Integral

The Inverse tangent integral is defined on the interval : $0 < z < 1$ by

{Ti}_{2} (z) = \int_{0}^{z} \frac{\tan^{- 1} x}{x} d x = \sum_{k = 0}^{\infty} (- 1)^{k} \frac{z^{2 k + 1}}{(2 k + 1)^{2}}

It has the following closed form in terms of the Clausen Function:

{Ti}_{2} (\tan θ) = θ \log (\tan θ) + \frac{1}{2} {Cl}_{2} (2 θ) + \frac{1}{2} {Cl}_{2} (π - 2 θ)

Proof of the Inverse Tangent Integral relation

From the integral definition of the Inverse tangent integral, we have

{Ti}_{2} (\tan θ) = \int_{0}^{\tan θ} \frac{\tan^{- 1} x}{x} d x

Performing an integration by parts

\int_{0}^{\tan θ} \frac{\tan^{- 1} x}{x} d x = \tan^{- 1} x \log x |_{0}^{\tan θ} - \int_{0}^{\tan θ} \frac{\log x}{1 + x^{2}} d x =

θ \log \tan θ - \int_{0}^{\tan θ} \frac{\log x}{1 + x^{2}} d x

Apply the substitution : $x = \tan y, y = \tan^{- 1} x, d y = \frac{d x}{1 + x^{2}}$ to obtain

θ \log \tan θ - \int_{0}^{θ} \log (\tan y) d y

For that last integral, apply the transform : $y = x / 2, d y = d x / 2$ to get

θ \log \tan θ - \frac{1}{2} \int_{0}^{2 θ} \log (\tan \frac{x}{2}) d x =

θ \log \tan θ - \frac{1}{2} \int_{0}^{2 θ} \log (\frac{\sin (x / 2)}{\cos (x / 2)}) d x =

θ \log \tan θ - \frac{1}{2} \int_{0}^{2 θ} \log (\frac{2 \sin (x / 2)}{2 \cos (x / 2)}) d x =

θ \log \tan θ - \frac{1}{2} \int_{0}^{2 θ} \log (2 \sin \frac{x}{2}) d x + \frac{1}{2} \int_{0}^{2 θ} \log (2 \cos \frac{x}{2}) d x =

θ \log \tan θ + \frac{1}{2} {Cl}_{2} (2 θ) + \frac{1}{2} \int_{0}^{2 θ} \log (2 \cos \frac{x}{2}) d x

Finally, as with the proof of the Duplication formula, the substitution $x = (π - y)$ reduces that last integral to

\int_{0}^{2 θ} \log (2 \cos \frac{x}{2}) d x = {Cl}_{2} (π - 2 θ) - {Cl}_{2} (π) = {Cl}_{2} (π - 2 θ)

Thus

{Ti}_{2} (\tan θ) = θ \log \tan θ + \frac{1}{2} {Cl}_{2} (2 θ) + \frac{1}{2} {Cl}_{2} (π - 2 θ) . ◻

Relation to the Barnes' G-function

For real : $0 < z < 1$ , the Clausen function of second order can be expressed in terms of the Barnes G-function and (Euler) Gamma function:

{Cl}_{2} (2 π z) = 2 π \log (\frac{G (1 - z)}{G (1 + z)}) - 2 π \log (\frac{\sin π z}{π})

Or equivalently

{Cl}_{2} (2 π z) = 2 π \log (\frac{G (1 - z)}{G (z)}) - 2 π \log Γ (z) - 2 π \log (\frac{\sin π z}{π})

Ref: See Adamchik, "Contributions to the Theory of the Barnes function", below.

Relation to the Polylogarithm

The Clausen functions represent the real and imaginary parts of the Polylogarithm, on the Unit Circle:

{Cl}_{2 m} (θ) = ℑ ({Li}_{2 m} (e^{i θ})), m \in ℤ \geq 1

{Cl}_{2 m + 1} (θ) = ℜ ({Li}_{2 m + 1} (e^{i θ})), m \in ℤ \geq 0

This is easily seen by appealing to the series definition of the Polylogarithm.

{Li}_{n} (z) = \sum_{k = 1}^{\infty} \frac{z^{k}}{k^{n}} \Rightarrow {Li}_{n} (e^{i θ}) = \sum_{k = 1}^{\infty} \frac{{(e^{i θ})}^{k}}{k^{n}} = \sum_{k = 1}^{\infty} \frac{e^{i k θ}}{k^{n}}

By Euler's Theorem,

e^{i θ} = \cos θ + i \sin θ

and by de Moivre's Theorem (DeMoivre's Formula)

(\cos θ + i \sin θ)^{k} = \cos k θ + i \sin k θ \Rightarrow {Li}_{n} (e^{i θ}) = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{n}} + i \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{n}}

Hence

{Li}_{2 m} (e^{i θ}) = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m}} + i \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m}} = {Sl}_{2 m} (θ) + i {Cl}_{2 m} (θ)

{Li}_{2 m + 1} (e^{i θ}) = \sum_{k = 1}^{\infty} \frac{\cos k θ}{k^{2 m + 1}} + i \sum_{k = 1}^{\infty} \frac{\sin k θ}{k^{2 m + 1}} = {Cl}_{2 m + 1} (θ) + i {Sl}_{2 m + 1} (θ)

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:

{Cl}_{2 m} (\frac{q π}{p}) = \frac{1}{(2 p)^{2 m} (2 m - 1)!} \sum_{j = 1}^{p} \sin (\frac{q j π}{p}) [ψ_{2 m - 1} (\frac{j}{2 p}) + (- 1)^{q} ψ_{2 m - 1} (\frac{j + p}{2 p})]

Let $p$ and $q$ be positive integers, such that $q / p$ is a rational number $0 < q / p < 1$ , then, by the series definition for the higher order Clausen function (of even index):

{Cl}_{2 m} (\frac{q π}{p}) = \sum_{k = 1}^{\infty} \frac{\sin (k q π / p)}{k^{2 m}}

We split this sum into exactly p-parts, so that the first series contains all, and only, those terms congruous to $k p + 1,$ the second series contains all terms congruous to $k p + 2,$ etc, up to the final p-th part, that contain all terms congruous to $k p + p$

{Cl}_{2 m} (\frac{q π}{p}) =

\sum_{k = 0}^{\infty} \frac{\sin [(k p + 1) \frac{q π}{p}]}{(k p + 1)^{2 m}} + \sum_{k = 0}^{\infty} \frac{\sin [(k p + 2) \frac{q π}{p}]}{(k p + 2)^{2 m}} + \sum_{k = 0}^{\infty} \frac{\sin [(k p + 3) \frac{q π}{p}]}{(k p + 3)^{2 m}} + \dots

+ \sum_{k = 0}^{\infty} \frac{\sin [(k p + p - 2) \frac{q π}{p}]}{(k p + p - 2)^{2 m}} + \sum_{k = 0}^{\infty} \frac{\sin [(k p + p - 1) \frac{q π}{p}]}{(k p + p - 1)^{2 m}} + \sum_{k = 0}^{\infty} \frac{\sin [(k p + p) \frac{q π}{p}]}{(k p + p)^{2 m}}

We can index these sums to form a double sum:

{Cl}_{2 m} (\frac{q π}{p}) = \sum_{j = 1}^{p} {\sum_{k = 0}^{\infty} \frac{\sin [(k p + j) \frac{q π}{p}]}{(k p + j)^{2 m}}} =

\sum_{j = 1}^{p} \frac{1}{p^{2 m}} {\sum_{k = 0}^{\infty} \frac{\sin [(k p + j) \frac{q π}{p}]}{(k + (j / p))^{2 m}}}

Applying the addition formula for the Sine function, $\sin (x + y) = \sin x \cos y + \cos x \sin y,$ the sine term in the numerator becomes:

\sin [(k p + j) \frac{q π}{p}] = \sin (k q π + \frac{q j π}{p}) = \sin k q π \cos \frac{q j π}{p} + \cos k q π \sin \frac{q j π}{p}

\sin m π \equiv 0, \cos m π \equiv (- 1)^{m} \Leftrightarrow m = 0, \pm 1, \pm 2, \pm 3, \dots

\sin [(k p + j) \frac{q π}{p}] = (- 1)^{k q} \sin \frac{q j π}{p}

Consequently,

{Cl}_{2 m} (\frac{q π}{p}) = \sum_{j = 1}^{p} \frac{1}{p^{2 m}} \sin (\frac{q j π}{p}) {\sum_{k = 0}^{\infty} \frac{(- 1)^{k q}}{(k + (j / p))^{2 m}}}

To convert the inner sum in the double sum into a non-alternating sum, split in two in parts in exactly the same way as the earlier sum was split into p-parts:

\sum_{k = 0}^{\infty} \frac{(- 1)^{k q}}{(k + (j / p))^{2 m}} = \sum_{k = 0}^{\infty} \frac{(- 1)^{(2 k) q}}{((2 k) + (j / p))^{2 m}} + \sum_{k = 0}^{\infty} \frac{(- 1)^{(2 k + 1) q}}{((2 k + 1) + (j / p))^{2 m}} =

\sum_{k = 0}^{\infty} \frac{1}{(2 k + (j / p))^{2 m}} + (- 1)^{q} \sum_{k = 0}^{\infty} \frac{1}{(2 k + 1 + (j / p))^{2 m}} =

\frac{1}{2^{p}} [\sum_{k = 0}^{\infty} \frac{1}{(k + (j / 2 p))^{2 m}} + (- 1)^{q} \sum_{k = 0}^{\infty} \frac{1}{(k + (\frac{j + p}{2 p}))^{2 m}}]

For $m \in ℤ \geq 1$ , the Polygamma function has the series representation

ψ_{m} (z) = (- 1)^{m + 1} m! \sum_{k = 0}^{\infty} \frac{1}{(k + z)^{m + 1}}

So, in terms of the Polygamma function, the previous inner sum becomes:

\frac{1}{2^{2 m} (2 m - 1)!} [ψ_{2 m - 1} (\frac{j}{2 p}) + (- 1)^{q} ψ_{2 m - 1} (\frac{j + p}{2 p})]

Plugging this back into the double sum gives the desired result:

{Cl}_{2 m} (\frac{q π}{p}) = \frac{1}{(2 p)^{2 m} (2 m - 1)!} \sum_{j = 1}^{p} \sin (\frac{q j π}{p}) [ψ_{2 m - 1} (\frac{j}{2 p}) + (- 1)^{q} ψ_{2 m - 1} (\frac{j + p}{2 p})]

Relation to the Generalized Logsine Integral

The Generalized Logsine Integral is defined by:

ℒ s_{n}^{m} (θ) = - \int_{0}^{θ} x^{m} \log^{n - m - 1} | 2 \sin \frac{x}{2} | d x

In this generalized notation, the Clausen function can be expressed in the form:

{Cl}_{2} (θ) = ℒ s_{2}^{0} (θ)

Kummer's relation

Ernst Kummer and Rogers give the relation

{Li}_{2} (e^{i θ}) = ζ (2) - θ (2 π - θ) / 4 + i {Cl}_{2} (θ)

valid for $0 \leq θ \leq 2 π$ .

Relation to the Lobachevsky function

The Lobachevsky function Λ or Л is essentially the same function with a change of variable:

Λ (θ) = - \int_{0}^{θ} \log | 2 \sin (t) | d t = {Cl}_{2} (2 θ) / 2

though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function

\int_{0}^{θ} \log | \sec (t) | d t = Λ (θ + π / 2) + θ \log 2 .

Relation to Dirichlet L-functions

For rational values of $θ / π$ (that is, for $θ / π = p / q$ for some integers p and q), the function $\sin (n θ)$ can be understood to represent a periodic orbit of an element in the cyclic group, and thus ${Cl}_{s} (θ)$ can be expressed as a simple sum involving the Hurwitz zeta function.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. This allows relations between certain Dirichlet L-functions to be easily computed.

Series acceleration

A series acceleration for the Clausen function is given by

\frac{{Cl}_{2} (θ)}{θ} = 1 - \log | θ | + \sum_{n = 1}^{\infty} \frac{ζ (2 n)}{n (2 n + 1)} {(\frac{θ}{2 π})}^{2 n}

which holds for $| θ | < 2 π$ . Here, $ζ (s)$ is the Riemann zeta function. A more rapidly convergent form is given by

\frac{{Cl}_{2} (θ)}{θ} = 3 - \log [| θ | (1 - \frac{θ^{2}}{4 π^{2}})] - \frac{2 π}{θ} \log (\frac{2 π + θ}{2 π - θ}) + \sum_{n = 1}^{\infty} \frac{ζ (2 n) - 1}{n (2 n + 1)} {(\frac{θ}{2 π})}^{n} .

Convergence is aided by the fact that $ζ (n) - 1$ approaches zero rapidly for large values of n. Both forms are obtainable through the types of resummation techniques used to obtain rational zeta series. (ref. Borwein, etal. 2000, below).

Special values

Some special values include

{Cl}_{2} (\frac{π}{2}) = G

{Cl}_{2} (\frac{π}{3}) = 3 π \log (\frac{G (\frac{2}{3})}{G (\frac{1}{3})}) - 3 π \log Γ (\frac{1}{3}) + π \log (\frac{2 π}{\sqrt{3}})

{Cl}_{2} (\frac{2 π}{3}) = 2 π \log (\frac{G (\frac{2}{3})}{G (\frac{1}{3})}) - 2 π \log Γ (\frac{1}{3}) + \frac{2 π}{3} \log (\frac{2 π}{\sqrt{3}})

{Cl}_{2} (\frac{π}{4}) = 2 π \log (\frac{G (\frac{7}{8})}{G (\frac{1}{8})}) - 2 π \log Γ (\frac{1}{8}) + \frac{π}{4} \log (\frac{2 π}{\sqrt{2 - \sqrt{2}}})

{Cl}_{2} (\frac{3 π}{4}) = 2 π \log (\frac{G (\frac{5}{8})}{G (\frac{3}{8})}) - 2 π \log Γ (\frac{3}{8}) + \frac{3 π}{4} \log (\frac{2 π}{\sqrt{2 + \sqrt{2}}})

{Cl}_{2} (\frac{π}{6}) = 2 π \log (\frac{G (\frac{11}{12})}{G (\frac{1}{12})}) - 2 π \log Γ (\frac{1}{12}) + \frac{π}{6} \log (\frac{2 π \sqrt{2}}{\sqrt{3} - 1})

{Cl}_{2} (\frac{5 π}{6}) = 2 π \log (\frac{G (\frac{7}{12})}{G (\frac{5}{12})}) - 2 π \log Γ (\frac{5}{12}) + \frac{5 π}{6} \log (\frac{2 π \sqrt{2}}{\sqrt{3} + 1})

Generalized special values

Some special values for higher order Clausen functions include

{Cl}_{2 m} (0) = {Cl}_{2 m} (π) = {Cl}_{2 m} (2 π) = 0

{Cl}_{2 m} (\frac{π}{2}) = β (2 m)

{Cl}_{2 m + 1} (0) = {Cl}_{2 m + 1} (2 π) = ζ (2 m + 1)

{Cl}_{2 m + 1} (π) = - η (2 m + 1) = - (\frac{2^{2 m} - 1}{2^{2 m}}) ζ (2 m + 1)

{Cl}_{2 m + 1} (\frac{π}{2}) = - \frac{1}{2^{2 m + 1}} η (2 m + 1) = - (\frac{2^{2 m} - 1}{2^{4 m + 1}}) ζ (2 m + 1)

where : $G = β (2)$ is Catalan's constant, : $β (x)$ is the Dirichlet beta function, : $η (x)$ is the Eta function (also called the alternating Zeta function), and : $ζ (x)$ is the Riemann Zeta function.

β (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{(2 k + 1)^{x}}

Integrals of the direct function

The following integrals are easily proven from the series representations of the Clausen function:

\int_{0}^{θ} {Cl}_{2 m} (x) d x = ζ (2 m + 1) - {Cl}_{2 m + 1} (θ)

\int_{0}^{θ} {Cl}_{2 m + 1} (x) d x = {Cl}_{2 m + 2} (θ)

\int_{0}^{θ} {Sl}_{2 m} (x) d x = {Sl}_{2 m + 1} (θ)

\int_{0}^{θ} {Sl}_{2 m + 1} (x) d x = ζ (2 m + 2) - {Cl}_{2 m + 2} (θ)

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 $G$ (Catalan's constant), $\log 2$ , and the special cases of the Zeta function, $ζ (2)$ and $ζ (3)$ .

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.

\int_{0}^{θ} \log (\sin x) d x = - \frac{1}{2} {Cl}_{2} (2 θ) - θ \log 2

\int_{0}^{θ} \log (\cos x) d x = \frac{1}{2} {Cl}_{2} (π - 2 θ) - θ \log 2

\int_{0}^{θ} \log (\tan x) d x = - \frac{1}{2} {Cl}_{2} (2 θ) - \frac{1}{2} {Cl}_{2} (π - 2 θ)

\int_{0}^{θ} \log (1 + \cos x) d x = 2 {Cl}_{2} (π - θ) - θ \log 2

\int_{0}^{θ} \log (1 - \cos x) d x = - 2 {Cl}_{2} (θ) - θ \log 2

\int_{0}^{θ} \log (1 + \sin x) d x = 2 G - 2 {Cl}_{2} (\frac{π}{2} + θ) - θ \log 2

\int_{0}^{θ} \log (1 - \sin x) d x = - 2 G + 2 {Cl}_{2} (\frac{π}{2} - θ) - θ \log 2

References

Template:AS ref
Template:Cite arXiv
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Leonard Lewin, (Ed.). Structural Properties of Polylogarithms (1991) American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Template:Cite web
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Template:Arxiv
Template:Cite arXiv
Template:Cite web

Odds ratio: Difference between revisions

Revision as of 21:43, 27 January 2014

Contents

Basic properties

General definition

Relation to the Bernoulli Polynomials

Duplication formula

Proof of the Duplication formula

Derivatives of general order Clausen functions

Relation to the Inverse Tangent Integral

Proof of the Inverse Tangent Integral relation

Relation to the Barnes' G-function

Relation to the Polylogarithm

Relation to the Polygamma function

Relation to the Generalized Logsine Integral

Kummer's relation

Relation to the Lobachevsky function

Relation to Dirichlet L-functions

Series acceleration

Special values

Generalized special values

Integrals of the direct function

Integral evaluations involving the direct function

References

Navigation menu

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+[[File:Clausen-function.png|thumbnail|Graph of the Clausen function Cl<sub>2</sub>(θ)]]
+In [[mathematics]], the '''Clausen function''' - introduced by {{harvs|txt|first=Thomas|last=Clausen|authorlink=Thomas Clausen (mathematician)|year=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 special functions. It is intimately connected with the [[Polylogarithm]], [[Inverse tangent integral]], [[Polygamma function]], [[Riemann Zeta function]], [[Eta function]], and [[Dirichlet beta function]].
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+The '''Clausen function of order 2''' - often referred to at ''the'' Clausen function, despite being but one of a class of many - is given by the integral:
+:<math>\operatorname{Cl}_2(\varphi)=-\int_0^{\varphi} \log\Bigg|2\sin\frac{x}{2} \Bigg|\, dx:</math>
+In the range :<math>0 < \varphi < 2\pi\, </math> 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:
+:<math>\operatorname{Cl}_2(\varphi)=\sum_{k=1}^{\infty}\frac{\sin k\varphi}{k^2} = \sin\varphi +\frac{\sin 2\varphi}{2^2}+\frac{\sin 3\varphi}{3^2}+\frac{\sin 4\varphi}{4^2}+ \, \cdots </math>
+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]], Central [[Binomial]] sums, 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 :<math>\pi, \,</math> since if :<math>k\in \mathbb{Z} \, </math> is an integer, :<math>\sin k\pi=0</math>
+:<math>\text{Cl}_2(m\pi) =0, \quad m= 0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots </math>
+It has maxima at :<math>\theta = \frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>
+:<math>\text{Cl}_2\left(\frac{\pi}{3}+2m\pi \right) =1.01494160 \cdots </math>
+and minima at :<math>\theta = -\frac{\pi}{3}+2m\pi \quad[m\in\mathbb{Z}]</math>
+:<math>\text{Cl}_2\left(-\frac{\pi}{3}+2m\pi \right) =-1.01494160 \cdots </math>
+The following properties are immediate consequences of the series definition:
+:<math>\text{Cl}_2(\theta+2m\pi) = \text{Cl}_2(\theta) </math>
+:<math>\text{Cl}_2(-\theta) = -\text{Cl}_2(\theta) </math>
+('''Ref''': See Lu and Perez, 1992, below for these results - although no proofs are given).
+==General definition==
+More generally, one defines the two generalized Clausen functions:
+:<math>\operatorname{S}_z(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta}{k^z}</math>
+:<math>\operatorname{C}_z(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta}{k^z}</math>
+which are valid for complex ''z'' with Re ''z'' &gt;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]]:
+:<math>\operatorname{Cl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}</math>
+:<math>\operatorname{Cl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}</math>
+:<math>\operatorname{Sl}_{2m+2}(\theta) = \sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}</math>
+:<math>\operatorname{Sl}_{2m+1}(\theta) = \sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}</math>
+N.B. The '''SL-type Clausen functions''' have the alternative notation  :<math>\operatorname{Gl}_m(\theta)\, </math> 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 <math>\, \theta\, </math>, and are closely related to the [[Bernoulli polynomials]]. This connection is apparent from the [[Fourier series]] representations of the Bernoulli Polynomials:
+:<math>B_{2n-1}(x)=\frac{2(-1)^n(2n-1)!}{(2\pi)^{2n-1}} \, \sum_{k=1}^{\infty}\frac{\sin 2\pi kx}{k^{2n-1}}</math>
+:<math>B_{2n}(x)=\frac{2(-1)^{n-1}(2n)!}{(2\pi)^{2n}} \, \sum_{k=1}^{\infty}\frac{\cos 2\pi kx}{k^{2n}}</math>
+Setting <math>\, x= \theta/2\pi \, </math> in the above, and then rearranging the terms gives the following closed form (polynomial) expressions:
+:<math>\text{Sl}_{2m}(\theta) = \frac{(-1)^{m-1}(2\pi)^{2m}}{2(2m)!} B_{2m}\left(\frac{\theta}{2\pi}\right)</math>
+:<math>\text{Sl}_{2m-1}(\theta) = \frac{(-1)^{m}(2\pi)^{2m-1}}{2(2m-1)!} B_{2m-1}\left(\frac{\theta}{2\pi}\right)</math>
+Where the [[Bernoulli polynomials]] <math>\, B_n(x)\,</math> are defined in terms of the [[Bernoulli numbers]] <math>\, B_n \equiv B_n(0)\, </math> by the relation:
+:<math>B_n(x)=\sum_{j=0}^n\binom{n}{j} B_jx^{n-j}</math>
+Explicit evaluations derived from the above include:
+:<math> \text{Sl}_1(\theta)= \frac{\pi}{2}-\frac{\theta}{2} </math>
+:<math> \text{Sl}_2(\theta)= \frac{\pi^2}{6}-\frac{\pi\theta}{2}+\frac{\theta^2}{4}  </math>
+:<math> \text{Sl}_3(\theta)= \frac{\pi^2\theta}{6} -\frac{\pi\theta^2}{4}+\frac{\theta^3}{12}  </math>
+:<math> \text{Sl}_4(\theta)= \frac{\pi^4}{90}-\frac{\pi^2\theta^2}{12}+\frac{\pi\theta^3}{12}-\frac{\theta^4}{48} </math>
+==Duplication formula==
+For :<math> 0 < \theta < \pi </math>, the duplication formula can be proven directly from the Integral definition (see also Lu and Perez, 1992, below for the result - although no proof is given):
+:<math>\operatorname{Cl}_{2}(2\theta) = 2\operatorname{Cl}_{2}(\theta) - 2\operatorname{Cl}_{2}(\pi-\theta) </math>
+Immediate consequences of the duplication formula, along with use of the special value :<math>\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=G</math>, include the relations:
+:<math>\operatorname{Cl}_2\left(\frac{\pi}{4}\right)- \operatorname{Cl}_2\left(\frac{3\pi}{4}\right)=\frac{G}{2}</math>
+:<math>2\operatorname{Cl}_2\left(\frac{\pi}{3}\right)= 3\operatorname{Cl}_2\left(\frac{2\pi}{3}\right)</math>
+For higher order Clausen functions, duplication formulae can be obtained from the one given above; simply replace <math> \, \theta \, </math> with the [[dummy variable]] <math>\, x \, </math>, and integrate over the interval <math> \, [0, \theta]. \, </math> Applying the same process repeatedly yields:
+:<math>\operatorname{Cl}_{3}(2\theta) = 4\operatorname{Cl}_{3}(\theta) + 4\operatorname{Cl}_{3}(\pi-\theta) </math>
+:<math>\operatorname{Cl}_{4}(2\theta) = 8\operatorname{Cl}_{4}(\theta) - 8\operatorname{Cl}_{4}(\pi-\theta) </math>
+:<math>\operatorname{Cl}_{5}(2\theta) = 16\operatorname{Cl}_{5}(\theta) + 16 \operatorname{Cl}_{5}(\pi-\theta) </math>
+:<math>\operatorname{Cl}_{6}(2\theta) = 32\operatorname{Cl}_{6}(\theta) - 32 \operatorname{Cl}_{6}(\pi-\theta) </math>
+And more generally, upon induction on <math>\, m, \, \, m \ge 1 </math>
+:<math>\operatorname{Cl}_{m+1}(2\theta) = 2^m\Bigg[\operatorname{Cl}_{m+1}(\theta) + (-1)^m \operatorname{Cl}_{m+1}(\pi-\theta) \Bigg]</math>
+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 <math>\, m \in \mathbb{Z} \ge 1\, </math>
+:<math>\text{Cl}_{2m}\left(\frac{\pi}{2}\right) = 2^{2m-1}\left[\text{Cl}_{2m}\left(\frac{\pi}{4}\right)- \text{Cl}_{2m}\left(\frac{3\pi}{4}\right) \right] = \beta(2m)</math>
+Where <math>\, \beta(x) \, </math> is the [[Dirichlet beta function]].
+==Proof of the Duplication formula==
+From the integral definition,
+:<math>\operatorname{Cl}_2(2\theta)=-\int_0^{2\theta} \log\Bigg| 2 \sin \frac{x}{2} \Bigg| \,dx</math>
+Apply the duplication formula for the [[Sine function]], :<math>\sin 2x = 2\sin\frac{x}{2}\cos\frac{x}{2}</math> to obtain
+:<math>-\int_0^{2\theta} \log\Bigg| \left(2 \sin \frac{x}{4} \right)\left(2 \cos \frac{x}{4}  \right) \Bigg| \,dx=</math>
+:<math>-\int_0^{2\theta} \log\Bigg| 2 \sin \frac{x}{4} \Bigg| \,dx -\int_0^{2\theta} \log\Bigg| 2 \cos \frac{x}{4} \Bigg| \,dx=</math>
+Apply the substitution <math>x=2y, dx=2\, dy</math> on both integrals
+<math>\Rightarrow</math>
+:<math>-2\int_0^{\theta} \log\Bigg| 2 \sin \frac{x}{2} \Bigg| \,dx -2\int_0^{\theta} \log\Bigg| 2 \cos \frac{x}{2} \Bigg| \,dx=</math>
+:<math>2\, \operatorname{Cl}_2(\theta) -2\int_0^{\theta} \log\Bigg| 2 \cos \frac{x}{2} \Bigg| \,dx</math>
+On that last integral, set <math>y=\pi-x, \, x= \pi-y, \, dx = -dy</math>, and use the trigonometric identity <math>\cos(x-y)=\cos x\cos y - \sin x\sin y</math> to show that:
+<math>\cos\left(\frac{\pi-y}{2}\right) = \sin \frac{y}{2} \Rightarrow </math>
+<math>\operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta) -2\int_0^{\theta} \log\Bigg| 2 \cos \frac{x}{2} \Bigg| \,dx=</math>
+<math>2\, \operatorname{Cl}_2(\theta) +2\int_{\pi}^{\pi-\theta} \log\Bigg| 2 \sin \frac{y}{2} \Bigg| \,dy= </math>
+<math>\, \operatorname{Cl}_2(\theta) -2\, \operatorname{Cl}_2(\pi-\theta) + 2\, \operatorname{Cl}_2(\pi)</math>
+<math>\operatorname{Cl}_2(\pi) = 0</math>
+Therefore
+<math>\operatorname{Cl}_2(2\theta)=2\, \operatorname{Cl}_2(\theta)-2\, \operatorname{Cl}_2(\pi-\theta)\, . \, \Box </math>
+==Derivatives of general order Clausen functions==
+Direct differentiation of the [[Fourier series]] expansions for the Clausen functions give:
+:<math>\frac{d}{d\theta}\operatorname{Cl}_{2m+2}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+2}}=\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}=\operatorname{Cl}_{2m+1}(\theta)</math>
+:<math>\frac{d}{d\theta}\operatorname{Cl}_{2m+1}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+1}}=-\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m}}=-\operatorname{Cl}_{2m}(\theta)</math>
+:<math>\frac{d}{d\theta}\operatorname{Sl}_{2m+2}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m+2}}= -\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}=-\operatorname{Sl}_{2m+1} (\theta)</math>
+:<math>\frac{d}{d\theta}\operatorname{Sl}_{2m+1}(\theta) = \frac{d}{d\theta}\sum_{k=1}^\infty \frac{\sin k\theta }{k^{2m+1}}=\sum_{k=1}^\infty \frac{\cos k\theta }{k^{2m}}=\operatorname{Sl}_{2m} (\theta)</math>
+By appealing to the [[First Fundamental Theorem Of Calculus]], we also have:
+:<math>\frac{d}{d\theta}\operatorname{Cl}_2(\theta) = \frac{d}{d\theta} \left[ -\int_0^{\theta} \log \Bigg| 2\sin \frac{x}{2}\Bigg| \,dx \, \right] = - \log \Bigg| 2\sin \frac{\theta}{2}\Bigg| = \operatorname{Cl}_1(\theta) </math>
+==Relation to the Inverse Tangent Integral==
+The [[Inverse tangent integral]] is defined on the interval :<math>0 < z < 1</math> by
+:<math>\operatorname{Ti}_2(z)=\int_0^z \frac{\tan^{-1}x}{x}\,dx = \sum_{k=0}^{\infty}(-1)^k\frac{z^{2k+1}}{(2k+1)^2}</math>
+It has the following closed form in terms of the Clausen Function:
+:<math>\operatorname{Ti}_2(\tan \theta)= \theta\log(\tan \theta) + \frac{1}{2}\operatorname{Cl}_2(2\theta) +\frac{1}{2}\operatorname{Cl}_2(\pi-2\theta)</math>
+==Proof of the Inverse Tangent Integral relation==
+From the integral definition of the [[Inverse tangent integral]], we have
+:<math>\operatorname{Ti}_2(\tan \theta) = \int_0^{\tan \theta}\frac{\tan^{-1}x}{x}\,dx</math>
+Performing an integration by parts
+:<math>\int_0^{\tan \theta}\frac{\tan^{-1}x}{x}\,dx= \tan^{-1}x\log x \, \Bigg|_0^{\tan \theta} - \int_0^{\tan \theta}\frac{\log x}{1+x^2}\,dx=</math>
+:<math>\theta \log{\tan \theta} - \int_0^{\tan \theta}\frac{\log x}{1+x^2}\,dx</math>
+Apply the substitution :<math>x=\tan y,\, y=\tan^{-1}x,\, dy=\frac{dx}{1+x^2}\,</math> to obtain
+:<math>\theta \log{\tan \theta} - \int_0^{\theta}\log(\tan y)\,dy</math>
+For that last integral, apply the transform :<math>y=x/2,\, dy=dx/2\,</math> to get
+:<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(\tan \frac{x}{2}\right)\,dx=</math>
+:<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{\sin (x/2) }{\cos (x/2)}\right)\,dx=</math>
+:<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(\frac{2\sin (x/2) }{2\cos (x/2)}\right)\,dx=</math>
+:<math>\theta \log{\tan \theta} - \frac{1}{2}\int_0^{2\theta}\log\left(2\sin \frac{x}{2}\right)\,dx+ \frac{1}{2}\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx=</math>
+:<math>\theta \log{\tan \theta} +\frac{1}{2}\operatorname{Cl}_2(2\theta)+ \frac{1}{2}\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx</math>
+Finally, as with the proof of the Duplication formula, the substitution <math>x=(\pi-y)\, </math> reduces that last integral to
+:<math>\int_0^{2\theta}\log\left(2\cos \frac{x}{2}\right)\,dx= \operatorname{Cl}_2(\pi-2\theta)- \operatorname{Cl}_2(\pi) = \operatorname{Cl}_2(\pi-2\theta)</math>
+Thus
+:<math>\operatorname{Ti}_2(\tan \theta) = \theta \log{\tan \theta} +\frac{1}{2}\operatorname{Cl}_2(2\theta)+ \frac{1}{2} \operatorname{Cl}_2(\pi-2\theta)\, . \, \Box </math>
+==Relation to the Barnes' G-function==
+For real :<math>0 < z < 1</math>, the Clausen function of second order can be expressed in terms of the [[Barnes G-function]] and (Euler) [[Gamma function]]:
+:<math>\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(1+z)} \right) -2\pi \log \left( \frac{\sin \pi z}{ \pi } \right) </math>
+Or equivalently
+:<math>\operatorname{Cl}_{2}(2\pi z) = 2\pi \log \left( \frac{G(1-z)}{G(z)} \right) -2\pi \log \Gamma(z)-2\pi \log \left( \frac{\sin \pi z}{ \pi } \right) </math>
+Ref: See '''Adamchik''', "Contributions to the Theory of the Barnes function", below.
+==Relation to the Polylogarithm==
+The Clausen functions represent the real and imaginary parts of the Polylogarithm, on the [[Unit Circle]]:
+:<math>\operatorname{Cl}_{2m}(\theta) = \Im (\operatorname{Li}_{2m}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 1</math>
+:<math>\operatorname{Cl}_{2m+1}(\theta) = \Re (\operatorname{Li}_{2m+1}(e^{i \theta})), \quad m\in\mathbb{Z} \ge 0</math>
+This is easily seen by appealing to the series definition of the [[Polylogarithm]].
+:<math>\text{Li}_n(z)=\sum_{k=1}^{\infty}\frac{z^k}{k^n} \quad \Rightarrow  \text{Li}_n\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\left(e^{i\theta}\right)^k}{k^n}= \sum_{k=1}^{\infty}\frac{e^{ik\theta}}{k^n}</math>
+By Euler's Theorem,
+:<math>e^{i\theta} = \cos \theta +i\sin \theta</math>
+and by de Moivre's Theorem ([[DeMoivre's Formula]])
+:<math>(\cos \theta +i\sin \theta)^k= \cos k\theta +i\sin k\theta \quad \Rightarrow \text{Li}_n\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^n}+ i \, \sum_{k=1}^{\infty}\frac{\sin k\theta}{k^n}</math>
+Hence
+:<math>\text{Li}_{2m}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{2m}}+ i \, \sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m}} = \text{Sl}_{2m}(\theta)+i\text{Cl}_{2m}(\theta)</math>
+:<math>\text{Li}_{2m+1}\left(e^{i\theta}\right)=\sum_{k=1}^{\infty}\frac{\cos k\theta}{k^{2m+1}}+ i \, \sum_{k=1}^{\infty}\frac{\sin k\theta}{k^{2m+1}} = \text{Cl}_{2m+1}(\theta)+i\text{Sl}_{2m+1}(\theta)</math>
+==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:
+:<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}(2m-1)!} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right]  </math>
+Let <math>\,p\,</math> and <math>\,q\,</math> be positive integers, such that <math>\,q/p\,</math> is a rational number <math>\,0 < q/p < 1\, </math>, then, by the series definition for the higher order Clausen function (of even index):
+:<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{k=1}^{\infty}\frac{\sin (kq\pi/p)}{k^{2m}} </math>
+We split this sum into exactly '''p'''-parts, so that the first series contains all, and only, those terms congruous to <math>\,kp+1,\, </math> the second series contains all terms congruous to <math>\,kp+2,\, </math> etc, up to the final '''p'''-th part, that contain all terms congruous to <math>\,kp+p\, </math>
+:<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)=</math>
+:<math>\sum_{k=0}^{\infty}\frac{\sin \left[(kp+1)\frac{q\pi}{p}\right]}{(kp+1)^{2m}} + \sum_{k=0}^{\infty}\frac{\sin \left[(kp+2)\frac{q\pi}{p}\right]}{(kp+2)^{2m}} +
+\sum_{k=0}^{\infty}\frac{\sin \left[(kp+3)\frac{q\pi}{p}\right]}{(kp+3)^{2m}} + \, \cdots \, </math>
+:<math>+ \sum_{k=0}^{\infty}\frac{\sin \left[(kp+p-2)\frac{q\pi}{p}\right]}{(kp+p-2)^{2m}} + \sum_{k=0}^{\infty}\frac{\sin \left[(kp+p-1)\frac{q\pi}{p}\right]}{(kp+p-1)^{2m}} +
+\sum_{k=0}^{\infty}\frac{\sin \left[(kp+p)\frac{q\pi}{p}\right]}{(kp+p)^{2m}}</math>
+We can index these sums to form a double sum:
+:<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^{p} \Bigg\{ \sum_{k=0}^{\infty}\frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(kp+j)^{2m}} \Bigg\} =</math>
+:<math>\sum_{j=1}^{p} \frac{1}{p^{2m}}\Bigg\{ \sum_{k=0}^{\infty}\frac{\sin \left[(kp+j)\frac{q\pi}{p}\right]}{(k+(j/p))^{2m}} \Bigg\} </math>
+Applying the addition formula for the [[Sine function]], <math>\,\sin(x+y)=\sin x\cos y+\cos x\sin y,\, </math> the sine term in the numerator becomes:
+:<math>\sin \left[(kp+j)\frac{q\pi}{p}\right]=\sin\left(kq\pi+\frac{qj\pi}{p}\right)=\sin kq\pi \cos \frac{qj\pi}{p}+\cos kq\pi \sin\frac{qj\pi}{p}</math>
+:<math>\sin m\pi \equiv 0, \quad \, \cos m\pi \equiv (-1)^m \quad \Leftrightarrow m=0,\, \pm 1,\, \pm 2,\, \pm 3,\, \cdots </math>
+:<math>\sin \left[(kp+j)\frac{q\pi}{p}\right]=(-1)^{kq}\sin\frac{qj\pi}{p}</math>
+Consequently,
+:<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \sum_{j=1}^{p} \frac{1}{p^{2m}} \sin\left(\frac{qj\pi}{p}\right)\, \Bigg\{ \sum_{k=0}^{\infty}\frac{(-1)^{kq}}{(k+(j/p))^{2m}} \Bigg\}  </math>
+To convert the '''inner sum''' in the double sum into a non-alternating sum, split in two in parts in exactly the same way as the earlier sum was split into '''p'''-parts:
+:<math>\sum_{k=0}^{\infty}\frac{(-1)^{kq}}{(k+(j/p))^{2m}}=\sum_{k=0}^{\infty}\frac{(-1)^{(2k)q}}{((2k)+(j/p))^{2m}}+ \sum_{k=0}^{\infty}\frac{(-1)^{(2k+1)q}}{((2k+1)+(j/p))^{2m}}=</math>
+:<math>\sum_{k=0}^{\infty}\frac{1}{(2k+(j/p))^{2m}}+ (-1)^q\, \sum_{k=0}^{\infty}\frac{1}{(2k+1+(j/p))^{2m}}=</math>
+:<math>\frac{1}{2^p}\left[ \sum_{k=0}^{\infty}\frac{1}{(k+(j/2p))^{2m}}+ (-1)^q\, \sum_{k=0}^{\infty}\frac{1}{(k+\left(\frac{j+p}{2p}\right))^{2m}} \right]</math>
+For <math>\,m \in\mathbb{Z} \ge 1\, </math>, the [[Polygamma function]] has the series representation
+:<math>\psi_{m}(z)=(-1)^{m+1}m! \, \sum_{k=0}^{\infty}\frac{1}{(k+z)^{m+1}} </math>
+So, in terms of the Polygamma function, the previous '''inner sum''' becomes:
+:<math>\frac{1}{2^{2m}(2m-1)!} \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right] </math>
+Plugging this back into the '''double sum''' gives the desired result:
+:<math>\text{Cl}_{2m}\left( \frac{q\pi}{p}\right)= \frac{1}{(2p)^{2m}(2m-1)!} \, \sum_{j=1}^{p} \sin\left(\tfrac{qj\pi}{p}\right)\, \left[\psi_{2m-1}\left(\tfrac{j}{2p}\right)+(-1)^q\psi_{2m-1}\left(\tfrac{j+p}{2p}\right)\right]  </math>
+==Relation to the Generalized Logsine Integral==
+The '''Generalized Logsine''' Integral is defined by:
+:<math>\mathcal{L}s_n^{m}(\theta) = -\int_0^{\theta} x^m \log^{n-m-1} \Bigg| 2\sin\frac{x}{2} \Bigg| \, dx</math>
+In this generalized notation, the Clausen function can be expressed in the form:
+:<math>\text{Cl}_2(\theta) = \mathcal{L}s_2^{0}(\theta) </math>
+==Kummer's relation==
+[[Ernst Kummer]] and Rogers give the relation
+:<math>\operatorname{Li}_2(e^{i \theta}) = \zeta(2) - \theta(2\pi-\theta)/4 + i\operatorname{Cl}_2(\theta)</math>
+valid for <math>0\leq \theta \leq 2\pi</math>.
+==Relation to the Lobachevsky function==
+The '''Lobachevsky function''' &Lambda; or Л is essentially the same function with a change of variable:
+:<math>\Lambda(\theta) = - \int_0^\theta \log|2 \sin(t)| \,dt = \operatorname{Cl}_2(2\theta)/2</math>
+though the name "Lobachevsky function" is not quite historically accurate, as Lobachevsky's formulas for hyperbolic volume used the slightly different function
+:<math>\int_0^\theta \log| \sec(t)| \,dt = \Lambda(\theta+\pi/2)+\theta\log 2.</math>
+==Relation to Dirichlet L-functions==
+For rational values of <math>\theta/\pi</math> (that is, for <math>\theta/\pi=p/q</math> for some integers ''p'' and ''q''), the function <math>\sin(n\theta)</math> can be understood to represent a periodic orbit of an element in the [[cyclic group]], and thus <math>\operatorname{Cl}_s(\theta)</math> can be expressed as a simple sum involving the [[Hurwitz zeta function]].{{citation needed|date=July 2013}} This allows relations between certain [[Dirichlet L-function]]s to be easily computed.
+==Series acceleration==
+A [[series acceleration]] for the Clausen function is given by
+:<math>\frac{\operatorname{Cl}_2(\theta)}{\theta} =
+-\log|\theta| +
+\sum_{n=1}^\infty \frac{\zeta(2n)}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^{2n}
+</math>
+which holds for <math>|\theta|<2\pi</math>. Here, <math>\zeta(s)</math> is the [[Riemann zeta function]]. A more rapidly convergent form is given by
+:<math>\frac{\operatorname{Cl}_2(\theta)}{\theta} =
+-\log\left[|\theta| \left(1-\frac{\theta^2}{4\pi^2}\right)\right]
+-\frac{2\pi}{\theta} \log \left( \frac{2\pi+\theta}{2\pi-\theta}\right)
++\sum_{n=1}^\infty \frac{\zeta(2n)-1}{n(2n+1)} \left(\frac{\theta}{2\pi}\right)^n.
+</math>
+Convergence is aided by the fact that <math>\zeta(n)-1</math> approaches zero rapidly for large values of ''n''.  Both forms are obtainable through the types of resummation techniques used to obtain [[rational zeta series]]. (ref. Borwein, ''etal.'' 2000, below).
+==Special values==
+Some special values include
+:<math>\operatorname{Cl}_2\left(\frac{\pi}{2}\right)=G</math>
+:<math>\operatorname{Cl}_2\left(\frac{\pi}{3}\right)=3\pi \log\left(
+\frac{G\left(\frac{2}{3}\right)}{ G\left(\frac{1}{3}\right)} \right)-3\pi \log
+\Gamma\left(\frac{1}{3}\right)+\pi \log \left(\frac{ 2\pi }{\sqrt{3}}\right)</math>
+:<math>\operatorname{Cl}_2\left(\frac{2\pi}{3}\right)=2\pi \log\left(
+\frac{G\left(\frac{2}{3}\right)}{ G\left(\frac{1}{3}\right)} \right)-2\pi \log
+\Gamma\left(\frac{1}{3}\right)+\frac{2\pi}{3} \log \left(\frac{ 2\pi
+}{\sqrt{3}}\right)</math>
+:<math>\operatorname{Cl}_2\left(\frac{\pi}{4}\right)=
+\pi\log \left( \frac{G\left(\frac{7}{8}\right)}{G\left(\frac{1}{8}\right)} \right) -2\pi
+\log \Gamma\left(\frac{1}{8}\right)+\frac{\pi}{4}\log \left( \frac{2\pi}{\sqrt{2-\sqrt{2}}}
+\right)</math>
+:<math>\operatorname{Cl}_2\left(\frac{3\pi}{4}\right)=
+\pi\log \left( \frac{G\left(\frac{5}{8}\right)}{G\left(\frac{3}{8}\right)} \right) -2\pi
+\log \Gamma\left(\frac{3}{8}\right)+\frac{3\pi}{4}\log \left( \frac{2\pi}{\sqrt{2+\sqrt{2}}}
+\right)</math>
+:<math>\operatorname{Cl}_2\left(\frac{\pi}{6}\right)=
+\pi\log \left( \frac{G\left(\frac{11}{12}\right)}{G\left(\frac{1}{12}\right)} \right) -2\pi
+\log \Gamma\left(\frac{1}{12}\right)+\frac{\pi}{6}\log \left( \frac{2\pi \sqrt{2}
+}{\sqrt{3}-1} \right)</math>
+:<math>\operatorname{Cl}_2\left(\frac{5\pi}{6}\right)=
+\pi\log \left( \frac{G\left(\frac{7}{12}\right)}{G\left(\frac{5}{12}\right)} \right) -2\pi
+\log \Gamma\left(\frac{5}{12}\right)+\frac{5\pi}{6}\log \left( \frac{2\pi \sqrt{2}
+}{\sqrt{3}+1} \right)</math>
+==Generalized special values==
+Some special values for higher order Clausen functions include
+:<math>\operatorname{Cl}_{2m}\left(0\right)=\operatorname{Cl}_{2m}\left(\pi\right)=\operatorname{Cl}_{2m}\left(2\pi\right)=0</math>
+:<math>\operatorname{Cl}_{2m}\left(\frac{\pi}{2}\right)=\beta(2m)</math>
+:<math>\operatorname{Cl}_{2m+1}\left(0\right)=\operatorname{Cl}_{2m+1}\left(2\pi\right)=\zeta(2m+1)</math>
+:<math>\operatorname{Cl}_{2m+1}\left(\pi\right)=-\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^{2m}}\right)\zeta(2m+1)</math>
+:<math>\operatorname{Cl}_{2m+1}\left(\frac{\pi}{2}\right)=-\frac{1}{2^{2m+1}}\eta(2m+1)=-\left(\frac{2^{2m}-1}{2^{4m+1}}\right)\zeta(2m+1)</math>
+where :<math>G = \beta(2)</math> is [[Catalan's constant]], :<math>\beta(x)</math> is the [[Dirichlet beta function]], :<math>\eta(x)</math> is the [[Eta function]] (also called the alternating Zeta function), and :<math>\zeta(x)</math> is the [[Riemann Zeta function]].
+:<math>\beta(x)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^x}</math>
+==Integrals of the direct function==
+The following integrals are easily proven from the series representations of the Clausen function:
+:<math>\int_0^{\theta} \operatorname{Cl}_{2m}(x)\,dx=\zeta(2m+1)-\operatorname{Cl}_{2m+1}(\theta)</math>
+:<math>\int_0^{\theta} \operatorname{Cl}_{2m+1}(x)\,dx=\operatorname{Cl}_{2m+2}(\theta)</math>
+:<math>\int_0^{\theta} \operatorname{Sl}_{2m}(x)\,dx=\operatorname{Sl}_{2m+1}(\theta)</math>
+:<math>\int_0^{\theta} \operatorname{Sl}_{2m+1}(x)\,dx=\zeta(2m+2)-\operatorname{Cl}_{2m+2}(\theta)</math>
+==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 <math>\, G \,</math> ([[Catalan's constant]]), <math>\, \log 2 \,</math>, and the special cases of the [[Zeta function]], <math>\, \zeta(2) \,</math> and <math>\, \zeta(3) \,</math>.
+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.
+:<math>\int_0^{\theta}\log(\sin x)\,dx=-\tfrac{1}{2}\text{Cl}_2(2\theta)-\theta\log 2</math>
+:<math>\int_0^{\theta}\log(\cos x)\,dx=\tfrac{1}{2}\text{Cl}_2(\pi-2\theta)-\theta\log 2</math>
+:<math>\int_0^{\theta}\log(\tan x)\,dx=-\tfrac{1}{2}\text{Cl}_2(2\theta)-\tfrac{1}{2}\text{Cl}_2(\pi-2\theta)</math>
+:<math>\int_0^{\theta}\log(1+\cos x)\,dx=2\text{Cl}_2(\pi-\theta)-\theta\log 2</math>
+:<math>\int_0^{\theta}\log(1-\cos x)\,dx=-2\text{Cl}_2(\theta)-\theta\log 2</math>
+:<math>\int_0^{\theta}\log(1+\sin x)\,dx=2G-2\text{Cl}_2\left(\frac{\pi}{2}+\theta\right)-\theta\log 2</math>
+:<math>\int_0^{\theta}\log(1-\sin x)\,dx=-2G+2\text{Cl}_2\left(\frac{\pi}{2}-\theta\right)-\theta\log 2</math>
+==References==
+* {{AS ref|27.8|1005}}
+* {{cite arXiv| first1=Viktor. S. | last1=Adamchik | eprint=math/0308086v1 | title=Contributions to the Theory of the Barnes Function}}
+*{{Cite journal | last1=Clausen | first1=Thomas | title=Über die Function sin φ + (1/2<sup>2</sup>) sin 2φ +  (1/3<sup>2</sup>) sin 3φ + etc. | url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0008 | year=1832 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=8 | pages=298–300 | ref=harv }}
+* {{cite journal | first1=Van E. | last1=Wood | title=Efficient calculation of Clausen's integral
+|journal=Math. Comp. | year=1968 | volume=22 | number=104 | pages=883–884 | mr=0239733
+|doi = 10.1090/S0025-5718-1968-0239733-9}}
+* Leonard Lewin, (Ed.). ''Structural Properties of Polylogarithms'' (1991)  American Mathematical Society, Providence, RI. ISBN 0-8218-4532-2
+* {{cite journal| first1=Kurt Siegfried | last1=K&ouml;lbig | title=Chebyshev coefficients for the Clausen function Cl<sub>2</sub>(x)
+|journal=J. Comput. Appl. Math. |year=1995 |volume=64 | number=3 |pages=295–297
+|mr=1365432 |doi=10.1016/0377-0427(95)00150-6}}
+* {{cite web|first1=Jonathan M. |last1=Borwein | first2=Armin |last2= Straub | url=http://www.thecarma.net/jon/nielsenrelations.pdf  | title=Relations for Nielsen Polylogarithms}}
+* {{cite journal|first1=Jonathan M. |last1=Borwein | first2=David M. |last2= Bradley |first3=Richard E. |last3=Crandall
+|title=Computational Strategies for the Riemann Zeta Function
+|journal=J. Comp. App. Math.
+|year=2000
+|volume=121 | mr=1780051
+|pages=247–296|ref=harv
+|url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf}}
+* {{cite journal|first1=Mikahil Yu. | last1=Kalmykov | first2=A. | last2=Sheplyakov
+|title=LSJK - a C++ library for arbitrary-precision numeric evaluation of the generalized log-sine integral
+|journal=Comput. Phys. Comm. |year=2005 | volume=172 | pages=45–59
+|doi=10.1016/j.cpc.2005.04.013 }} {{arxiv| archive=hep-ph | id=0411100}}
+* {{cite arXiv| first1=R. J. | last1=Mathar | eprint=1309.7504 | title=A C99 implementation of the Clausen sums}}
+* {{cite web| first1=Hung Jung | last1=Lu | first2=Christopher A. | last2=Perez |url=http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-5809.pdf | title=Massless one-loop scalar three-point integral and associated Clausen, Glaisher, and L-functions | year=1992}}
+[[Category:Zeta and L-functions]]

Odds ratio: Difference between revisions

Revision as of 21:43, 27 January 2014

Basic properties

General definition

Relation to the Bernoulli Polynomials

Duplication formula

Proof of the Duplication formula

Derivatives of general order Clausen functions

Relation to the Inverse Tangent Integral

Proof of the Inverse Tangent Integral relation

Relation to the Barnes' G-function

Relation to the Polylogarithm

Relation to the Polygamma function

Relation to the Generalized Logsine Integral

Kummer's relation

Relation to the Lobachevsky function

Relation to Dirichlet L-functions

Series acceleration

Special values

Generalized special values

Integrals of the direct function

Integral evaluations involving the direct function

References

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