LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle \beta(x;s)= 2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}= \frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix}) }

${\displaystyle \beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi inx)}{(2\pi n)^{s}}}={\frac {2\Gamma (s+1)}{(2\pi )^{s}}}{\mbox{Li}}_{s}(e^{2\pi ix})}$

... is translated to the CAS output ...

Semantic latex: \beta(x ; s) = 2 \EulerGamma@{s + 1} \sum_{n=1}^\infty \frac{\exp(2 \cpi \iunit nx){(2 \cpi n)^s}} = \frac{2 \EulerGamma@{s + 1}}{(2 \cpi)^s}{Li}_s(\expe^{2 \cpi \iunit x})

Confidence: 0.62884815014109

Mathematica

Translation: \[Beta][x ; s] == 2*Gamma[s + 1]*Sum[Divide[Exp[2*Pi*I*n*x]*(2*Pi*n)^(s),==]*Divide[2*Gamma[s + 1],(2*Pi)^(s)]*L*Subscript[i, s]*(Exp[2*Pi*I*x]), {n, 1, Infinity}, GenerateConditions->None]

Information

Sub Equations

\[Beta][x ; s] = 2*Gamma[s + 1]*Sum[Divide[Exp[2*Pi*I*n*x]*(2*Pi*n)^(s),==]*Divide[2*Gamma[s + 1],(2*Pi)^(s)]*L*Subscript[i, s]*(Exp[2*Pi*I*x]), {n, 1, Infinity}, GenerateConditions->None]

Free variables

L

Subscript[i, s]

\[Beta]

s

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Exponential function; Example: \exp@@{z}

Will be translated to: Exp[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 Mathematica: https://reference.wolfram.com/language/ref/Exp.html

Pi was translated to: Pi

Recognizes e with power as the exponential function. It was translated as a function.

Imaginary unit was translated to: I

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: Gamma[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Gamma.html

Tests

Symbolic

Numeric

SymPy

Translation:

Information

Symbol info

(LaTeX -> SymPy) No translation possible for given token: Cannot extract information from feature set: \EulerGamma [\EulerGamma]

Tests

Symbolic

Numeric

Maple

Translation: beta(x ; s) = 2*GAMMA(s + 1)*sum((exp(2*Pi*I*n*x)*(2*Pi*n)^(s))/(=)*(2*GAMMA(s + 1))/((2*Pi)^(s))*L*i[s]*(exp(2*Pi*I*x)), n = 1..infinity)

Information

Sub Equations

beta(x ; s) = 2*GAMMA(s + 1)*sum((exp(2*Pi*I*n*x)*(2*Pi*n)^(s))/(=)*(2*GAMMA(s + 1))/((2*Pi)^(s))*L*i[s]*(exp(2*Pi*I*x)), n = 1..infinity)

Free variables

L

beta

i[s]

s

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Exponential function; Example: \exp@@{z}

Will be translated to: exp($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E19 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=LinearAlgebra/Trace

Pi was translated to: Pi

Recognizes e with power as the exponential function. It was translated as a function.

Imaginary unit was translated to: I

Euler Gamma function; Example: \EulerGamma@{z}

Will be translated to: GAMMA($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$\Gamma$

$\beta$

$1$

${\text{Li}}_{s}(z)$

$n$

$\beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi inx)}{(2\pi n)^{s}}}={\frac {2\Gamma (s+1)}{(2\pi )^{s}}}{\mbox{Li}}_{s}(e^{2\pi ix})$

$s$

Is part of

$\beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi inx)}{(2\pi n)^{s}}}={\frac {2\Gamma (s+1)}{(2\pi )^{s}}}{\mbox{Li}}_{s}(e^{2\pi ix})$

Complete translation information:

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    "SymPy" : {
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  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	In [[mathematics]], the '''Hurwitz zeta function''', named after [[Adolf Hurwitz]], is one of the many [[zeta function]]s. It is formally defined for [[complex number\|complex]] arguments ''s'' with Re(''s'') > 1 and ''q'' with Re(''q'') > 0 by :<math>\zeta(s,q) = \sum_{n=0}^\infty \frac{1}{(n+q)^{s}}.</math> This series is [[absolutely convergent]] for the given values of ''s'' and ''q'' and can be extended to a [[meromorphic function]] defined for all ''s''≠1. The [[Riemann zeta function]] is ζ(''s'',1). [[File:Hurwitza1ov3v2.png\|right\|thumb\|Hurwitz zeta function corresponding to {{nowrap\|1=''q'' = 1/3}}. It is generated as a [[Matplotlib]] plot using a version of the [[Domain coloring]] method.<ref>http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb</ref>]] ==Analytic continuation== [[File:Hurwitza24ov25v2.png\|right\|thumb\|Hurwitz zeta function corresponding to {{nowrap\|1=''q'' = 24/25}}.]] If <math>\mathrm{Re}(s) \leq 1</math> the Hurwitz zeta function can be defined by the equation :<math>\zeta (s,q)=\Gamma(1-s)\frac{1}{2 \pi i} \int_C \frac{z^{s-1}e^{qz}}{1-e^{z}}dz</math> where the [[Contour integration\|contour]] <math>C</math> is a loop around the negative real axis. This provides an analytic continuation of <math>\zeta (s,q)</math>. The Hurwitz zeta function can be extended by [[analytic continuation]] to a [[meromorphic function]] defined for all complex numbers <math>s</math> with <math>s \neq 1</math>. At <math>s = 1</math> it has a [[simple pole]] with [[residue (complex analysis)\|residue]] <math>1</math>. The constant term is given by :<math>\lim_{s\to 1} \left[ \zeta (s,q) - \frac{1}{s-1}\right] = \frac{-\Gamma'(q)}{\Gamma(q)} = -\psi(q)</math> where <math>\Gamma</math> is the [[gamma function]] and <math>\psi</math> is the [[digamma function]]. ==Series representation== [[File:HurwitzofAz3p4j.png\|right\|thumb\| Hurwitz zeta function as a function of ''q'' with {{nowrap\|1=''s'' = 3+4''i''}}.]] A convergent [[Newton series]] representation defined for (real) ''q'' > 0 and any complex ''s'' ≠ 1 was given by [[Helmut Hasse]] in 1930:<ref>{{Citation \|first=Helmut \|last=Hasse \|title=Ein Summierungsverfahren für die Riemannsche ζ-Reihe \|year=1930 \|journal=[[Mathematische Zeitschrift]] \|volume=32 \|issue=1 \|pages=458–464 \|doi=10.1007/BF01194645 \| jfm=56.0894.03 \| url=https://eudml.org/doc/168238 }}</ref> :<math>\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.</math> This series converges uniformly on [[compact subset]]s of the ''s''-plane to an [[entire function]]. The inner sum may be understood to be the ''n''th [[forward difference]] of <math>q^{1-s}</math>; that is, :<math>\Delta^n q^{1-s} = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (q+k)^{1-s}</math> where Δ is the [[forward difference operator]]. Thus, one may write :<math>\begin{align} \zeta(s, q) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n q^{1-s}\\ &= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} q^{1-s} \end{align}</math> Other series converging globally include these examples :<math>\zeta(s,v-1)=\frac{1}{s-1}\sum_{n=0}^\infty H_{n+1}\sum_{k=0}^n (-1)^k \binom{n}{k}(k+v)^{1-s}</math> :<math>\zeta(s,v)=\frac{k!}{(s-k)_k}\sum_{n=0}^\infty \frac{1}{(n+k)!}\left[{n+k \atop n}\right]\sum_{l=0}^{n+k-1}\!(-1)^l \binom{n+k-1}{l} (l+v)^{k-s},\quad k=1, 2, 3,\ldots</math> :<math>\zeta(s,v)=\frac{v^{1-s}}{s-1} + \sum_{n=0}^\infty \|G_{n+1}\| \sum_{k=0}^{n}(-1)^k \binom{n}{k}(k+v)^{-s}</math> :<math>\zeta(s,v)=\frac{(v-1)^{1-s}}{s-1} - \sum_{n=0}^\infty C_{n+1}\sum_{k=0}^{n} (-1)^k \binom{n}{k}(k+v)^{-s}</math> :<math>\zeta(s,v)\big(v-\tfrac12\big)=\frac{s-2}{s-1}\zeta(s-1,v) + \sum_{n=0}^\infty (-1)^n G_{n+2}\sum_{k=0}^{n} (-1)^k \binom{n}{k} (k+v)^{-s}</math> :<math>\zeta(s,v)=-\sum_{l=1}^{k-1} \frac{(k-l+1)_l}{(s-l)_l} \zeta(s-l,v) + \sum_{l=1}^{k} \frac{(k-l+1)_l}{(s-l)_l} v^{l-s} + k \sum_{n=0}^\infty (-1)^n G_{n+1}^{(k)}\sum_{k=0}^{n}(-1)^k \binom{n}{k} (k+v)^{-s}</math> where {{math\|''H''<sub>''n''</sub>}} are the [[Harmonic numbers]], <math>\left[{\cdot \atop \cdot}\right]</math> are the [[Stirling numbers of the first kind]], <math>(\ldots)_{\ldots}</math> is the [[Pochhammer symbol]], {{math\|''G''<sub>''n''</sub>}} are the [[Gregory coefficients]], {{math\|''G''{{su\|b=''n''\|p=(''k'')}}}} are the [[Gregory coefficients]] of higher order and {{math\|''C''<sub>''n''</sub>}} are the Cauchy numbers of the second kind ({{math\|''C''<sub>1</sub> {{=}} 1/2}}, {{math\|''C''<sub>2</sub> {{=}} 5/12}}, {{math\|''C''<sub>3</sub> {{=}} 3/8}},...), see Blagouchine's paper.<ref>{{cite journal \| last = Blagouchine \| first = Iaroslav V. \| arxiv = 1606.02044 \| url = http://math.colgate.edu/~integers/vol18a.html \| journal = INTEGERS: The Electronic Journal of Combinatorial Number Theory \| pages = 1–45 \| title = Three Notes on Ser's and Hasse's Representations for the Zeta-functions \| volume = 18A \| year = 2018\| bibcode = 2016arXiv160602044B}}</ref> ==Integral representation== The function has an integral representation in terms of the [[Mellin transform]] as :<math>\zeta(s,q)=\frac{1}{\Gamma(s)} \int_0^\infty \frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt</math> for <math>\Re s>1</math> and <math>\Re q >0.</math> ==Hurwitz's formula== Hurwitz's formula is the theorem that :<math>\zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s) + e^{i\pi s/2} \beta(1-x;s) \right]</math> where :<math>\beta(x;s)= 2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}= \frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix}) </math> is a representation of the zeta that is valid for <math>0\le x\le 1</math> and s > 1. Here, <math>\text{Li}_s (z)</math> is the [[polylogarithm]]. ==Functional equation== The [[functional equation]] relates values of the zeta on the left- and right-hand sides of the complex plane. For integers <math>1\leq m \leq n </math>, :<math>\zeta \left(1-s,\frac{m}{n} \right) = \frac{2\Gamma(s)}{ (2\pi n)^s } \sum_{k=1}^n \left[\cos \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\; \zeta \left( s,\frac {k}{n} \right)\right] </math> holds for all values of ''s''. ==Some finite sums== Closely related to the functional equation are the following finite sums, some of which may be evaluated in a closed form :<math> \sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right) \cos\dfrac{2\pi rk}{m} =\frac{m \Gamma(1-s)}{(2\pi m)^{1-s}} \sin\frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) + \zeta\left(1-s,1-\frac{k}{m}\right) \right\} - \zeta(s) </math> :<math> \sum_{r=1}^{m-1} \zeta\left(s,\frac{r}{m}\right) \sin\dfrac{2\pi rk}{m}= \frac{m \Gamma(1-s)}{(2\pi m)^{1-s}} \cos \frac{\pi s}{2} \cdot \left\{\zeta\left(1-s,\frac{k}{m}\right) - \zeta\left(1-s,1-\frac{k}{m}\right)\right\} </math> :<math> \sum_{r=1}^{m-1} \zeta^2\left(s,\frac{r}{m}\right) = \big(m^{2s-1}-1 \big)\zeta^2(s) + \frac{2m\Gamma^2(1-s)}{(2\pi m)^{2-2s}} \sum_{l=1}^{m-1} \left\{\zeta\left(1-s,\frac{l}{m}\right) - \cos\pi s \cdot \zeta\left(1-s,1-\frac{l}{m}\right)\right\} \zeta\left(1-s,\frac{l}{m}\right) </math> where ''m'' is positive integer greater than 2 and ''s'' is complex, see e.g. Appendix B in.<ref>{{cite journal\|doi=10.1016/j.jnt.2014.08.009 \|first=I.V. \|last=Blagouchine \|title=A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations \|journal=Journal of Number Theory \|publisher=Elsevier \|volume=148 \|pages=537–592 \|date=2014 \|arxiv=1401.3724}}</ref> ==Taylor series== The derivative of the zeta in the second argument is a [[sheffer sequence\|shift]]: :<math>\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).</math> Thus, the [[Taylor series]] can be written as: :<math>\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!} \frac {\partial^k} {\partial x^k} \zeta (s,x) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).</math> Alternatively, :<math>\zeta(s, q) = \frac{1}{q^s} + \sum_{n=0}^{\infty} (-q)^n {s + n - 1 \choose n} \zeta(s + n),</math> with <math>\|q\| < 1</math>.<ref>{{cite journal \|last=Vepstas \|first=Linas \|authorlink= \|arxiv=math/0702243 \|title=An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions \|class= \|year=2007 \|doi=10.1007/s11075-007-9153-8 \|volume=47 \|issue=3 \|journal=Numerical Algorithms \|pages=211–252\|bibcode=2008NuAlg..47..211V}}</ref> Closely related is the '''Stark–Keiper''' formula: :<math>\zeta(s,N) = \sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right] {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N) </math> which holds for integer ''N'' and arbitrary ''s''. See also [[Faulhaber's formula]] for a similar relation on finite sums of powers of integers. ==Laurent series== The [[Laurent series]] expansion can be used to define [[Stieltjes constants]] that occur in the series :<math>\zeta(s,q)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n(q) \; (s-1)^n.</math> Specifically <math>\gamma_0(q) = -\psi(q)</math> and <math>\gamma_0(1) = -\psi(1) = \gamma_0 = \gamma</math>. ==Fourier transform== The [[discrete Fourier transform]] of the Hurwitz zeta function with respect to the order ''s'' is the [[Legendre chi function]]. ==Relation to Bernoulli polynomials== The function <math>\beta</math> defined above generalizes the [[Bernoulli polynomials]]: :<math>B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right] </math> where <math>\Re z</math> denotes the real part of ''z''. Alternately, :<math>\zeta(-n,x)=-{B_{n+1}(x) \over n+1}.</math> In particular, the relation holds for <math>n=0</math> and one has :<math>\zeta(0,x)= \frac{1}{2} -x.</math> ==Relation to Jacobi theta function== If <math>\vartheta (z,\tau)</math> is the Jacobi [[theta function]], then :<math>\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}= \pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]</math> holds for <math>\Re s > 0</math> and ''z'' complex, but not an integer. For ''z''=''n'' an integer, this simplifies to :<math>\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}= 2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s) =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).</math> where ζ here is the [[Riemann zeta function]]. Note that this latter form is the [[functional equation]] for the Riemann zeta function, as originally given by Riemann. The distinction based on ''z'' being an integer or not accounts for the fact that the Jacobi theta function converges to the periodic [[Dirac delta function\|delta function]], or [[Dirac comb]] in ''z'' as <math>t\rightarrow 0</math>. ==Relation to Dirichlet ''L''-functions== At rational arguments the Hurwitz zeta function may be expressed as a linear combination of [[Dirichlet L-function]]s and vice versa: The Hurwitz zeta function coincides with [[Riemann zeta function\|Riemann's zeta function]] ζ(''s'') when ''q'' = 1, when ''q'' = 1/2 it is equal to (2<sup>''s''</sup>−1)ζ(''s''),<ref name=Dav73/> and if ''q'' = ''n''/''k'' with ''k'' > 2, (''n'',''k'') > 1 and 0 < ''n'' < ''k'', then<ref name=MM13>{{cite web\|last=Lowry\|first=David\|title=Hurwitz Zeta is a sum of Dirichlet L functions, and vice-versa\|url=http://mixedmath.wordpress.com/2013/02/08/hurwitz-zeta-is-a-sum-of-dirichlet-l-functions-and-vice-versa/\|work=mixedmath\|accessdate=8 February 2013}}</ref> :<math>\zeta(s,n/k)=\frac{k^s}{\varphi(k)}\sum_\chi\overline{\chi}(n)L(s,\chi),</math> the sum running over all [[Dirichlet character]]s mod ''k''. In the opposite direction we have the linear combination<ref name=Dav73/> :<math>L(s,\chi)=\frac {1}{k^s} \sum_{n=1}^k \chi(n)\; \zeta \left(s,\frac{n}{k}\right).</math> There is also the [[multiplication theorem]] :<math>k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),</math> of which a useful generalization is the ''distribution relation''<ref>{{cite book \| first1=Daniel S. \| last1=Kubert \| authorlink1=Daniel Kubert \| first2=Serge \| last2=Lang \| authorlink2=Serge Lang \| title=Modular Units \| series= Grundlehren der Mathematischen Wissenschaften \| volume=244 \| publisher=[[Springer-Verlag]] \| year=1981 \| isbn=0-387-90517-0 \| zbl=0492.12002 \| page=13 }}</ref> :<math>\sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^s\,\zeta(s,qa).</math> (This last form is valid whenever ''q'' a natural number and 1 − ''qa'' is not.) ==Zeros== If ''q''=1 the Hurwitz zeta function reduces to the [[Riemann zeta function]] itself; if ''q''=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument ''s'' (''vide supra''), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<''q''<1 and ''q''≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1<Re(''s'')<1+ε for any positive real number ε. This was proved by [[Harold Davenport\|Davenport]] and [[Hans Heilbronn\|Heilbronn]] for rational or transcendental irrational ''q'',<ref>{{Citation \|last=Davenport \|first=H. \|name-list-style=amp \|last2=Heilbronn \|first2=H. \|title=On the zeros of certain Dirichlet series \|journal=[[Journal of the London Mathematical Society]] \|volume=11 \|issue=3 \|year=1936 \|pages=181–185 \|doi=10.1112/jlms/s1-11.3.181 \| zbl=0014.21601 }}</ref> and by [[J. W. S. Cassels\|Cassels]] for algebraic irrational ''q''.<ref name=Dav73>Davenport (1967) p.73</ref><ref>{{Citation \|last=Cassels \|first=J. W. S. \|title=Footnote to a note of Davenport and Heilbronn \|journal=Journal of the London Mathematical Society \|volume=36 \|issue=1 \|year=1961 \|pages=177–184 \|doi=10.1112/jlms/s1-36.1.177 \| zbl=0097.03403 }}</ref> ==Rational values== The Hurwitz zeta function occurs in a number of striking identities at rational values.<ref>Given by {{Citation \|first=Djurdje \|last=Cvijović \|name-list-style=amp \|first2=Jacek \|last2=Klinowski \|title=Values of the Legendre chi and Hurwitz zeta functions at rational arguments \|journal=Mathematics of Computation \|volume=68 \|issue=228 \|year=1999 \|pages=1623–1630 \|doi=10.1090/S0025-5718-99-01091-1\|bibcode=1999MaCom..68.1623C \|doi-access=free }}</ref> In particular, values in terms of the [[Euler polynomial]]s <math>E_n(x)</math>: :<math>E_{2n-1}\left(\frac{p}{q}\right) = (-1)^n \frac{4(2n-1)!}{(2\pi q)^{2n}} \sum_{k=1}^q \zeta\left(2n,\frac{2k-1}{2q}\right) \cos \frac{(2k-1)\pi p}{q}</math> and :<math>E_{2n}\left(\frac{p}{q}\right) = (-1)^n \frac{4(2n)!}{(2\pi q)^{2n+1}} \sum_{k=1}^q \zeta\left(2n+1,\frac{2k-1}{2q}\right) \sin \frac{(2k-1)\pi p}{q}</math> One also has :<math>\zeta\left(s,\frac{2p-1}{2q}\right) = 2(2q)^{s-1} \sum_{k=1}^q \left[ C_s\left(\frac{k}{q}\right) \cos \left(\frac{(2p-1)\pi k}{q}\right) + S_s\left(\frac{k}{q}\right) \sin \left(\frac{(2p-1)\pi k}{q}\right) \right]</math> which holds for <math>1\le p \le q</math>. Here, the <math>C_\nu(x)</math> and <math>S_\nu(x)</math> are defined by means of the [[Legendre chi function]] <math>\chi_\nu</math> as :<math>C_\nu(x) = \operatorname{Re}\, \chi_\nu (e^{ix})</math> and :<math>S_\nu(x) = \operatorname{Im}\, \chi_\nu (e^{ix}).</math> For integer values of ν, these may be expressed in terms of the Euler polynomials. These relations may be derived by employing the functional equation together with Hurwitz's formula, given above. ==Applications== Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in [[number theory]], where its theory is the deepest and most developed. However, it also occurs in the study of [[fractals]] and [[dynamical systems]]. In applied [[statistics]], it occurs in [[Zipf's law]] and the [[Zipf–Mandelbrot law]]. In [[particle physics]], it occurs in a formula by [[Julian Schwinger]],<ref>{{Citation \|last=Schwinger \|first=J. \|title=On gauge invariance and vacuum polarization \|journal=[[Physical Review]] \|volume=82 \|issue=5 \|year=1951 \|pages=664–679 \|doi=10.1103/PhysRev.82.664 \|bibcode=1951PhRv...82..664S}}</ref> giving an exact result for the [[pair production]] rate of a [[Paul Dirac\|Dirac]] [[Dirac equation#Electromagnetic interaction\|electron]] in a uniform electric field. ==Special cases and generalizations== The Hurwitz zeta function with a positive integer ''m'' is related to the [[polygamma function]]: :<math>\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z) \ .</math> For negative integer −''n'' the values are related to the [[Bernoulli polynomials]]:<ref name=Ap264>Apostol (1976) p.264</ref> :<math>\zeta(-n,x) = - \frac{B_{n+1}(x)}{n+1} \ . </math> The [[Barnes zeta function]] generalizes the Hurwitz zeta function. The [[Lerch transcendent]] generalizes the Hurwitz zeta: :<math>\Phi(z, s, q) = \sum_{k=0}^\infty \frac { z^k} {(k+q)^s}</math> and thus :<math>\zeta (s,q)=\Phi(1, s, q).\,</math> [[Hypergeometric function]] :<math>\zeta(s,a)=a^{-s}\cdot{}_{s+1}F_s(1,a_1,a_2,\ldots a_s;a_1+1,a_2+1,\ldots a_s+1;1)</math> where <math>a_1=a_2=\ldots=a_s=a\text{ and }a\notin\N\text{ and }s\in\N^+.</math> [[Meijer G-function]] :<math>\zeta(s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1 \; \left\| \; \begin{matrix}0,1-a,\ldots,1-a\\0,-a,\ldots,-a\end{matrix}\right)\right.\qquad\qquad s\in\N^+.</math> ==Notes== <references/> ==References== {{dlmf\|id=25.11\|first=T. M. \|last=Apostol}} See chapter 12 of {{Apostol IANT}} * Milton Abramowitz and Irene A. Stegun, ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions]]'', (1964) Dover Publications, New York. {{ISBN\|0-486-61272-4}}. ''(See [http://www.math.sfu.ca/~cbm/aands/page_260.htm Paragraph 6.4.10] for relationship to polygamma function.)'' * {{cite book \| last=Davenport \| first=Harold \| authorlink=Harold Davenport \| title=Multiplicative number theory \| publisher=Markham \| series=Lectures in advanced mathematics \| volume=1 \| location=Chicago \| year=1967 \| zbl=0159.06303 }} * {{cite journal \|first1=Jeff \|last1=Miller \|first2=Victor S. \|last2=Adamchik \|url=http://www-2.cs.cmu.edu/~adamchik/articles/hurwitz.htm \|title= Derivatives of the Hurwitz Zeta Function for Rational Arguments \|journal= Journal of Computational and Applied Mathematics \|volume=100 \|issue=2 \|year=1998 \|pages=201–206 \|doi=10.1016/S0377-0427(98)00193-9 \|doi-access=free }} * {{cite web \|first1=Linas \|last1=Vepstas \|url=http://www.linas.org/math/gkw.pdf \|title= The Bernoulli Operator, the Gauss–Kuzmin–Wirsing Operator, and the Riemann Zeta }} * {{cite journal \|first1=István \|last1=Mező \|first2=Ayhan \|last2=Dil \|doi=10.1016/j.jnt.2009.08.005 \|title=Hyperharmonic series involving Hurwitz zeta function \|journal=Journal of Number Theory \|year=2010 \|volume=130 \|number=2 \|pages=360–369 \|hdl=2437/90539 \|hdl-access=free }} ==External links== * {{mathworld\|urlname=HurwitzZetaFunction\|title=Hurwitz Zeta Function\|author=Jonathan Sondow and Eric W. Weisstein}} [[Category:Zeta and L-functions]]
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