{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.clausen-function","code":"thing-q823290","url":"https://ver.cy/models/thing/q823290/","name":"Clausen function","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"The Clausen function usually denotes the real, odd, 2π-periodic special function Cl₂(θ) = Σₙ₌₁∞ sin(nθ)/n², equivalently the imaginary part of the dilogarithm Li₂(eⁱθ).","purpose":"Enable an agent to recognize Clausen-function expressions, resolve their conventions, and select justified symbolic or numerical operations.","scope":{"in":["Identification of the classical Cl_2 function and its defining normalization","Fourier-series, logarithmic-integral, and dilogarithm representations","Real-argument domains, periodicity, symmetry, and endpoint behavior","Evaluation methods, error requirements, and validation identities","Notation and scope decisions for generalized Clausen functions"],"out":["The general theory of polylogarithms and dilogarithms","Clausen's hypergeometric identities","Independent models of Catalan's constant and other special values","Full models of physical or geometric problems using Clausen functions","General-purpose numerical-library architecture"],"boundaries":[]},"distinguishingFeatures":["For real theta, the classical function has the identifying series Cl_2(theta) = sum_{k>=1} sin(k theta)/k^2; record both the sine numerator and order-two denominator. See [mpmath definitions](https://mpmath.org/doc/current/functions/zeta.html#clsin).","On 0 < theta < 2*pi, it equals -integral_0^theta log(2*sin(t/2)) dt, with normalization Cl_2(0)=0. See [GSL Clausen functions](https://www.gnu.org/software/gsl/doc/html/specfunc.html#clausen-functions).","For real theta, it is Im Li_2(exp(i theta)), rather than the full complex dilogarithm. See [GSL Clausen functions](https://www.gnu.org/software/gsl/doc/html/specfunc.html#clausen-functions).","A generalized sine function, a complementary cosine function, and the classical Cl_2 require explicit differentiation by formula and order. See [mpmath Clausen functions](https://mpmath.org/doc/current/functions/zeta.html#clsin)."],"structure":{"bundles":[{"id":"identity-and-conventions","name":"Identity and conventions","description":"Determine exactly which function a Clausen label denotes.","layers":[{"id":"classical-definition","name":"Classical definition","description":"Establish the identifying formula and normalization of Cl_2.","findings":[{"id":"defining-formula","name":"Defining formula","description":"Record the formula that establishes identity, including order, sign, argument scaling, and normalization.","questions":[{"text":"Does the expression match sum_{k>=1} sin(k theta)/k^2, or does it introduce a different sign, scale, or order?","id":"defining-formula-q01","kind":"definition"},{"text":"Which reference establishes the adopted definition and normalization?","id":"defining-formula-q02","kind":"provenance"}]}]},{"id":"generalized-notation","name":"Generalized notation","description":"Resolve higher-order and sine-versus-cosine naming conventions.","findings":[{"id":"family-membership","name":"Family membership","description":"Treat broader Clausen terminology as a scope decision requiring a defining formula.","questions":[{"text":"Does this source use Clausen function for Cl_2 alone, a sine family, or an order-dependent sine/cosine convention?","id":"family-membership-q01","kind":"definition"},{"text":"Should the cited generalization be represented within this entry or linked to a separately registered concept?","id":"family-membership-q02","kind":"boundary"}]}]}]},{"id":"representations-and-validity","name":"Representations and validity","description":"Connect interchangeable formulas while preserving their mathematical conditions.","layers":[{"id":"series-and-integral","name":"Series and integral","description":"Specify convergence and logarithm conventions for real-variable definitions.","findings":[{"id":"real-representation-conditions","name":"Real representation conditions","description":"Capture the argument interval, convergence justification, and treatment of logarithmic endpoints.","questions":[{"text":"On what domain is the selected Fourier series convergent and equal to the intended Clausen function?","id":"real-representation-conditions-q01","kind":"boundary"},{"text":"When extending the logarithmic integral beyond its initial interval, is an absolute value or another explicit convention required?","id":"real-representation-conditions-q02","kind":"definition"}]}]},{"id":"dilogarithm-connection","name":"Dilogarithm connection","description":"Control use of unit-circle polylogarithm representations.","findings":[{"id":"continuation-and-branches","name":"Continuation and branches","description":"Separate the real-argument imaginary-part identity from a proposed complex continuation.","questions":[{"text":"Are the assumptions needed to replace Cl_2(theta) by Im Li_2(exp(i theta)) satisfied?","id":"continuation-and-branches-q01","kind":"action"},{"text":"If the argument is complex, which analytic expression, branches, and cuts define the continuation?","id":"continuation-and-branches-q02","kind":"definition"}]}]}]},{"id":"real-argument-behavior","name":"Real-argument behavior","description":"Describe transformations and local behavior relevant to real Cl_2 arguments.","layers":[{"id":"symmetry-and-reduction","name":"Symmetry and reduction","description":"Establish periodicity and reflection rules from the adopted definition.","findings":[{"id":"argument-equivalence","name":"Argument equivalence","description":"Record the derivation and sign bookkeeping for equivalent real arguments.","questions":[{"text":"How do the defining sine series establish oddness and 2*pi periodicity for classical Cl_2?","id":"argument-equivalence-q01","kind":"definition"},{"text":"Which reduced argument and sign preserve the requested value while improving evaluation?","id":"argument-equivalence-q02","kind":"action"}]}]},{"id":"logarithmic-endpoints","name":"Logarithmic endpoints","description":"Distinguish finite function values from singular derivative behavior.","findings":[{"id":"endpoint-regularity","name":"Endpoint regularity","description":"Require separate evidence for continuity, differentiability, and local approximations near multiples of 2*pi.","questions":[{"text":"What limits and differentiability statements follow from the logarithmic integral at multiples of 2*pi?","id":"endpoint-regularity-q01","kind":"boundary"},{"text":"Which local expansion and remainder control should be used near an endpoint?","id":"endpoint-regularity-q02","kind":"measurement"}]}]}]},{"id":"numerical-evaluation","name":"Numerical evaluation","description":"Make computed Clausen values reproducible and assessable.","layers":[{"id":"method-selection","name":"Method selection","description":"Choose an evaluator compatible with the function convention and argument regime.","findings":[{"id":"evaluator-contract","name":"Evaluator contract","description":"Record the implementation, precision, input interpretation, and supported function member.","questions":[{"text":"Does the selected routine evaluate classical Cl_2 or a generalized family with an explicit order parameter?","id":"evaluator-contract-q01","kind":"definition"},{"text":"Which method meets the error target for this argument, especially near zeros or after reducing a very large angle?","id":"evaluator-contract-q02","kind":"action"}]}]},{"id":"error-and-cross-checks","name":"Error and cross-checks","description":"Distinguish reported digits from verified accuracy.","findings":[{"id":"accuracy-evidence","name":"Accuracy evidence","description":"Attach meaningful error information and independent checks to evaluated values.","questions":[{"text":"Is the uncertainty a rigorous bound, an implementation estimate, or agreement observed at increased precision?","id":"accuracy-evidence-q01","kind":"measurement"},{"text":"Do independent representations agree within the claimed absolute error, including at symmetry-related arguments?","id":"accuracy-evidence-q02","kind":"measurement"}]}]}]},{"id":"symbolic-use-and-special-values","name":"Symbolic use and special values","description":"Support exact transformations and use in larger mathematical expressions.","layers":[{"id":"special-value-evidence","name":"Special-value evidence","description":"Record exact values and relations at distinguished arguments.","findings":[{"id":"exact-versus-approximate","name":"Exact versus approximate","description":"Keep proofs, symbolic representations, and decimal evaluations distinguishable.","questions":[{"text":"Which values at integer or rational multiples of pi have an established exact expression under this normalization?","id":"exact-versus-approximate-q01","kind":"definition"},{"text":"What derivation or reference proves each claimed relation to a named constant?","id":"exact-versus-approximate-q02","kind":"provenance"}]}]},{"id":"integral-transformations","name":"Integral transformations","description":"Use the logarithmic-integral representation in a surrounding calculation.","findings":[{"id":"substitution-conditions","name":"Substitution conditions","description":"Preserve integration limits, scale factors, additive constants, and logarithm conventions.","questions":[{"text":"Which substitution converts the target logarithmic-trigonometric integral to the adopted Cl_2 definition?","id":"substitution-conditions-q01","kind":"action"},{"text":"What interval restrictions, endpoint limits, and integration constants must accompany the resulting expression?","id":"substitution-conditions-q02","kind":"boundary"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of Clausen function this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of Clausen function this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of Clausen function this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of Clausen function this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of Clausen function this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Evaluate and explain the Clausen function, its series and its relation to the dilogarithm.","Check numerical values against reliable references."],"mustNot":["Present an unverified identity involving the function as proven.","Use a slowly converging series without error bounds in a result others rely on.","Confuse the two-index Clausen functions or the standard and alternate notations."],"requiresHuman":["Publishing a new result about the function under a person's name."]},"ethics":{"considerations":["Accurate attribution and verified results matter in mathematical work."],"affectedParties":["Students","Mathematicians","Physicists using its values"]},"owners":{"steward":"Nobody owns it; it is a mathematical function.","roles":[],"masterSystems":[]},"relations":[{"target":"Q11348","type":"parent","note":"registry parent class"},{"target":"Q134326403","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.clausen-function","Wikidata: Q823290 (https://www.wikidata.org/wiki/Q823290)"]},"properties":{"applicability":"not-applicable","items":["Function convention: classical Cl_2; generalized sine; generalized cosine; other explicitly sourced convention","Order: dimensionless; 2 for the classical function; otherwise explicitly specified","Argument: real angle in radians; complex argument only under a declared continuation","Representation: Fourier series; logarithmic integral; polylogarithmic expression; numerical approximation","Continuation convention: real-domain definition; specified complex branch; unresolved","Evaluation accuracy: absolute error; relative error where meaningful; working precision","Defining authority: reference or implementation documenting the exact convention"]},"recognition":{"applicability":"optional","items":["Written Cl2 of theta, a periodic wave-like curve that is zero at multiples of pi."]},"capabilities":{"applicability":"required","items":["Recognize a Clausen expression from its formula and resolve ambiguous notation.","Convert between representations after checking argument and branch conditions.","Reduce real arguments using justified symmetry and periodicity.","Evaluate a specified function member to a declared error target.","Validate computed values against independent representations and special cases.","Express suitable logarithmic-trigonometric integrals using Cl_2 with explicit substitutions and limits."]},"hazards":{"applicability":"required","items":["Truncated series give inaccurate values near the peaks."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["The Clausen function occurs in special-function theory, Fourier analysis and evaluations of definite integrals.","It helps express results involving dilogarithms on the unit circle, with the order and convention checked before symbolic or numerical evaluation."]}},"sources":[{"title":"Wikidata item Q823290: Clausen function","url":"https://www.wikidata.org/wiki/Q823290","note":"identity and sense of the item"},{"title":"Wikipedia: Clausen function","url":"https://en.wikipedia.org/wiki/Clausen_function","note":"general description of the item"}],"openQuestions":["Does the registry's intended source identify this entry with classical Cl_2 alone or the broader Clausen family?","Which higher-order naming convention should be canonical when references use different sine/cosine conventions?","Should complex continuations be owned by this model, and which branch convention would be authoritative?","Which independently verified special values, identities, and numerical error guarantees are sufficient for publication?","Does an existing Vercy world model already cover this concept and require this registry entry to link to it?"],"resources":{"spec":"/models/things/publications/thing-q823290/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q823290/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T08:45:01Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:33+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No specific standard, law or official classification applies to this mathematical function."},"score":1.0}}