Clausen function

Defining formula

Record the formula that establishes identity, including order, sign, argument scaling, and normalization.

1. Does the expression match sum_{k>=1} sin(k theta)/k^2, or does it introduce a different sign, scale, or order? definition
2. Which reference establishes the adopted definition and normalization? provenance

Family membership

Treat broader Clausen terminology as a scope decision requiring a defining formula.

1. Does this source use Clausen function for Cl_2 alone, a sine family, or an order-dependent sine/cosine convention? definition
2. Should the cited generalization be represented within this entry or linked to a separately registered concept? boundary

Real representation conditions

Capture the argument interval, convergence justification, and treatment of logarithmic endpoints.

1. On what domain is the selected Fourier series convergent and equal to the intended Clausen function? boundary
2. When extending the logarithmic integral beyond its initial interval, is an absolute value or another explicit convention required? definition

Continuation and branches

Separate the real-argument imaginary-part identity from a proposed complex continuation.

1. Are the assumptions needed to replace Cl_2(theta) by Im Li_2(exp(i theta)) satisfied? action
2. If the argument is complex, which analytic expression, branches, and cuts define the continuation? definition

Argument equivalence

Record the derivation and sign bookkeeping for equivalent real arguments.

1. How do the defining sine series establish oddness and 2*pi periodicity for classical Cl_2? definition
2. Which reduced argument and sign preserve the requested value while improving evaluation? action

Endpoint regularity

Require separate evidence for continuity, differentiability, and local approximations near multiples of 2*pi.

1. What limits and differentiability statements follow from the logarithmic integral at multiples of 2*pi? boundary
2. Which local expansion and remainder control should be used near an endpoint? measurement

Evaluator contract

Record the implementation, precision, input interpretation, and supported function member.

1. Does the selected routine evaluate classical Cl_2 or a generalized family with an explicit order parameter? definition
2. Which method meets the error target for this argument, especially near zeros or after reducing a very large angle? action

Accuracy evidence

Attach meaningful error information and independent checks to evaluated values.

1. Is the uncertainty a rigorous bound, an implementation estimate, or agreement observed at increased precision? measurement
2. Do independent representations agree within the claimed absolute error, including at symmetry-related arguments? measurement

Exact versus approximate

Keep proofs, symbolic representations, and decimal evaluations distinguishable.

1. Which values at integer or rational multiples of pi have an established exact expression under this normalization? definition
2. What derivation or reference proves each claimed relation to a named constant? provenance

Substitution conditions

Preserve integration limits, scale factors, additive constants, and logarithm conventions.

1. Which substitution converts the target logarithmic-trigonometric integral to the adopted Cl_2 definition? action
2. What interval restrictions, endpoint limits, and integration constants must accompany the resulting expression? boundary

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Clausen function this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these kinds and varieties hold for the sense of Clausen function this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Clausen function this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Clausen function this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. 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? provenance