hyperbolic function
Enable an AI agent to recognise a hyperbolic function, assess its mathematical and computational validity, and choose justified transformations, evaluations or applications.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
Researched by: Codex
Purpose and description
Enable an AI agent to recognise a hyperbolic function, assess its mathematical and computational validity, and choose justified transformations, evaluations or applications.
It can be Classify a function expression as a canonical hyperbolic member or a parameterised form.; Evaluate values and limits using formulas appropriate to the argument magnitude and proximity to poles.; Rewrite exponential and hyperbolic expressions while preserving domains and parameter assumptions.; Differentiate or integrate expressions with the required chain factors and domain conditions.; Solve equations or select an inverse after checking ranges, injectivity and complex branches.; Check whether a proposed hyperbola parameterisation or differential-equation solution satisfies its defining conditions..
Distinguishing features
A canonical member can be expressed using sinh(z) = (exp(z) - exp(-z))/2 and cosh(z) = (exp(z) + exp(-z))/2, or their specified quotients and reciprocals.
The foundational identity is cosh(z)^2 - sinh(z)^2 = 1; confusing it with the circular identity changes the sign and invalidates deductions.
Canonical hyperbolic functions have no nonzero real period; their complex extensions have imaginary periods, unlike the real periods of circular trigonometric functions.
An inverse hyperbolic function solves for an argument of a hyperbolic function and requires inverse-domain or branch information; it is not a reciprocal family member.
For real t, the pair (cosh(t), sinh(t)) lies on the right branch of the unit hyperbola; this interpretation applies to the pair and does not identify every hyperbolic function with a hyperbola.
Scope
+ Identification of sinh, cosh, tanh, coth, sech and csch, including explicitly parameterised forms
+ Real or complex argument domains, excluded points and output ranges
+ Exponential definitions, hyperbolic identities and relationships among family members
+ Zeros, poles, parity, periodicity, asymptotes and real monotonicity
+ Conditions for differentiation, integration, inversion and stable numerical evaluation
- Circular trigonometric functions as independently modelled functions
- Inverse hyperbolic functions as independently modelled functions with their own branch conventions
- Hyperbolas and other conic sections as geometric objects
- Hyperbolic geometry, manifolds and curvature
- Physical systems or differential equations that use hyperbolic functions
- Software libraries and numerical algorithms as independently maintained implementations
Characteristics
- Canonical family member
- sinh | cosh | tanh | coth | sech | csch Determines the defining expression, singularities and applicable identities.
- Argument setting
- real scalar | complex scalar; other settings require an explicit extension Controls whether ordering, monotonicity, poles and imaginary periodicity are relevant.
- Argument normalisation
- argument expression, parameter meanings and dimensionless normalisation Prevents treating hyperbolic arguments as degree-valued angles or applying identities to mismatched inputs.
- Domain restrictions
- admissible set with excluded points and parameter-dependent restrictions Separates valid operations from evaluation at poles or outside a selected restriction.
- Real output range
- subset of the real numbers for a stated real domain Supports equation feasibility checks and inverse selection.
- Parity
- even | odd | neither on the stated domain Supports symmetry checks and reductions while accounting for shifts or restrictions.
- Singularity structure
- locations and orders of poles; entire when applicable Constrains continuation, simplification and numerical evaluation.
- Injectivity on selected domain
- injective | non-injective | not established Determines whether a single-valued inverse can be used without further restriction.
- Evaluation error
- absolute or relative error bound, with precision and argument region stated Determines whether a computed value is adequate for the intended decision.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 10 layers · 10 findings · 23 questions.
Family identity Establishes which hyperbolic function is represented and how its argument is formed.
Names and notation alone cannot distinguish reciprocals, inverses and transformed functions.
Canonical definition
Connects the named function to its exponential or quotient definition.
Member identification
Record the canonical family member and a defining expression that can be checked.
- Which of sinh, cosh, tanh, coth, sech or csch is represented, and what defining expression establishes that identity? definition
- Does any notation such as sinh^-1 mean a reciprocal, an inverse function or a power in this source? boundary
Argument and transformation
Separates the canonical function from scaling, translation and composition.
Normalised argument
Record the actual argument and external transformations before transferring canonical properties.
- For an expression such as A sinh(bx + c) + d, which symbols are variables and which are fixed parameters? definition
- If the input represents a physical quantity, what scale makes the exponential argument dimensionless? boundary
Domain and singularities Defines where the function exists and which outputs or inverses are admissible.
Quotient family members have excluded points, and real restrictions differ from complex analytic domains.
Admissible inputs
Records the argument setting and exclusions induced by denominators or imposed restrictions.
Domain and poles
Distinguish entire members from meromorphic members and identify restrictions on the represented expression.
- Is the function considered over the reals or complexes, and what additional domain restriction is imposed? boundary
- Which zeros of sinh or cosh produce denominator zeros here, and are they poles or removable singularities of the full expression? definition
Range and inversion
Records attainable values and the restrictions needed to recover an argument.
Inverse admissibility
Make range and injectivity checks explicit before invoking an inverse hyperbolic function.
- What is the output range on the selected real domain, including whether limiting endpoints are attained? boundary
- What domain restriction or linked complex inverse branch makes solving for the argument single-valued? action
Qualitative behaviour Captures symmetry, growth and local behaviour needed to recognise or assess the function.
These properties expose wrong family assignments and determine feasible bounds or solution intervals.
Real behaviour
Describes signs, extrema and trends on connected portions of the real domain.
Real shape
Record parity, zeros, monotonic intervals and asymptotic behaviour for the actual function form.
- What parity, real zeros, sign intervals and extrema hold after accounting for argument and output transformations? definition
- On each connected real-domain interval, where is the function increasing or decreasing, and what limits hold at its boundaries? boundary
Complex behaviour
Describes imaginary periodicity and analytic features unavailable from a real graph.
Imaginary periods and zeros
Record the complex zeros, periods and pole patterns relevant to continuation and equation solving.
- What imaginary periods or sign-changing imaginary shifts does this member have, and how does argument scaling alter them? definition
- Which complex zeros and poles must be included when enumerating solutions or choosing a region of analysis? boundary
Identities and calculus Specifies transformations and calculus operations that preserve mathematical meaning.
Hyperbolic identities have characteristic signs and domain conditions that are easily confused with circular analogues.
Algebraic relations
Connects members through reciprocal, quotient, addition and exponential identities.
Domain-preserving rewrites
Record applicable identities together with denominator exclusions and any continuation introduced by a rewrite.
- Which reciprocal, quotient, addition or double-argument identities justify the proposed rewrite? action
- Does cancellation or conversion to exponentials change the apparent domain, requiring excluded inputs to remain recorded? boundary
Differential relations
Captures derivative, antiderivative and differential-equation relationships.
Calculus preconditions
Record derivative rules, chain factors and conditions on primitives or differential-equation claims.
- What derivative follows for the complete expression, including argument scaling and composition? action
- If an antiderivative uses a logarithm, what real interval or complex branch conditions make the expression valid? boundary
- If the function is used in a solution of y'' = y or a scaled variant, which coefficients and initial conditions establish the claimed solution? definition
Evaluation and use Connects mathematical properties to reliable computation and justified interpretation.
Correct symbolic definitions can still yield unreliable numerical values or unsupported application claims.
Numerical evaluation
Records accuracy needs and argument-dependent computational hazards.
Stable value computation
Assess cancellation near zero, exponential overflow at large arguments and sensitivity near poles.
- What argument region and absolute or relative error tolerance must the computed values support? measurement
- Does direct exponential evaluation risk cancellation, overflow or an indeterminate quotient, and which equivalent formula or series avoids it? action
- Near a zero or pole, how does uncertainty in the argument affect the usefulness of the reported value? measurement
Interpretation and checks
Records the specific role played by the function and checks that can validate that role.
Application contract
Link the function to its application without treating mathematical availability as evidence of physical adequacy.
- Is the function being used as a hyperbola coordinate, a differential-equation solution, a bounded response or another explicitly stated construction? definition
- Which identity, boundary condition or independently computed value can check the proposed use? action
- Which source establishes the interpretation and normalisation of the argument in the linked application? provenance
What the second pass must settle
- Does this registry entry intend the six canonical direct functions only, or also inverse hyperbolic functions under one family-level model?
- Which authoritative references will support the definitions, complex pole patterns, identities and numerical evaluation guidance in a researched publication?
- Should matrix and operator hyperbolic functions be represented as extensions here or linked to separate functional-calculus models?
- What level of parameterised composition remains an instance of this model before ownership passes to a general function or expression model?
- Which numerical precision regimes and application contexts must a completed model support?