Mathematical Function Index

Oakfield Mathematical Reference

Structured entry point for mathematical functions used across Oakfield models.

Work in Progress

This reference is actively being developed. More functions, integrators, and detailed mathematical content are being added regularly. Stay tuned for updates!

Trigonometric & Periodic FunctionsGaussian & Window FunctionsRandom & Stochastic FunctionsPolynomial & Power-Law FunctionsExponential & Logarithmic FunctionsHyperbolic & Sigmoid-Type FunctionsFourier & Spectral AnalysisDifferential & Integral OperatorsComplex-Valued FunctionsStatistical & Averaging FunctionsSpecial Functionsq-Analogs & DeformationsHyperexponential & Warp FunctionsNumerical Integration FunctionsUtility & Shaping FunctionsContext APIField APIOperator APIIntegrator APILogging APIExamples

Trigonometric & Periodic Functions

9 entries

• Checkerboard Periodic Patterns

  Two-dimensional periodic tilings alternating sign or phase on adjacent cells.

• Chirps

  Signals whose instantaneous frequency or wavenumber varies over time or space.

• Complex Exponential

  Complex exponential mapping phase variables to unit-magnitude complex values.

• Cosine Function

  Even periodic sinusoid given by the real component of the complex exponential.

• Multi-Tone Sinusoidal Superpositions

  Linear combinations of sinusoids at distinct frequencies producing rich periodic structure.

• Phase-Shifted Sinusoids

  Sinusoids offset by fixed phase to control relative timing and interference.

• Sine Function

  Odd periodic sinusoid defined as the imaginary component of the complex exponential.

• Square Waves

  Piecewise-constant periodic signals alternating between two levels with sharp transitions.

• Standing Waves

  Also: stationary waves

  Superposition of equal and opposite traveling waves producing stationary nodes and antinodes.

Gaussian & Window Functions

7 entries

• Gabor Function

  Gaussian window modulated by a complex exponential to form a localized analytic atom.

• Gaussian Convolution Kernel

  Normalized Gaussian kernel used for smoothing and scale-space filtering.

• Gaussian Function

  Real-valued Gaussian envelope with unit peak amplitude and variance parameter σ^2.

• Gaussian Window

  Unit-energy Gaussian taper used to localize signals in time or space.

• Moving Gaussian Envelope

  Gaussian amplitude profile translated through space or time to track a moving focus.

• Rotated Gaussian

  Anisotropic Gaussian kernel oriented at an angle to model directional smoothing.

• Softplus Function

  Smooth rectifier that approximates a hinge while remaining differentiable.

Random & Stochastic Functions

8 entries

• Box–Muller Gaussian Transform

  Transforms pairs of uniform random variables into independent standard Gaussians.

• Fractional Brownian Motion

  Self-similar Gaussian process with Hurst exponent H controlling long-range dependence.

• Fractional Ornstein–Uhlenbeck Noise

  Mean-reverting stochastic process with long-memory correlations set by a fractional exponent.

• Gaussian White Noise

  Zero-mean generalized Gaussian process with delta-correlated covariance.

• Laplace Distribution

  Double-exponential distribution with sharp peak and heavy tails relative to a Gaussian.

• Ornstein-Uhlenbeck Process

  Also: ou-process

  Mean-reverting Gaussian process driven by white noise with exponential autocorrelation.

• Random Fourier Feature Expansions

  Approximate shift-invariant kernels using randomized sinusoidal basis functions.

• Uniform Random Variables

  Baseline random samples drawn evenly over a finite interval.

Polynomial & Power-Law Functions

5 entries

• Absolute Value

  Maps inputs to their magnitude, discarding sign while preserving scale.

• Binomial Coefficients

  Coefficients counting combinations and weighting polynomial expansions.

• Fractional Finite-Difference Kernels

  Discrete convolution weights that approximate derivatives of non-integer order.

• Fractional Powers

  Raises inputs to non-integer exponents to interpolate between linear and root-like behavior.

• Power Function

  Raises inputs to a real exponent for shaping, scaling, or growth/decay models.

Exponential & Logarithmic Functions

5 entries

• Exponential Function

  Maps inputs to exponential growth or decay with a constant relative rate.

• Log-Absolute Value

  Applies a logarithm to magnitudes to compress dynamic range while ignoring sign.

• Log-Sum-Exp

  Smooth maximum defined as the logarithm of a summed exponential field.

• Logarithm

  Inverse of the exponential function, converting products into sums.

• Stretched Exponential

  Exponentially decaying profile with fractional power on time or space.

Hyperbolic & Sigmoid-Type Functions

3 entries

• Hyperbolic Tangent

  Odd sigmoidal map saturating to ±1 with smooth slope at the origin.

• Smooth Saturation Functions

  Differentiable maps that taper magnitude toward finite bounds.

• Soft Clamping Functions

  Smoothly saturating clamp from lower to upper bounds using logistic transitions.

Fourier & Spectral Analysis

8 entries

• Dispersion Relations

  Relations between frequency and wavenumber describing phase and group propagation.

• Fourier Series

  Representation of periodic functions as sums of trigonometric modes.

• Fourier Transform

  Integral transform mapping functions between physical and frequency domains.

• Inverse Fourier Transform

  Integral transform recovering a function from its frequency-domain representation.

• Spectral Bandwidth

  Root second moment of spectral energy measuring wavenumber spread.

• Spectral Entropy

  Entropy of normalized spectral energy measuring concentration across modes.

• Spectral Filtering

  Multiplier-based attenuation or truncation applied to Fourier coefficients.

• Spectral Phase Modulation

  Phase-only transformation of Fourier coefficients via prescribed phase shifts.

Differential & Integral Operators

8 entries

• Convolution Operators

  Integral transforms combining a signal with a kernel via shifts.

• Finite Differences

  Discrete differential approximations using forward, backward, and central stencils.

• Fractional Integrals

  Nonlocal integrals of order α with power-law kernels generalizing repeated integration.

• Fractional Laplacian

  Nonlocal operator defined via spectral multiplier acting on the Fourier transform.

• Gradient Operators

  Vector of partial derivatives mapping scalar fields to directional rates of change.

• Riemann-Liouville Fractional Derivative

  Nonlocal derivative of order α defined via a singular convolution followed by differentiation.

• Subordination Integrals

  Integral mixtures over operational time linking semigroups via Bernstein densities.

• Temporal Memory Kernels

  Convolutional time operators weighting past states with prescribed kernels.

Complex-Valued Functions

6 entries

• Complex Magnitude

  Euclidean modulus of a complex number or field.

• Complex Multiplication and Division

  Product and quotient of complex numbers via polar form composition.

• Complex Phase

  Principal argument of a complex number capturing angular information.

• Phase Coherence Measures

  Order parameters quantifying alignment of phase angles.

• Phase-Preserving Nonlinearities

  Amplitude-only mappings that retain complex phase.

• Polar-Cartesian Transforms

  Bidirectional conversion between rectangular and polar complex coordinates.

Statistical & Averaging Functions

6 entries

• Energy Norms

  Quadratic norms measuring total energy of scalar or vector fields.

• Exponential Moving Average

  Recursively smoothed average with exponential forgetting.

• Mean

  Arithmetic average of a collection or field.

• Root-Mean-Square

  Quadratic mean measuring magnitude irrespective of sign.

• Variance

  Second central moment measuring spread about the mean.

• Welford Online Statistics

  Single-pass updates for mean and variance with numerical stability.

Special Functions

7 entries

• Digamma Function

  Logarithmic derivative of the Gamma function, encoding harmonic growth and analytic continuation.

• Gamma Function

  Analytic continuation of the factorial, fundamental to special functions and complex analysis.

• Mortici Accelerated Series

  Shifted Stirling-type expansion of log-gamma with improved convergence.

• Polygamma Functions

  Successive derivatives of the digamma capturing higher-order logarithmic curvature.

• Riemann Zeta Function

  Analytic continuation of a Dirichlet series encoding the distribution of prime numbers.

• Special Functions API

  Lua entry points for digamma, trigamma, hyperexponential, and q-deformed functions.

• Stirling Series

  Asymptotic expansion of log-gamma for large arguments using Bernoulli numbers.

q-Analogs & Deformations

4 entries

• q-Digamma Function

  Logarithmic derivative of the q-Gamma function, generalizing ψ to q-deformed calculus.

• q-Exponential

  Basic hypergeometric exponential defined by a q-deformed power series.

• q-Number

  q-deformed scalar reducing to the identity as q → 1.

• q-Zeta Function

  q-analog of the Riemann zeta defined via q-numbers.

Hyperexponential & Warp Functions

7 entries

• Analytic Warp Map

  Complex-analytic coordinate warp defined by a convergent power series around the origin.

• Complex Analytic Warps

  Conformal maps acting on complex planes with holomorphic derivatives.

• Hyperexponential Derivatives

  Parameter derivatives of the hyperexponential warp via polygamma differences.

• Hyperexponential Function

  Finite-sum warp built from shifted reciprocal terms and digamma differences.

• Phase-Space Warps

  Canonical transformations acting on positions and momenta while preserving symplectic structure.

• Power-Law Warp

  Monotone amplitude warp mapping

• Warp Gradients

  Jacobians of warp maps capturing local stretching and contraction.

Numerical Integration Functions

7 entries

• Backward Euler

  Implicit first-order time step evaluating drift at the new level.

• Crank–Nicolson Integration

  Implicit trapezoidal time step averaging drift between levels n and n+1.

• Euler Method

  First-order explicit time step using the current drift.

• Heun Method

  Two-stage explicit trapezoidal integrator with second-order accuracy.

• Implicit Fixed-Point Iteration

  Picard fixed-point solve for implicit timestep equations.

• Runge–Kutta 4

  Classical four-stage explicit integrator with fourth-order accuracy.

• Runge–Kutta–Fehlberg 4(5)

  Embedded fourth–fifth order explicit Runge–Kutta pair with built-in error estimate.

Utility & Shaping Functions

6 entries

• Clamping

  Hard bounding of a value within fixed lower and upper limits.

• Energy Thermostats

  Rescaling steps that drive system energy toward a target level.

• Gain Control

  Amplitude scaling with optional soft knee to prevent overload.

• Linear Interpolation

  Affine blend between two values parameterized by α.

• Normalization

  Rescaling of vectors or fields to unit norm or prescribed magnitude.

• Thresholding

  One-sided cutoff that zeroes or clips values below a threshold.

Context API

5 entries

• Context Creation

  Creating and managing simulation contexts in Oakfield.

• Introspection

  Inspect fields, operators, parameters, and runtime faults inside a context.

• Running Simulations

  Choose between CLI-driven step loops and GUI-driven runs.

• Time & Timestep

  Control timestep policy, read clocks, and inspect per-step metrics.

• Visual Mode

  Switch visualization presets for contexts driving the GUI/vis bridge.

Field API

3 entries

• Field Creation

  Allocate real or complex fields, seed values, and control layout options.

• Field Examples

  Patterns for seeding, inspecting, and reusing fields across operators.

• Field Handles & Methods

  Inspect rank, shape, components, and values from Lua Field userdata.

Operator API

10 entries

• Advection Operators

  Phase-aware analytic warps and spatial derivatives

• Coupling Operators

  Blend two fields with linear, modulation, and nonlinear mixes.

• Custom Operators

  Defining and using user-defined operators.

• Diffusion Operators

  Linear dissipative/dispersion kernels and fractional memory primitives.

• Measurement Operators

  Feature extraction, small-kernel convolution, and spatial smoothing.

• Noise Operators

  Stochastic drivers and procedural noise stimuli.

• Potential & Stimulus Operators

  Sinusoidal, Gaussian, spectral-line, checkerboard, and Gabor stimuli.

• Reaction Operators

  Nonlinear residue measurement between warped and reference fields.

• Thermostat Operators

  Energy regulation via soft lambda or additive/multiplicative feedback.

• Utility Operators

  Utility Operators

Integrator API

2 entries

• Integrator Creation

  Create and attach time integrators for a context.

• Integrator Examples

  Practical integrator setups for fixed and adaptive stepping.

Logging API

1 entries

• Logging & Profiling

  Print structured messages from Lua and emit async logs to disk.

Examples

6 entries

• Advanced Examples

  Heavier Lua orchestration, parameter sweeps, and custom operators.

• Basic Examples

  Smallest possible scripts to learn the API surface.

• Benchmark Examples

  Measure performance and convergence of operators and integrators.

• Intermediate Examples

  Useful patterns that stay readable while showing power.

• Mathematics Examples

  Classic math and recreational patterns built with the API.

• Physics Examples

  Toy setups for diffusion, waves, and energy regulation.