Discrete Calculus

Discrete calculus is the calculus of sequences, a.k.a. discrete time signals. Discrete calculus is the foundation for continuous calculus and used to derive numerical algorithms for it. It is the calculus used for discrete-time signal processing, discrete-time control systems and digital image processing. It is also a calculus used for combinatorics, discrete probability, finance and analysis of algorithms.  The Wolfram Language provides extensive coverage of discrete calculus and its applications, including a large library of special sequences and sequence guessing from data; state-of-the-art solvers for sums, difference equations and many more; and full support for its many application domains, from signal processing and control systems to probability and number theory.

Special Sequences »

Fibonacci  ▪  FactorialPower  ▪  BernoulliB  ▪  StirlingS1  ▪  HarmonicNumber  ▪  PolyGamma  ▪  Zeta  ▪  QFactorial  ▪  ...

LinearRecurrence generate a linear recurrence sequence from a kernel

DifferenceRoot symbolic representation of solutions to linear difference equations

Sequence Recognition

FindSequenceFunction find formulas for sequences from data

FindLinearRecurrence  ▪  FindGeneratingFunction

Sequence Visualization »

DiscretePlot plot 1D sequences given by formulas

DiscretePlot3D plot 2D sequences given by formulas

ListPlot, ListStepPlot plot sequences given as lists

Sums & Differences

Sum () symbolic definite and indefinite sums, single and multiple...

NSum numeric approximate sum

AsymptoticSum asymptotic approximate sum

DifferenceDelta () difference , inverse of sum

DifferenceQuotient difference quotients , approximate derivative

SumConvergence test for convergence of a sum

TruncateSum truncate the sums in an expression

Products & Ratios

Product () symbolic definite and indefinite products, single and multiple...

NProduct numeric approximate product

AsymptoticProduct asymptotic approximate product

DiscreteRatio () ratio , inverse of product

Difference Equations

RSolve, RSolveValue symbolic solutions of recurrences and discrete functional equations

RecurrenceTable numeric solutions of recurrences

AsymptoticRSolveValue asymptotic solution of recurrences

DifferenceRoot  ▪  DifferenceRootReduce  ▪  Casoratian

Summation Transforms »

ZTransform  ▪  GeneratingFunction  ▪  FourierSequenceTransform  ▪  ...

Sequence Limits

DiscreteLimit limits of sequences including recurrence and number theory

DiscreteMinLimit, DiscreteMaxLimit lower and upper limits

Sequence Properties »

FunctionPeriod  ▪  FunctionSign  ▪  FunctionMonotonicity  ▪  ...

Sequence Optimization »

Minimize find the global optimum of a sequence

Maximize  ▪  MinValue  ▪  MaxValue  ▪  ArgMin  ▪  ArgMax

Other Sequence Operations

DiscreteShift shift

ContinuedFractionK construct a continued fraction from a formula for terms

Finite Sequence Operations »

Differences  ▪  Ratios  ▪  Accumulate  ▪  Table  ▪  ...

Numerical Discrete Calculus

NSum  ▪  NProduct