Discrete Calculus

With origins stretching back several centuries, discrete calculus is now an increasingly central methodology for many problems related to discrete systems and algorithms. Building on a large body of original research at Wolfram Research, the Wolfram Language for the first time delivers a comprehensive system for discrete calculus.

Symbolic Operations

Sum, Product definite and indefinite sums and products

DiscreteShift shift

DifferenceDelta difference

DiscreteRatio ratio etc

DifferenceQuotient difference quotients etc.

SumConvergence test for convergence of a sum

ContinuedFractionK construct a continued fraction from a formula for terms

Difference Equations

RSolve symbolic solutions of recurrences and discrete functional equations

DifferenceRoot  ▪  DifferenceRootReduce  ▪  Casoratian

RecurrenceTable tables of values from recurrences and functional equations

RSolveValue find an expression for the solution of a difference equation

Summation Transforms »

ZTransform  ▪  GeneratingFunction  ▪  FourierSequenceTransform  ▪  ...

Sequence Limits

DiscreteLimit limits of sequences including recurrence and number theory

DiscreteMinLimit, DiscreteMaxLimit lower and upper limits

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 Properties »

FunctionPeriod  ▪  FunctionSign  ▪  FunctionMonotonicity  ▪  ...

Asymptotics »

AsymptoticSum asymptotic approximation to sum

AsymptoticRSolveValue asymptotic approximation to difference equations

Sequence Recognition

FindSequenceFunction try to find functional forms for sequences

FindLinearRecurrence  ▪  FindGeneratingFunction

Continuous Calculus »

Integrate definite and indefinite integrals

D  ▪  Limit  ▪  DSolve  ▪  ...

Sequence Visualization »

DiscretePlot plot discrete sequences specified by formulas

DiscretePlot3D plot 2D discrete sequences

ListPlot, ListStepPlot plot sequences given as lists

Operations on Explicit Lists »

Differences  ▪  Ratios  ▪  Accumulate  ▪  Table  ▪  ...

Numerical Discrete Calculus

NSum  ▪  NProduct