# Operators

## Sums and products

Symbol: Sum $\sum_{n} f(n)$ Sum
Symbol: Product $\prod_{n} f(n)$ Product
Symbol: PrimeSum $\sum_{p} f(p)$ Sum over primes
Symbol: PrimeProduct $\prod_{p} f(p)$ Product over primes
Symbol: DivisorSum $\sum_{k \mid n} f(k)$ Sum over divisors
Symbol: DivisorProduct $\prod_{k \mid n} f(k)$ Product over divisors

## Solutions and zeros

Symbol: Zeros $\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$ Zeros (roots) of function
Symbol: UniqueZero $\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x)$ Unique zero (root) of function
Symbol: Solutions $\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$ Solution set
Symbol: UniqueSolution $\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$ Unique solution

## Extreme values

Symbol: Supremum $\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$ Supremum of a set or function
Symbol: Infimum $\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$ Infimum of a set or function
Symbol: Minimum $\mathop{\min}\limits_{x \in S} f(x)$ Minimum value of a set or function
Symbol: Maximum $\mathop{\max}\limits_{x \in S} f(x)$ Maximum value of a set or function
Symbol: ArgMin $\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)$ Locations of minimum value
Symbol: ArgMax $\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x)$ Locations of maximum value
Symbol: ArgMinUnique $\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x)$ Unique location of minimum value
Symbol: ArgMaxUnique $\mathop{\operatorname{arg\,max*}}\limits_{x \in S} f(x)$ Unique location of maximum value

## Limits

Symbol: Limit $\lim_{x \to a} f(x)$ Limiting value
Symbol: SequenceLimit $\lim_{n \to a} f(n)$ Limiting value of sequence
Symbol: RealLimit $\lim_{x \to a} f(x)$ Limiting value, real variable
Symbol: LeftLimit $\lim_{x \to {a}^{-}} f(x)$ Limiting value, from the left
Symbol: RightLimit $\lim_{x \to {a}^{+}} f(x)$ Limiting value, from the right
Symbol: ComplexLimit $\lim_{z \to a} f(z)$ Limiting value, complex variable
Symbol: MeromorphicLimit $\lim_{z \to a} f(z)$ Limiting value, allowing poles
Symbol: SequenceLimitInferior $\liminf_{n \to a} f(n)$ Limit inferior of sequence
Symbol: SequenceLimitSuperior $\limsup_{n \to a} f(n)$ Limit superior of sequence

## Derivatives

Symbol: Derivative $\frac{d}{d z}\, f\!\left(z\right)$ Derivative
Symbol: RealDerivative $\frac{d}{d x}\, f\!\left(x\right)$ Real derivative
Symbol: ComplexDerivative $\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Symbol: ComplexBranchDerivative $\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative, allowing branch cuts
Symbol: MeromorphicDerivative $\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative, allowing poles

## Integrals

Symbol: Integral $\int_{a}^{b} f(x) \, dx$ Integral

## Indefinite integrals

Symbol: IndefiniteIntegralEqual $\int f(x) \, dx = g(x) + \mathcal{C}$ Indefinite integral
Symbol: RealIndefiniteIntegralEqual $\int f(x) \, dx = g(x) + \mathcal{C}$ Indefinite integral, real derivative
Symbol: ComplexIndefiniteIntegralEqual $\int f(x) \, dx = g(x) + \mathcal{C}$ Indefinite integral, complex derivative

## Holomorphic functions

Symbol: IsHolomorphic $f(z) \text{ is holomorphic at } z = c$ Holomorphic predicate
Symbol: IsMeromorphic $f(z) \text{ is meromorphic at } z = c$ Meromorphic predicate
Symbol: ComplexZeroMultiplicity $\mathop{\operatorname{ord}}\limits_{z=c} f(z)$ Multiplicity (order) of complex zero
Symbol: Residue $\mathop{\operatorname{res}}\limits_{z=c} f(z)$ Complex residue

## Paths and analytic continuation

Symbol: Path $a \rightsquigarrow b \rightsquigarrow c$ Line path
Symbol: CurvePath $\left(f(t),\, t : a \rightsquigarrow b\right)$ Path along a curve
Symbol: AnalyticContinuation $\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$ Analytic continuation

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC