Fungrim home page

Operators

Table of contents: Sums and products - Solutions and zeros - Extreme values - Limits - Derivatives - Integrals - Indefinite integrals - Holomorphic functions - Paths and analytic continuation

Sums and products

044e42
Symbol: Sum nf(n)\sum_{n} f(n) Sum
1e2755
Symbol: Product nf(n)\prod_{n} f(n) Product
9f703a
Symbol: PrimeSum pf(p)\sum_{p} f(p) Sum over primes
2a896d
Symbol: PrimeProduct pf(p)\prod_{p} f(p) Product over primes
8baf79
Symbol: DivisorSum knf(k)\sum_{k \mid n} f(k) Sum over divisors
5830eb
Symbol: DivisorProduct knf(k)\prod_{k \mid n} f(k) Product over divisors

Solutions and zeros

f7ce46
Symbol: Zeros zerosxSf(x)\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x) Zeros (roots) of function
d2714b
Symbol: UniqueZero zero*xSf(x)\mathop{\operatorname{zero*}\,}\limits_{x \in S} f(x) Unique zero (root) of function
5862bb
Symbol: Solutions solutionsxSQ(x)\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x) Solution set
f5ae93
Symbol: UniqueSolution solution*xSQ(x)\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x) Unique solution

Extreme values

6ec976
Symbol: Supremum supxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
bbeb35
Symbol: Infimum infxSf(x)\mathop{\operatorname{inf}}\limits_{x \in S} f(x) Infimum of a set or function
d0cb24
Symbol: Minimum minxSf(x)\mathop{\min}\limits_{x \in S} f(x) Minimum value of a set or function
65ccf2
Symbol: Maximum maxxSf(x)\mathop{\max}\limits_{x \in S} f(x) Maximum value of a set or function
0a3e5a
Symbol: ArgMin arg minxSf(x)\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x) Locations of minimum value
617fe3
Symbol: ArgMax arg maxxSf(x)\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x) Locations of maximum value
f4fbb8
Symbol: ArgMinUnique arg min*xSf(x)\mathop{\operatorname{arg\,min*}}\limits_{x \in S} f(x) Unique location of minimum value
be4926
Symbol: ArgMaxUnique arg max*xSf(x)\mathop{\operatorname{arg\,max*}}\limits_{x \in S} f(x) Unique location of maximum value

Limits

26ea9f
Symbol: Limit limxaf(x)\lim_{x \to a} f(x) Limiting value
1d2ee5
Symbol: SequenceLimit limnaf(n)\lim_{n \to a} f(n) Limiting value of sequence
6fe5c1
Symbol: RealLimit limxaf(x)\lim_{x \to a} f(x) Limiting value, real variable
c8a5f0
Symbol: LeftLimit limxaf(x)\lim_{x \to {a}^{-}} f(x) Limiting value, from the left
afd5ca
Symbol: RightLimit limxa+f(x)\lim_{x \to {a}^{+}} f(x) Limiting value, from the right
05a3ee
Symbol: ComplexLimit limzaf(z)\lim_{z \to a} f(z) Limiting value, complex variable
2be0b5
Symbol: MeromorphicLimit limzaf(z)\lim_{z \to a} f(z) Limiting value, allowing poles
6cd302
Symbol: SequenceLimitInferior lim infnaf(n)\liminf_{n \to a} f(n) Limit inferior of sequence
fdf152
Symbol: SequenceLimitSuperior lim supnaf(n)\limsup_{n \to a} f(n) Limit superior of sequence

Derivatives

1b6a57
Symbol: Derivative ddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Derivative
452407
Symbol: RealDerivative ddxf ⁣(x)\frac{d}{d x}\, f\!\left(x\right) Real derivative
b4b319
Symbol: ComplexDerivative ddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
96f695
Symbol: ComplexBranchDerivative ddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing branch cuts
4c6780
Symbol: MeromorphicDerivative ddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative, allowing poles

Integrals

c285c7
Symbol: Integral abf(x)dx\int_{a}^{b} f(x) \, dx Integral

Indefinite integrals

2e4fbc
Symbol: IndefiniteIntegralEqual f(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral
0be77d
Symbol: RealIndefiniteIntegralEqual f(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral, real derivative
fb2790
Symbol: ComplexIndefiniteIntegralEqual f(x)dx=g(x)+C\int f(x) \, dx = g(x) + \mathcal{C} Indefinite integral, complex derivative

Holomorphic functions

499bdf
Symbol: IsHolomorphic f(z) is holomorphic at z=cf(z) \text{ is holomorphic at } z = c Holomorphic predicate
0895b1
Symbol: IsMeromorphic f(z) is meromorphic at z=cf(z) \text{ is meromorphic at } z = c Meromorphic predicate
231a99
Symbol: ComplexZeroMultiplicity ordz=cf(z)\mathop{\operatorname{ord}}\limits_{z=c} f(z) Multiplicity (order) of complex zero
c54261
Symbol: Residue resz=cf(z)\mathop{\operatorname{res}}\limits_{z=c} f(z) Complex residue

Paths and analytic continuation

4d0e14
Symbol: Path abca \rightsquigarrow b \rightsquigarrow c Line path
bf8f37
Symbol: CurvePath (f(t),t:ab)\left(f(t),\, t : a \rightsquigarrow b\right) Path along a curve
457aaa
Symbol: AnalyticContinuation Continuationz:abf(z)\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.

2019-11-11 15:50:15.016492 UTC