Power

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Exponentiation: raise a base to a power.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaExponentiation↗MathWorldExponentiation↗WikidataQ33456↗FungrimPow↗Wolfram LanguagePower!↗

Power(complex | infinity, complex | signed_infinity) -> numberExponentiation: raise a base to a power.

mpmathenumeratio_pow(complex | infinity, complex | signed_infinity)Juliaenumeratio_pow(complex | infinity, complex | signed_infinity)Rustpower(complex | infinity, complex | signed_infinity)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | signed_infinity) -> number	adelesreplaces numerals	on Adele, Idele, ProfiniteNumber
(complex | infinity, complex | signed_infinity) -> number	analyticreplaces compute-engine	on Around, CenteredInterval, Interval
(complex | infinity, complex | signed_infinity) -> number	compute-engine	always
(complex | infinity, complex | signed_infinity) -> number	evaluationreplaces compute-engine	always
(complex | infinity, complex | signed_infinity) -> number	hypercomplexreplaces analytic	for symbols ^(?:i|j|epsilon|e|f|theta|epsilonSymbol|varepsilon|thetaSymbol|vartheta)_\d+$
(complex | infinity, complex | signed_infinity) -> number	numeralsreplaces residues	on AdicNumeral
(complex | infinity, complex | signed_infinity) -> number	residuesreplaces hypercomplex	on ResidueClass

Domain: Compute engine

Details

• Differs from Wolfram: $Exp(x)\,E^y$ stays a product of two powers; Wolfram combines it into $E^{x+y}$.

Plot

ComplexPlot

real slice

Examples

3 more kept as data, checked against the oracles

Basic #

Power(2, ComplexInfinity) is Indeterminate, the same direction-dependence as e^z

live ↗

Known value · Same reasoning as e^z, for any positive real base != 1

Exact integer power, a 31-digit result

live ↗

In$2^{100}$Out$1\,267\,650\,600\,228\,229\,401\,496\,703\,205\,376$

Scope #

Power(1/2, ComplexInfinity) is a type error, since the exponent excludes ComplexInfinity

live ↗

Power(x, ComplexInfinity) is a type error, since the exponent excludes ComplexInfinity

live ↗

A 50-digit approximation of a rational

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In$N(\frac{248\,832}{1\,419\,857}, 50)$Out$0.175\,251\,451\,378\,554\,319\,202\,567\,582\,510\,069\,676\,030\,755\,209\,855\,64$

Threads over a list of exponents

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In$2^{\bigl\lbrack1, 2, 3\bigr\rbrack}$Out$\bigl\lbrack2, 4, 8\bigr\rbrack$

Interval arithmetic cubes the endpoints

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In$\mathrm{Interval}(\frac{1}{3}, 2)^3$Out$\mathrm{Interval}(\frac{1}{27}, 8)$

Matrix exponents act entry by entry, with exact roots

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In$2^{\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack-\frac{5}{3}, \frac{1}{2}\bigr\rbrack\bigr\rbrack}$Out$\bigl\lbrack\bigl\lbrack\sqrt{2}, \frac{1}{2}\bigr\rbrack, \bigl\lbrack2^{-\frac{5}{3}}, \sqrt{2}\bigr\rbrack\bigr\rbrack$

Continuity of $x^n$ on $x \ge 0$, restated in canonical order

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In$\mathrm{FullSimplify}(\mathrm{FunctionContinuous}(\bigl\lbrack x^{n}, x≥0\bigr\rbrack, x))$Out$\mathrm{FunctionContinuous}(\bigl\lbrack x^{n}, 0≤x\bigr\rbrack, x)$

The power rule: $\frac{d}{dx} x^n = n x^{n-1}$

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In$\frac{\mathrm{d}x^{n}}{\mathrm{d}x}$Out$nx^{n-1}$

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In$\sqrt[3]{-1}$Out$-1$

The matrix power function of a non-diagonal matrix, expressed through its eigenvalues

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\operatorname{\_1}↦2^{\operatorname{\_1}}, \bigl\lbrack\bigl\lbrack1/2, -1\bigr\rbrack, \bigl\lbrack-5/3, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}×2^{(1+(2×15^{1/2})/3)/2}+\frac{1}{2}×2^{(1-(2×15^{1/2})/3)/2}, \frac{1}{10}\sqrt{15}×2^{(1-(2×15^{1/2})/3)/2}-\frac{1}{10}\sqrt{15}×2^{(1+(2×15^{1/2})/3)/2}\bigr\rbrack, \bigl\lbrack\frac{1}{6}\sqrt{15}×2^{(1-(2×15^{1/2})/3)/2}-\frac{1}{6}\sqrt{15}×2^{(1+(2×15^{1/2})/3)/2}, \frac{1}{2}×2^{(1+(2×15^{1/2})/3)/2}+\frac{1}{2}×2^{(1-(2×15^{1/2})/3)/2}\bigr\rbrack\bigr\rbrack$

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In$N(2^{i})$Out$0.769\,238\,901\,363\,972\,126\,578+0.638\,961\,276\,313\,634\,801\,15i$

Possible issues #

Power(1, ComplexInfinity) is Indeterminate: $1^\infty$ is an indeterminate form, as $1^{\pm\infty}$ is

live ↗

Known value · 1^Infinity is an indeterminate form

About

Examples	17
Checked against	Wolfram Language 8 agree9 differSageMath 7 agree1 differ1 unevaluated5 errorSymPy 8 agree6 differ1 unevaluatedmpmath 6 agree1 differ1 unevaluatedJulia 6 agree1 differ1 unevaluatedRust 2 agree4 error
Crosswalk	WikipediaMathWorldWikidataFungrimWolfram LanguagempmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

Python's int negative int is a double (10-29 makes 1 + 256*10**-29 equal 1); the helper keeps it at working precision.

mappedsagepackages/oracle/src/mappings.ts

mappedoscarpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts