Exp

The natural exponential function $e^x$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaExponential function↗MathWorldExponentialFunction↗WikidataQ168698↗Britannicatopic/exponential-function↗DLMF4.2.E19↗FungrimExp✓↗Fungrim entryWolfram LanguageExp!↗

Exp(x)$e^x$, the natural exponential of x.

SageMathexp(x)✓↗SymPyexp(x)↗mpmathexp(x)↗Juliaenumeratio_ball(exp, x)Rustexp(x)

Typed signatures

Type	Package	Applies
(number) -> number	compute-engine	always

Domain: Elementary functions

Details

• Inverse of Ln: $\ln(e^x) = x$ and $e^{\ln x} = x$ for $x > 0$.
• Once the argument is symbolic, compute-engine represents Exp(x) as Power(ExponentialE, x) -- the same object as $e^x$ written with Power.
• Applied to a matrix (or any nested list) it broadcasts element-wise, same as Wolfram's Exp -- this is NOT the matrix exponential, which is MatrixExp.

Plot

Examples

4 more kept as data, checked against the oracles

Basic #

live ↗

In$e^0$Out$1$

Stays the exact constant rather than decimalizing

live ↗

In$e^1$Out$e$

compute-engine represents Exp(x) as $e^x$ once the argument is symbolic

live ↗

In$e^{x}$Out$e^{x}$

A machine-precision argument evaluates numerically

live ↗

In$e^{3.4}$Out$29.964\,100\,047\,397\,013\,348\,2$

live ↗

In$\operatorname{Series}(e^{x}, x, 0, 10)$Out$1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}+\frac{x^6}{720}+\frac{x^7}{5\,040}+\frac{x^8}{40\,320}+\frac{x^9}{362\,880}+\frac{x^{10}}{3\,628\,800}+O\left(x^{11}\right)$

Scope #

Broadcasts element-wise, giving $\begin{pmatrix}1 & e\\ e & 1\end{pmatrix}$ -- NOT the matrix exponential $\begin{pmatrix}\cosh 1 & \sinh 1\\ \sinh 1 & \cosh 1\end{pmatrix}$, which is MatrixExp

live ↗

In$e^{\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack}$Out$\bigl\lbrack\bigl\lbrack1, e\bigr\rbrack, \bigl\lbrack e, 1\bigr\rbrack\bigr\rbrack$

Arbitrary precision via N(...)

live ↗

In$N(e^{\frac{8}{5}})$Out$4.953\,032\,424\,395\,114\,803\,65$

Evaluates at a complex argument

live ↗

In$e^{2.5+i}$Out$6.582\,229\,578\,192\,772\,984\,4+10.251\,215\,190\,529\,404i$

Euler's identity $e^{i\pi} = -1$

live ↗

In$e^{i π}$Out$-1$

live ↗

In$\sqrt{e}^{i π}$Out$i$

A special imaginary argument folds through $e^{i\theta} = \cos\theta + i\sin\theta$

live ↗

In$e^{\frac{i π}{5}}$Out$\frac{1}{4}+\frac{\sqrt{5}}{4}+\frac{\sqrt{2}}{4}i\sqrt{5-\sqrt{5}}$

live ↗

In$e^{\ln(2)}$Out$2$

Inverse of Ln symbolically

live ↗

In$e^{\ln(x)}$Out$x$

$e^{\ln(1+2i)} = 1+2i$ -- compute-engine's native $e^{\ln z} = z$ fold holds for complex z too

live ↗

In$e^{\ln(1+2i)}$Out$1+2i$

live ↗

In$e^{-∞}$Out$0$

live ↗

In$e^{∞}$Out$∞$

Element-wise over a matrix (not the matrix exponential -- see above)

live ↗

In$e^{\bigl\lbrack\bigl\lbrack1, (i π)/2\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack}$Out$\bigl\lbrack\bigl\lbrack e, i\bigr\rbrack, \bigl\lbrack1, e\bigr\rbrack\bigr\rbrack$

Interval arithmetic: $\exp[-1, \ln 2] = [e^{-1}, 2]$

live ↗

In$e^{\mathrm{Interval}(-1, \ln(2))}$Out$\mathrm{Interval}(\frac{1}{e}, 2)$

Uncertainty propagation: $e^{2 \pm 0.01} \approx 7.389 \pm 0.074$

live ↗

In$e^{\mathrm{Around}(2, 0.01)}$Out$\mathrm{Around}(7.389\,056\,098\,930\,65, 0.073\,890\,560\,989\,306\,5)$

Its own derivative

live ↗

In$\frac{\mathrm{d}e^{x}}{\mathrm{d}x}$Out$e^{x}$

Its own antiderivative

live ↗

In$\int\!e^{x}\, \mathrm{d}x$Out$e^{x}$

Taylor series about 0

live ↗

In$\operatorname{Series}(e^{x}, x, 0, 4)$Out$1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+O\left(x^5\right)$

$N()$ still decimalizes a combined product

live ↗

In$N(e^1e^2)$Out$20.085\,536\,923\,187\,668$

Agrees numerically with the plain product at a complex point

live ↗

In$N(e^{1.3+0.7i}e^{-0.4+2.1i})$Out$-2.317\,493\,000\,510\,215+0.823\,937\,896\,324\,175\,7i$

To 50 significant digits, every one correctly rounded

live ↗

In$N(e^1, 50)$Out$2.718\,281\,828\,459\,045\,235\,360\,287\,471\,352\,662\,497\,757\,247\,093\,7$

Matrix exponential of an upper triangular Jordan-like block

live ↗

In$\mathrm{MatrixFunction}(\mathrm{Exp}, \bigl\lbrack\bigl\lbrack1, (1/2)i π\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e, \frac{1}{2}ie π\bigr\rbrack, \bigl\lbrack0, e\bigr\rbrack\bigr\rbrack$

The domain is all of the reals

live ↗

In$\mathrm{FunctionDomain}(e^{x}, x)$Out$\mathrm{True}$

The range is the positive reals: $y > 0$

live ↗

In$\mathrm{FunctionRange}(e^{x}, x, y)$Out$0\lt y$

Entire, so analytic everywhere

live ↗

In$\mathrm{FunctionAnalytic}(e^{x}, x)$Out$\mathrm{True}$

Strictly increasing everywhere (monotonicity 1)

live ↗

In$\mathrm{FunctionMonotonicity}(e^{x}, x)$Out$1$

One-to-one on the reals

live ↗

In$\mathrm{FunctionInjective}(e^{x}, x)$Out$\mathrm{True}$

Not onto the reals, since it never reaches 0 or below

live ↗

In$\mathrm{FunctionSurjective}(e^{x}, x)$Out$\mathrm{False}$

Always positive, sign 1

live ↗

In$\mathrm{FunctionSign}(e^{x}, x)$Out$1$

No singularities anywhere

live ↗

In$\mathrm{FunctionSingularities}(e^{x}, x)$Out$\mathrm{False}$

No discontinuities anywhere

live ↗

In$\mathrm{FunctionDiscontinuities}(e^{x}, x)$Out$\mathrm{False}$

Convex everywhere (convexity 1)

live ↗

In$\mathrm{FunctionConvexity}(e^{x}, x)$Out$1$

live ↗

In$N(e^{\frac{8}{5}}, 50)$Out$4.953\,032\,424\,395\,114\,803\,654\,286\,356\,423\,964\,256\,413\,031\,123\,556\,6$

live ↗

In$e^{\frac{9}{2}i π}$Out$i$

live ↗

In$\mathrm{ExpToTrig}(e^{(1/5)i π})$Out$\frac{1}{4}+\frac{\sqrt{5}}{4}+\frac{\sqrt{2}}{4}i\sqrt{5-\sqrt{5}}$

live ↗

In$\mathrm{Assuming}(n∈ℤ, \mathrm{Refine}(e^{2πi n}))$Out$e^{2i πn}$

live ↗

In$\mathrm{Assuming}(n∈ℤ, \mathrm{Refine}(e^{2πi(n+1/2)}))$Out$e^{2i π(n+\frac{1}{2})}$

Applications #

Threads element-wise over a list

live ↗

In$e^{\bigl\lbrack0, 1\bigr\rbrack}$Out$\bigl\lbrack1, e\bigr\rbrack$

Properties #

Inverse of Ln: $e^{\ln x} = x$

live ↗

In$e^{\ln(5)}=5$Out$\mathrm{True}$

$e^a \, e^b = e^{a+b}$

live ↗

In$e^2e^3=e^5$Out$\mathrm{True}$

$e$ as the limit $\lim_{x\to\infty}(1 + 1/x)^x$

live ↗

In$\lim_{x→∞}(1+\frac{1}{x})^{x}$Out$e$

$\lim_{x\to0}\frac{e^x - 1}{x} = 1$

live ↗

In$\lim_{x→0}\frac{e^{x}-1}{x}$Out$1$

A product of exponentials combines: $e^x e^y = e^{x+y}$

live ↗

In$e^{x}e^{y}$Out$e^{x+y}$

Combines past two factors

live ↗

In$e^{a}e^{b}e^{c}$Out$e^{a+b+c}$

Combines past extra, non-exponential factors in the product

live ↗

In$2e^{x}e^{y}z$Out$2ze^{x+y}$

$e^x e^{-x}$ cancels to 1, past what compute-engine's own like-term merge catches

live ↗

In$e^{x}e^{-x}$Out$1$

Euler's formula $e^{ix} = \cos x + i\sin x$ via ExpToTrig

live ↗

In$\mathrm{ExpToTrig}(e^{i x})$Out$i\sin(x)+\cos(x)$

Possible issues #

$e^{\ln(-2)}$ is left as $e^{i\pi + \ln 2}$ rather than folding to $-2$

live ↗

In$e^{\ln(-2)}$Out$e^{i π+\ln(2)}$

A lone Exp factor is untouched -- the combine rule needs at least two

live ↗

In$e^{x}y$Out$ye^{x}$

$2^x 2^y$ is also an identity, but this rule only touches ExponentialE -- left to compute-engine's own rules

live ↗

In$2^{x}×2^{y}$Out$2^{x}×2^{y}$

Exp(ComplexInfinity) is Indeterminate -- Exp(z) canonicalizes to Power(ExponentialE, z) before this head's own evaluate ever runs, so the fix lives on Power's exponent side. See Power.

live ↗

Known value · DLMF 4.2, via Power(ExponentialE, ComplexInfinity)

About

Examples	59
Checked against	Wolfram Language 56 agree3 differSageMath 38 agree1 errorSymPy 38 agree2 differ1 unevaluatedmpmath 24 agreeJulia 23 agree3 errorRust 12 agree1 differ3 error
Crosswalk	WikipediaMathWorldWikidataBritannicaDLMFFungrimFungrim entry ×144Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Ln, Log, Sinh, MatrixExp