ExpIntegralE

The generalized exponential integral $E_n(z) = \int_1^\infty e^{-zt}/t^n\,dt$. Built on this package's generalized incomplete Gamma: $E_n(z) = z^{n-1}\,\Gamma(1-n, z)$.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageExpIntegralE!↗

ExpIntegralE(n, z)$E_n(z)$, for any order $n$ (real, complex, or non-integer).

SageMathexp_integral_e(n, z)SymPyexpint(n, z)mpmathexpint(n, z)Juliaenumeratio_ball(exp_integral_e, n, z)Rustexp_integral_e(n, z)

Typed signatures

Type	Package	Applies
(number, number) -> number	analytic	always

Domain: Special functions

Details

• compute-engine has no ExpIntegralE; this reduces it entirely to Gamma's generalized incomplete form, already extended here for complex operands.
• Two identities are kept exact ahead of the general formula, which hits a genuine $0 \cdot \infty$ at each: $E_0(z) = e^{-z}/z$ (since $\Gamma(1,z) = e^{-z}$ exactly, symbolic $z$ included), and $E_n(0) = 1/(n-1)$ for $\operatorname{Re}(n) > 1$ — the removable limit the $z^{n-1}$ factor can't see through when $z$ actually is 0.
• Non-integer order (e.g. $n = 1/2$) works the same way, since the incomplete Gamma it reduces to does.
• N(x, d) past a double's digits sums the convergent series for integer $n \ge 1$ and real $0 < z \le 100$, in enough extra digits to cover its cancellation; other arguments stay unevaluated there.
• FunctionExpand rewrites $E_n$ by order, as Wolfram does: $e^{-z}$ times a Laurent polynomial for an integer $n \le 0$; $e^{-z}$ times a polynomial plus a multiple of $z^{n-1}E_1(z)$ for an integer $n \ge 2$; the same with $\sqrt\pi\,\operatorname{erfc}(\sqrt z)$ in place of $E_1$ for a half-integer $n$; and $z^{n-1}\,\Gamma(1-n, z)$ for any other order, symbolic included. $E_1$ itself and floating-point orders are left alone.

Plot

Examples

9 more kept as data, checked against the oracles

Basic #

$E_1(3/2) = \Gamma(0, 3/2)$

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In$\mathrm{ExpIntegralE}(1, 1.5)$Out$0.100\,019\,582\,406\,632\,65$

Known value · Wolfram 15.0.0 ExpIntegralE

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In$\mathrm{ExpIntegralE}(2, 1.5)$Out$0.073\,100\,786\,538\,480\,84$

Known value · Wolfram 15.0.0 ExpIntegralE

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

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In$\operatorname{Series}(\mathrm{ExpIntegralE}(1, x), x, 0, 4)$Out$\operatorname{Series}(\mathrm{ExpIntegralE}(1, x), x, 0, 4)$

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In$\operatorname{Normal}(\operatorname{Series}(\mathrm{ExpIntegralE}(n, x), x, ∞, 4))$Out$\operatorname{Series}(\mathrm{ExpIntegralE}(n, x), x, ∞, 4)$

Scope #

Non-integer order

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In$\mathrm{ExpIntegralE}(0.5, 2.5)$Out$0.028\,414\,299\,709\,289\,756$

Known value · Wolfram 15.0.0 ExpIntegralE

Exact arguments under N()

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In$N(\mathrm{ExpIntegralE}(1, 1))$Out$0.219\,383\,934\,395\,520\,29$

Complex order and argument evaluate to a complex float

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In$N(\mathrm{ExpIntegralE}(1+i, 2+3i))$Out$-0.018\,943\,830\,508\,373\,671\,865\,807\,130\,229\,991\,161\,44+0.020\,443\,770\,102\,830\,021\,668\,024\,586\,210\,364\,454\,22i$

Order 0 reduces to elementary form $e^{-z}/z$

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In$\mathrm{ExpIntegralE}(0, z)$Out$\frac{e^{-z}}{z}$

Series coefficients of $E_0$ are deferred to the elementary form

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In$\mathrm{SeriesCoefficient}(\mathrm{ExpIntegralE}(0, x), \bigl\lbrack x, 0, n\bigr\rbrack)$Out$\mathrm{SeriesCoefficient}(\frac{e^{-x}}{x}, \bigl\lbrack x, 0, n\bigr\rbrack)$

$E_0(\infty)=0$

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In$\mathrm{ExpIntegralE}(0, ∞)$Out$0$

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In$N(\mathrm{ExpIntegralE}(3, \frac{11}{5}), 50)$Out$0.023\,520\,656\,648\,276\,500\,007\,831\,719\,911\,764\,625\,915\,312\,610\,516\,519$

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In$\mathrm{Simplify}(\mathrm{FunctionExpand}(\bigl\lbrack\mathrm{ExpIntegralE}(5/2, z), \mathrm{ExpIntegralE}(4, z), \mathrm{ExpIntegralE}(-4, z)\bigr\rbrack))$Out$\bigl\lbrack-\frac{1}{3}×4ze^{-z}+\frac{2e^{-z}}{3}+\frac{1}{3}×4\operatorname{erfc}(\sqrt{z})\sqrt{π}\sqrt{z}^{3}, (\frac{z^2}{6}-\frac{z}{6}+\frac{1}{3})e^{-z}-\frac{1}{6}\mathrm{ExpIntegralE}(1, z)z^3, (\frac{4}{z^2}+\frac{12}{z^3}+\frac{24}{z^4}+\frac{24}{z^5}+\frac{1}{z})e^{-z}\bigr\rbrack$

A half-integer order reduces to $e^{-z}$ and $\operatorname{erfc}(\sqrt z)$

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In$\mathrm{FunctionExpand}(\mathrm{ExpIntegralE}(\frac{3}{2}, z))$Out$2e^{-z}-2\operatorname{erfc}(\sqrt{z})\sqrt{π}\sqrt{z}$

A positive integer order keeps $E_1$ beside $e^{-z}$ times a polynomial

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In$\mathrm{FunctionExpand}(\mathrm{ExpIntegralE}(3, z))$Out$\frac{1}{2}\mathrm{ExpIntegralE}(1, z)z^2+(\frac{1}{2}-\frac{z}{2})e^{-z}$

A non-positive integer order is elementary

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In$\mathrm{FunctionExpand}(\mathrm{ExpIntegralE}(-2, z))$Out$(\frac{2}{z^2}+\frac{2}{z^3}+\frac{1}{z})e^{-z}$

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In$\operatorname{Series}(\mathrm{ExpIntegralE}(n, x), x, 0, 7)$Out$\mathrm{ExpIntegralE}(n, 0)+x\mathrm{ExpIntegralE}^{(0, 1)}(n, 0)+\frac{1}{2}\mathrm{ExpIntegralE}^{(0, 2)}(n, 0)x^2+\frac{1}{6}\mathrm{ExpIntegralE}^{(0, 3)}(n, 0)x^3+\frac{1}{24}\mathrm{ExpIntegralE}^{(0, 4)}(n, 0)x^4+\frac{1}{120}\mathrm{ExpIntegralE}^{(0, 5)}(n, 0)x^5+\frac{1}{720}\mathrm{ExpIntegralE}^{(0, 6)}(n, 0)x^6+\frac{1}{5\,040}\mathrm{ExpIntegralE}^{(0, 7)}(n, 0)x^7+O\left(x^8\right)$

Properties #

$E_n(0) = 1/(n-1)$ for $n > 1$

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In$\mathrm{ExpIntegralE}(2, 0)$Out$1$

$E_0(x) = e^{-x}/x$

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In$\mathrm{ExpIntegralE}(0, x)$Out$\frac{e^{-x}}{x}$

$E_0(0)$ is a pole

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In$\mathrm{ExpIntegralE}(0, 0)$Out$\tilde∞$

Known value · Wolfram MathematicalFunctionData, ExpIntegralE

$E_n(0) = \frac{1}{n-1}$, so $\frac12$ for $n=3$

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In$\mathrm{ExpIntegralE}(3, 0)$Out$\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, ExpIntegralE

About

Examples	30
Checked against	Wolfram Language 21 agree3 differ3 unevaluatedSageMath 17 agree1 differ1 unevaluated3 errorSymPy 19 agree2 differ2 unevaluatedmpmath 14 agree2 differ1 unevaluatedJulia 14 agree1 unevaluated2 errorRust 9 agree5 error
Crosswalk	Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/exp-integral-e.ts

z^(n-1)·Γ(1-n, z), with the n = 0 and z = 0 removable cases handled ahead of it.

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

A non-negative integer order and x >= 0 only.

See also: Gamma, GammaRegularized