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).

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.

Examples

4 more kept as data, checked against the oracles

Basic #

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

In

\mathrm{ExpIntegralE}(1, 1.5)

Out

0.100\,019\,582\,406\,632\,65

Known value · Wolfram 15.0.0 ExpIntegralE

In

\mathrm{ExpIntegralE}(2, 1.5)

Out

0.073\,100\,786\,538\,480\,84

Known value · Wolfram 15.0.0 ExpIntegralE

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)

In

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

Out

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

In

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

Out

\operatorname{Series}(\mathrm{ExpIntegralE}(n, x), x, ∞, 4)

Scope #

Non-integer order

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()

In

N(\mathrm{ExpIntegralE}(1, 1))

Out

0.219\,383\,934\,395\,520\,29

Complex order and argument evaluate to a complex float

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 $e^{-z}/z$

In

\mathrm{ExpIntegralE}(0, z)

Out

\frac{\exp(-z)}{z}

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

In

\mathrm{SeriesCoefficient}(\mathrm{ExpIntegralE}(0, x), \bigl\lbrack x, 0, n\bigr\rbrack)

Out

\mathrm{SeriesCoefficient}(\frac{\exp(-x)}{x}, \bigl\lbrack x, 0, n\bigr\rbrack)

$E_0(\infty)=0$

In

\mathrm{ExpIntegralE}(0, ∞)

Out

0

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

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}(4z\exp(-z))+\frac{2\exp(-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})\exp(-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})\exp(-z)\bigr\rbrack

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

In

\mathrm{FunctionExpand}(\mathrm{ExpIntegralE}(\frac{3}{2}, z))

Out

2\exp(-z)-2\operatorname{erfc}(\sqrt{z})\sqrt{π}\sqrt{z}

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

In

\mathrm{FunctionExpand}(\mathrm{ExpIntegralE}(3, z))

Out

\frac{1}{2}(\mathrm{ExpIntegralE}(1, z)z^2)+(\frac{1}{2}-\frac{z}{2})\exp(-z)

A non-positive integer order is elementary

In

\mathrm{FunctionExpand}(\mathrm{ExpIntegralE}(-2, z))

Out

(\frac{2}{z^2}+\frac{2}{z^3}+\frac{1}{z})\exp(-z)

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 $n > 1$

In

\mathrm{ExpIntegralE}(2, 0)

Out

1

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

In

\mathrm{ExpIntegralE}(0, x)

Out

\frac{\exp(-x)}{x}

$E_0(0)$ is a pole

In

\mathrm{ExpIntegralE}(0, 0)

Out

\tilde∞

Known value · Wolfram MathematicalFunctionData, ExpIntegralE

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

In

\mathrm{ExpIntegralE}(3, 0)

Out

\frac{1}{2}

Known value · Wolfram MathematicalFunctionData, ExpIntegralE

About

Examples	25
Checked against	Wolfram Language 21 agree1 differ3 unevaluated
Crosswalk	Wolfram Language

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.

See also: Gamma, GammaRegularized