Exponential Integral Eₙ(x) Calculator

Enter an order n and a positive x to compute the exponential integral Eₙ(x)=∫₁^∞ e^(-x t)/tⁿ dt with high accuracy, using series and continued fractions, plus neighboring orders and ratios.

Input

Compute the exponential integral Eₙ(x)=∫₁^∞ e^(-x t)/tⁿ dt from an order n and a positive x.

Order n

Enter a non-negative integer.

Value of x

Enter a real number greater than 0.

Result

Value of E[2](1)

0.1484955068

E[1](1) (previous order)

0.2193839344

E[3](1) (next order)

0.1096919672

Previous order ÷ next order

Eₙ(x) versus the order n

Eₙ(x) for nearby orders at x=1

Order n	Value of Eₙ(x)
n = 0	0.3678794412
n = 1	0.2193839344
n = 2	0.1484955068
n = 3	0.1096919672
n = 4	0.0860624913

How it works

The exponential integral Eₙ(x) is defined by Eₙ(x)=∫₁^∞ e^(-x t)/tⁿ dt for positive x and a non-negative integer order n.
For n=0 it reduces to the closed form E₀(x)=e^(-x)/x.
For x greater than 1 the tool evaluates Eₙ(x) directly with a continued fraction using the modified Lentz algorithm.
For x at most 1 with n at least 1 it uses a series expansion that includes the logarithmic term ln x.
Neighboring orders are linked by the recurrence Eₙ₊₁(x)=(e^(-x) − x·Eₙ(x))/n for n at least 1.
The highlighted statistic shows E₍n-1₎(x) divided by E₍n+1₎(x).
Results are accurate approximations; the value is undefined when x is zero or negative, or when n is negative or non-integer.

Reviews

Tell us what you think of this calculator.