Exponential Integral Ei(x) Calculator

Compute the exponential integral Ei(x) (principal value) with a convergent series and asymptotic expansion. Shows its relation to E1(x) and a curve graph.

Input

Enter x to compute the exponential integral Ei(x), the principal value integral, using a series and asymptotic expansion.

Value of x

Any real number except 0. Examples: 1, 2.5, -1

Result

Value of Ei(1)

1.8951178164

E1(1) = -Ei(-x)

0.2193839344

Input x

The exponential integral E1 is related by E1(x) = -Ei(-x).

Graph of Ei(x)

How it works

The exponential integral Ei(x) is the principal value integral defined by Ei(x) = γ + ln|x| + Σ x^n/(n·n!), where γ is the Euler-Mascheroni constant.
For small to moderate |x| a convergent series is used, and for large positive x the asymptotic expansion Ei(x) ≈ (e^x/x)·Σ k!/x^k is applied.
At x = 0 the function has a singularity and diverges to negative infinity, so x = 0 cannot be entered.
The related exponential integral E1(x) (for positive x) is connected by E1(x) = -Ei(-x).
The asymptotic series is divergent, so it is truncated near its smallest term to keep a reasonable accuracy.

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