What is the exponential integral Ei(x)?

Ei(x) is defined as the Cauchy principal value of the integral ∫ (eᵗ/t) dt from −∞ to x, a function that has no elementary closed form and must be evaluated via series or numerical methods.

Why is Ei(x) undefined at x = 0?

The integrand eᵗ/t has a non-integrable singularity at t = 0, and the ln|x| term in the series expansion also diverges as x approaches 0, so Ei(0) is undefined.

What is the Euler–Mascheroni constant?

It's the constant γ ≈ 0.5772156649, defined as the limiting difference between the harmonic series and the natural logarithm — it appears throughout number theory and analysis, including in this series for Ei(x).

Does the series converge for all x?

Yes — because the series has k! in the denominator, it converges for every finite nonzero x by the ratio test, though very large |x| requires many more terms to reach full precision, which is why this calculator caps |x| for reliability.

How does this differ from E₁(x)?

E₁(x), the complementary exponential integral commonly used in engineering, is related by E₁(x) = −Ei(−x) for x > 0 — this calculator computes Ei(x) directly, from which E₁(x) can be derived with a sign and argument flip.

Why does this calculator limit |x| to 50?

Beyond that range, the series requires increasingly many terms to converge to full double-precision accuracy, and floating-point rounding in intermediate powers of x starts to degrade reliability — 50 comfortably covers the range most practical uses need.

Where is the exponential integral used in real life?

It appears in heat conduction and radiative transfer problems, groundwater flow modeling (the Theis well function), electrical engineering, and evaluating certain probability distributions.

Is Ei(x) related to the natural logarithm for large x?

For large positive x, Ei(x) grows roughly like eˣ/x (much faster than any logarithm), while the ln|x| term in its series is just one small piece of the full asymptotic behavior.

Exponential Integral Calculator — Compute Ei(x)

Our Exponential Integral Calculator computes Ei(x) = γ + ln|x| + Σ xᵏ/(k·k!) for any nonzero real x, using a convergent series expansion around the Euler–Mascheroni constant γ, giving accurate results for a wide range of input magnitudes.

Quick Answer

Ei(x) = γ + ln|x| + Σ xᵏ/(k·k!). Enter any nonzero x below to instantly compute the exponential integral via its convergent series.

Enter x, then click Calculate.

How to Use the Exponential Integral Calculator — Ei(x) Online

1
Enter any nonzero real number x.
2
Click 'Calculate' to compute Ei(x) via its convergent series expansion.
3
Review the formula and approximate result shown in the steps.

Why Use Exponential Integral Calculator — Ei(x) Online?

The exponential integral Ei(x) arises naturally whenever you try to integrate eᵗ/t, an integral with no elementary closed form. Instead, it's expressed through a convergent series built around the Euler–Mascheroni constant γ ≈ 0.5772156649 — a series that, unlike a Taylor series with a limited radius of convergence, converges for every nonzero real x since the k! in the denominator eventually overwhelms any xᵏ growth. This calculator sums enough terms of that series to reach double-precision accuracy for the supported input range.

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