Exponential Integral Calculator

The Exponential Integral Ei(x) is a special mathematical function defined as the Cauchy principal value of the integral of eᵗ/t from −∞ to x. It arises naturally in physics, engineering, and number theory, particularly in problems involving heat conduction, antenna theory, and the prime counting function. It is related to, but distinct from, other exponential integral forms such as E₁(x).

Ei(x) and E₁(x) are two closely related but different exponential integral functions. For positive real x, the relationship is E₁(x) = −Ei(−x). Ei(x) is defined for real x ≠ 0, while E₁(x) is typically defined for x > 0 and is more common in applied mathematics. They share the same logarithmic singularity at x = 0 but differ in sign conventions and domains.

The integrand eᵗ/t in the definition of Ei(x) has a non-integrable singularity at t = 0. As x approaches 0, the logarithmic term ln|x| in the power series expansion diverges to −∞ (from the left) or +∞ (from the right). The Cauchy principal value is used to handle this singularity in a well-defined way, but Ei(0) itself is undefined.

For large positive x, the power series converges very slowly and the asymptotic expansion Ei(x) ≈ (eˣ/x)(1 + 1!/x + 2!/x² + 3!/x³ + …) becomes highly accurate. This asymptotic series is divergent in the classical sense but provides an excellent numerical approximation when only a finite number of terms are used near the optimal truncation point. This calculator uses the power series for moderate x and reports the asymptotic approximation alongside it for comparison.

Yes. For x < 0, Ei(x) is real-valued and negative (approaching 0 from below as x → −∞). The power series Ei(x) = γ + ln|x| + Σ xᵏ/(k·k!) remains valid for x < 0. Note that some sources use the notation −E₁(−x) to represent this quantity for negative arguments, so be mindful of the sign convention when comparing results from different references.

The power series expansion is Ei(x) = γ + ln|x| + Σₖ₌₁^∞ xᵏ/(k·k!), where γ ≈ 0.5772156649 is the Euler–Mascheroni constant. This series converges for all x ≠ 0, though convergence becomes slower for large |x|. This calculator sums terms until the contribution of each new term falls below a tolerance threshold determined by your chosen decimal precision.

Ei(x) appears in many areas of science and engineering. In physics it is used in radiation transport and plasma physics. In electrical engineering it arises in antenna radiation integrals and transient circuit analysis. In number theory it is closely related to the logarithmic integral Li(x), which approximates the prime counting function π(x). It also appears in solutions to certain differential equations and in the evaluation of improper integrals.

This calculator uses a convergent power series with up to 15 significant decimal places, summing terms until numerical precision is reached. For most values of x in the range [−20, 20], the result is accurate to the selected precision. For very large |x|, floating-point arithmetic limitations may reduce accuracy slightly, and the asymptotic approximation shown alongside provides a useful cross-check.