Exponential Integral Calculator

What Is the Exponential Integral?

The exponential integral $\text{Ei}(x)$ is a classical special function that appears in heat transfer, fluid flow in porous media, and electromagnetic theory. It is defined as a Cauchy principal value integral because the integrand is singular at $t = 0$.

Definition

For $x \neq 0$, the power series expansion is $\text{Ei}(x) = \gamma + \ln|x| + \sum_{k=1}^{\infty} x^k / (k \cdot k!)$, where $\gamma \approx 0.5772156649$ is the Euler-Mascheroni constant.

Key Properties

• Derivative: $\frac{d}{dx}\text{Ei}(x) = e^x / x$
• Logarithmic singularity at $x = 0$
• For large positive $x$: $\text{Ei}(x) \sim e^x / x$
• Related to $E_1(x)$: $\text{Ei}(-x) = -E_1(x)$ for $x > 0$

Explore related special functions with the Error Function Calculator or the Natural Log Calculator.