Exponential integral ($E_1,\mathrm{Ei}$)

For real $x>0$, the decaying exponential integral isThe other standard real convention satisfies $\mathrm{Ei}(-x)=-E_1(x)$. The NIST definitions specify the analytic continuation and branch conventions.

As $x\downarrow 0$, the exponential integral haswhere $\gamma$ is the Euler--Mascheroni constant. Integrating the Taylor series of $E_1^{\prime }(x)=-e^{-x}/x$ gives every nonconstant coefficient; the constant follows from the limiting definition of $\gamma$. A logarithm of a stretched variable can create a switchback term in a matched asymptotic expansion.