Small-argument expansion of the exponential integral
= Small-argument expansion of the exponential integral
As $x\downarrow0$, the <exponential integral> has
$$
E_1(x)=-\gamma-\log x+x-\frac{x^2}4+O(x^3),
$$
where $\gamma$ is the <Euler--Mascheroni constant>. Integrating the <Taylor series> of $E_1'(x)=-e^{-x}/x$ 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>.