exponential integral
/ the symbol Ei is read 'E-eye' /
The simplest-looking integrand e^t over t, or e^{-t} over t, refuses to have an elementary antiderivative — dividing the exponential by its own variable already pushes us off the map of polynomials, logs and trig. The exponential integral is the named function that captures this combination, recording the accumulated area of an exponential weighted by one over its argument.
Two standard versions appear. For positive argument, Ei(x) = the (principal-value) integral from minus infinity to x of e^t over t dt, where the principal value is needed because the integrand blows up at t = 0. For the decaying side, E_1(x) = integral from x to infinity of e^{-t} over t dt, which is finite and clean for x greater than 0. Near the origin both behave like a logarithm plus a constant: E_1(x) is approximately minus gamma_E minus ln x plus x minus ..., where gamma_E approximately 0.5772 is the Euler-Mascheroni constant. For large x the function has an asymptotic (and ultimately divergent) tail, E_1(x) is approximately e^{-x} over x times (1 minus 1/x plus 2/x^2 minus ...), accurate when truncated early.
The exponential integral is the backbone of radiative transfer (it sums the attenuation of light through an absorbing medium over all angles), groundwater hydrology (the Theis well-drawdown solution is an exponential integral), neutron transport, and antenna theory. It is also the bridge to the logarithmic and sine and cosine integrals, which are the same idea with the exponential replaced by a logarithm or a sinusoid.
E_1(1) = integral from 1 to infinity of e^{-t}/t dt is approximately 0.21938. The leading asymptotic guess e^{-1}/1 is approximately 0.368 is too crude here; the series is only useful once x is large.
The exponential-integral asymptotic series is excellent for large x but useless at x = 1, where the series approach has not yet kicked in.
Ei needs a Cauchy principal value because of the pole at t = 0; do not confuse Ei(x) (a single principal-value integral) with E_1(x), the version for the decaying exponential.