Questionable statement in Abramowitz and Stegun: Exponential Integral Function

Question

I have in front of me the ninth (1972) printing of Abramowitz and Stegun's Handbook of Mathematical Functions (1964).

In Section 3 of the introduction, "Auxiliary Functions and Arguments" they discuss the Exponential Integral Function, which is presented in the following form:

The exponential integral of positive argument is given by

$$\operatorname{Ei} (x) = \int_{-\infty}^x \frac {e^u} u du$$

This looks suspect to me: the domain encompasses an obvious pole at $u = 0$ which does not at first glance look removable.

The way I have been accustomed to seeing this defined (as found in the Schaum Mathematical Handbook of Formulas and Tables (1968) Chapter $35$ is:

$$\operatorname{Ei} (x) = \int_x^{+\infty} \frac {e^{-u} } u d u$$

which makes more sense.

I am wondering whether the statement "The exponential integral of positive argument" is incorrect, and that perhaps it should say for negative argument?

The power series expansion of the first of these is given in Abramowitz and Stegun as:

$$\operatorname{Ei} (x) = \gamma + \ln x + \frac x {1 \times 1!} + \frac {x^2} {2 \times 2!} + \frac {x^3} {3 \times 3!} + \dots$$

while from the Schaum Handbook definition you can derive:

$$\operatorname{Ei} (x) = -\gamma + -\ln x + \frac x {1 \times 1!} - \frac {x^2} {2 \times 2!} + \frac {x^3} {3 \times 3!} - \dots$$

Can anyone shed any light on this? I would work out the derivation from the A&S definition except for the fact that I really don't know how to handle the singularity.

Answer 1

Abramowitz and Stegun on page 228 states (with \pvint instead of \int)

$$ 5.1.2 \qquad \mathrm{Ei}(x)=-\,\int_{-x}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t=% \int_{-\infty}^{x}\frac{e^{t}}{t}\mathrm{d}t \qquad (x>0) $$

where the integrals are Cauchy principal value integrals because of the obvious pole. What was missing on page X was the slight difference in the regular integral sign and the Cauchy principal value integral sign. The DLMF $6.2.5$ states exactly the same thing as A&S $5.1.2$.

Tthe correct series valid for all real $\,x< 0\,$ or $\,x>0\,$ (cf. DLMF $6.6.1$ with $x$ instead of $|x|$) is

$$ \mathrm{Ei} (x)=\gamma +\ln |x| +\sum_{n=1}^\infty \frac{x^n}{n\, n!} $$

where for some reason the DLMF version is qualified by $\,x>0.$ Note that the power series converges for all real $\,x\,$ and the derivative of the expression is $\,\frac{e^x}x\,$ for all $\,x\ne 0\,$ and thus the only difficulty is justifying the choice of the constant $\,\gamma.$

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Note that while $\,\mathrm{Ei}(x)\,$ is defined only for real $\,x\ne 0,\,$ there is a closely related function as given in DLMF $6.2.1$ as

The principal value of the exponential integral $\,E_1(z)\,$ is defined by $$ 6.2.1 \qquad \qquad \qquad E_1(z) = \int_{z}^\infty \frac{e^{-t}}t \mathrm{d}t,\qquad\qquad z\ne 0, $$ where the path does not cross the negative real axis or pass through the origin. As in the case of the logarithm ($\S4.2$(i)) there is a cut along the interval $\,(-\infty,0]\,$ and the principal value is two-valued on $\,(-\infty,0).$