Integral exponential function

The special function defined for real $ x \neq 0 $ by the equation

$$ \mathop{\rm Ei} ( x) = \ \int\limits _ {- \infty } ^ { x } \frac{e ^ {t} }{t} d t = - \int\limits _ { - x} ^ \infty \frac{e ^ {-t} }{t} d t . $$

The graph of the integral exponential function is illustrated in Fig..

Figure: i051440a

Graphs of the functions $ y = \mathop{\rm Ei} ( - x ) $, $ y = \mathop{\rm Ei} ^ {*} ( x) $ and $ y = \mathop{\rm Li} ( x) $.

For $ x > 0 $, the function $ e ^ {t} / t $ has an infinite discontinuity at $ t = 0 $, and the integral exponential function is understood in the sense of the principal value of this integral:

$$ \mathop{\rm Ei} ( x) = \ \lim\limits _ {\epsilon \rightarrow + 0 } \ \left \{ \int\limits _ {- \infty } ^ \epsilon \frac{e ^ {t} }{t} d t + \int\limits _ \epsilon ^ { x } \frac{e ^ {t} }{t} d t \right \} . $$

The integral exponential function can be represented by the series

$$ \tag{1 } \mathop{\rm Ei} ( x) = \ c + \mathop{\rm ln} ( - x ) + \sum_{k=1}^ \infty \frac{x ^ {k} }{k!k} ,\ \ x < 0 , $$

and

$$ \tag{2 } \mathop{\rm Ei} ( x) = c + \mathop{\rm ln} ( x) + \sum_{k=1}^ \infty \frac{x ^ {k} }{k!k} ,\ \ x > 0 , $$

where $ c = 0.5772 \dots $ is the Euler constant.

There is an asymptotic representation:

$$ \mathop{\rm Ei} ( - x ) \approx \frac{e ^ {- x} }{x} \left ( 1 - \frac{1!}{x} + \frac{2!} {x ^ {2} } - \frac{3!} {x ^ {3} } + \dots \right ) ,\ \ x \rightarrow + \infty . $$

As a function of the complex variable $ z $, the integral exponential function

$$ \mathop{\rm Ei} ( z) = \ C + \mathop{\rm ln} ( - z ) + \sum_{k=1}^ \infty \frac{z ^ {k} }{k!k} ,\ \ | \mathop{\rm arg} ( - z ) | < \pi , $$

is a single-valued analytic function in the $ z $- plane slit along the positive real semi-axis $ ( 0 < \mathop{\rm arg} z < 2 \pi ) $; here the value of $ \mathop{\rm ln} ( - z) $ is chosen such that $ - \pi < { \mathop{\rm Im} \mathop{\rm ln} } (- z) < \pi $. The behaviour of $ \mathop{\rm Ei} ( z) $ close to the slit is described by the limiting relations:

$$ \left . \begin{array}{c} \lim\limits _ {\eta \downarrow 0 } \ \mathop{\rm Ei} ( z + i \eta ) = \ \mathop{\rm Ei} ( z) - i \pi , \\ \lim\limits _ {\eta \downarrow 0 } \ \mathop{\rm Ei} ( z - i \eta ) = \ \mathop{\rm Ei} ( z) + i \pi , \\ \end{array} \right \} \ \ z = x + i y. $$

The asymptotic representation in the region $ 0 < \mathop{\rm arg} z < 2 \pi $ is:

$$ \mathop{\rm Ei} ( z) \sim \ \frac{e ^ {z} }{z} \left ( \frac{1!}{z} + \frac{2!}{z ^ {2} } + \dots \frac{k!} {z ^ {k} } + \dots \right ) ,\ \ | z | \rightarrow \infty . $$

The integral exponential function is related to the integral logarithm $ \mathop{\rm li} ( x) $ by the formulas

$$ \mathop{\rm Ei} ( x) = \ \mathop{\rm li} ( e ^ {x} ) ,\ \ x < 0 , $$

$$ \mathop{\rm Ei} ( \mathop{\rm ln} x ) = \mathop{\rm li} ( x) ,\ x < 1 ; $$

and to the integral sine $ \mathop{\rm Si} ( x) $ and the integral cosine $ \mathop{\rm Ci} ( x) $ by the formulas:

$$ \mathop{\rm Ei} ( \pm i x ) = \ \mathop{\rm Ci} ( x) \pm i \mathop{\rm Si} ( x) \mps \frac{\pi i }{2} ,\ \ x > 0 . $$

The differentiation formula is:

$$ \frac{d ^ {n} \mathop{\rm Ei} ( - x ) }{d x ^ {n} } = \ ( - 1 ) ^ {n-1} ( n - 1 ) ! x ^ {- x} e ^ {-x} e _ {n-1} ( x) ,\ \ n = 1 , 2 , . . . . $$

The following notations are sometimes used:

$$ \operatorname{\rm Ei}^+ ( z) = \ \mathop{\rm Ei} ( z + i 0 ) ,\ \ \operatorname{\rm Ei}^- ( z) = \ \mathop{\rm Ei} ( z - i 0 ) , $$

$$ \operatorname{\rm Ei} ^ {*} ( z) = { \mathop{\rm Ei} ( z) } bar = \mathop{\rm Ei} ( z) + \pi i . $$

References

[1]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2]	E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[3]	A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)
[4]	N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)

Comments

The function $\mathop{\rm Ei}$ is usually called the exponential integral.

Instead of by the series representation, for complex values of $ z $( $ x $ not positive real) the function $ \mathop{\rm Ei} ( z) $ can be defined by the integal (as for real $ x \neq 0 $); since the integrand is analytic, the integral is path-independent in $ \mathbf C \setminus \{ {x \in \mathbf R } : {x \geq 0 } \} $.

Formula (1) with $ x $ replaced by $ z $ then holds for $ | \mathop{\rm arg} ( - z ) | < \pi $, and the function defined by (2) (for $ x > 0 $) is also known as the modified exponential integral.

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