Integral exponential function
The special function defined for real by the equation
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The graph of the integral exponential function is illustrated in Fig..
Figure: i051440a
Graphs of the functions ,
and
.
For , the function
has an infinite discontinuity at
, and the integral exponential function is understood in the sense of the principal value of this integral:
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The integral exponential function can be represented by the series
![]() | (1) |
and
![]() | (2) |
where is the Euler constant.
There is an asymptotic representation:
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As a function of the complex variable , the integral exponential function
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is a single-valued analytic function in the -plane slit along the positive real semi-axis
; here the value of
is chosen such that
. The behaviour of
close to the slit is described by the limiting relations:
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The asymptotic representation in the region is:
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The integral exponential function is related to the integral logarithm by the formulas
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and to the integral sine and the integral cosine
by the formulas:
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The differentiation formula is:
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The following notations are sometimes used:
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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 is usually called the exponential integral.
Instead of by the series representation, for complex values of (
not positive real) the function
can be defined by the integal (as for real
); since the integrand is analytic, the integral is path-independent in
.
Formula (1) with replaced by
then holds for
, and the function defined by (2) (for
) is also known as the modified exponential integral.
Integral exponential function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_exponential_function&oldid=47371