Difference between revisions of "Integral exponential function"
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\int\limits _ { - x} ^ \infty | \int\limits _ { - x} ^ \infty | ||
− | \frac{e ^ {-} | + | \frac{e ^ {-t} }{t} |
d t . | d t . | ||
$$ | $$ | ||
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\mathop{\rm Ei} ( x) = \ | \mathop{\rm Ei} ( x) = \ | ||
c + \mathop{\rm ln} ( - x ) + | c + \mathop{\rm ln} ( - x ) + | ||
− | \ | + | \sum_{k=1}^ \infty |
\frac{x ^ {k} }{k!k} | \frac{x ^ {k} }{k!k} | ||
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\mathop{\rm Ei} ( x) = c + | \mathop{\rm Ei} ( x) = c + | ||
\mathop{\rm ln} ( x) + | \mathop{\rm ln} ( x) + | ||
− | \ | + | \sum_{k=1}^ \infty |
\frac{x ^ {k} }{k!k} | \frac{x ^ {k} }{k!k} | ||
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$$ | $$ | ||
− | \mathop{\rm Ei} ( - x ) \approx | + | \mathop{\rm Ei} ( - x ) \approx \frac{e ^ {- x} }{x} |
− | |||
− | \frac{e ^ {-} | ||
\left ( | \left ( | ||
− | 1 - 1! | + | 1 - \frac{1!}{x} + |
− | 2! | + | \frac{2!} {x ^ {2} } - |
− | 3! | + | \frac{3!} {x ^ {3} } + \dots |
\right ) ,\ \ | \right ) ,\ \ | ||
x \rightarrow + \infty . | x \rightarrow + \infty . | ||
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\mathop{\rm Ei} ( z) = \ | \mathop{\rm Ei} ( z) = \ | ||
C + \mathop{\rm ln} ( - z ) + | C + \mathop{\rm ln} ( - z ) + | ||
− | \ | + | \sum_{k=1}^ \infty |
\frac{z ^ {k} }{k!k} | \frac{z ^ {k} }{k!k} | ||
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\left ( | \left ( | ||
− | 1! | + | \frac{1!}{z} + |
− | 2! | + | \frac{2!}{z ^ {2} } + \dots |
− | + | \frac{k!} {z ^ {k} } + \dots | |
\right ) ,\ \ | \right ) ,\ \ | ||
| z | \rightarrow \infty . | | z | \rightarrow \infty . | ||
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\frac{d ^ {n} \mathop{\rm Ei} ( - x ) }{d x ^ {n} } | \frac{d ^ {n} \mathop{\rm Ei} ( - x ) }{d x ^ {n} } | ||
= \ | = \ | ||
− | ( - 1 ) ^ {n-} | + | ( - 1 ) ^ {n-1} |
− | ( n - 1 ) ! x ^ {-} | + | ( n - 1 ) ! x ^ {- x} |
− | e ^ {-} | + | e ^ {-x} e _ {n-1} ( x) ,\ \ |
n = 1 , 2 , . . . . | n = 1 , 2 , . . . . | ||
$$ | $$ | ||
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$$ | $$ | ||
− | \ | + | \operatorname{\rm Ei}^+ ( z) = \ |
\mathop{\rm Ei} ( z + i 0 ) ,\ \ | \mathop{\rm Ei} ( z + i 0 ) ,\ \ | ||
− | \ | + | \operatorname{\rm Ei}^- ( z) = \ |
\mathop{\rm Ei} ( z - i 0 ) , | \mathop{\rm Ei} ( z - i 0 ) , | ||
$$ | $$ | ||
$$ | $$ | ||
− | \ | + | \operatorname{\rm Ei} ^ {*} ( z) = { \mathop{\rm Ei} ( z) } bar = \mathop{\rm Ei} ( z) + \pi i . |
$$ | $$ | ||
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====Comments==== | ====Comments==== | ||
− | The function | + | The function \mathop{\rm Ei} is usually called the exponential integral. |
− | is usually called the exponential integral. | ||
Instead of by the series representation, for complex values of z ( | Instead of by the series representation, for complex values of z ( | ||
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and the function defined by (2) (for x > 0 ) | and the function defined by (2) (for x > 0 ) | ||
is also known as the modified exponential integral. | is also known as the modified exponential integral. | ||
+ | |||
+ | {{OldImage}} |
Latest revision as of 19:20, 11 January 2024
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.
Integral exponential function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_exponential_function&oldid=52747