Special function defined by an integral
Plot of the exponential integral function E n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics, the exponential integral Ei is a
special function on the
complex plane .
It is defined as one particular
definite integral of the ratio between an
exponential function and its
argument .
Definitions
For real non-zero values of x , the exponential integral Ei(x ) is defined as
Ei
(
x
)
=
−
∫
−
x
∞
e
−
t
t
d
t
=
∫
−
∞
x
e
t
t
d
t
.
{\displaystyle \operatorname {Ei} (x)=-\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,dt=\int _{-\infty }^{x}{\frac {e^{t}}{t}}\,dt.}
The
Risch algorithm shows that Ei is not an
elementary function . The definition above can be used for positive values of x , but the integral has to be understood in terms of the
Cauchy principal value due to the singularity of the integrand at zero.
For complex values of the argument, the definition becomes ambiguous due to
branch points at 0 and
∞
{\displaystyle \infty }
.
[1] Instead of Ei, the following notation is used,
[2]
E
1
(
z
)
=
∫
z
∞
e
−
t
t
d
t
,
|
A
r
g
(
z
)
|
<
π
{\displaystyle E_{1}(z)=\int _{z}^{\infty }{\frac {e^{-t}}{t}}\,dt,\qquad |{\rm {Arg}}(z)|<\pi }
Plot of the exponential integral function Ei(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
For positive values of x , we have
−
E
1
(
x
)
=
Ei
(
−
x
)
{\displaystyle -E_{1}(x)=\operatorname {Ei} (-x)}
.
In general, a
branch cut is taken on the negative real axis and E 1 can be defined by
analytic continuation elsewhere on the complex plane.
For positive values of the real part of
z
{\displaystyle z}
, this can be written
[3]
E
1
(
z
)
=
∫
1
∞
e
−
t
z
t
d
t
=
∫
0
1
e
−
z
/
u
u
d
u
,
ℜ
(
z
)
≥
0.
{\displaystyle E_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,dt=\int _{0}^{1}{\frac {e^{-z/u}}{u}}\,du,\qquad \Re (z)\geq 0.}
The behaviour of E 1 near the branch cut can be seen by the following relation:
[4]
lim
δ
→
0
+
E
1
(
−
x
±
i
δ
)
=
−
Ei
(
x
)
∓
i
π
,
x
>
0.
{\displaystyle \lim _{\delta \to 0+}E_{1}(-x\pm i\delta )=-\operatorname {Ei} (x)\mp i\pi ,\qquad x>0.}
Properties
Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above.
Convergent series
Plot of
E
1
{\displaystyle E_{1}}
function (top) and
Ei
{\displaystyle \operatorname {Ei} }
function (bottom).
For real or complex arguments off the negative real axis,
E
1
(
z
)
{\displaystyle E_{1}(z)}
can be expressed as
[5]
E
1
(
z
)
=
−
γ
−
ln
z
−
∑
k
=
1
∞
(
−
z
)
k
k
k
!
(
|
Arg
(
z
)
|
<
π
)
{\displaystyle E_{1}(z)=-\gamma -\ln z-\sum _{k=1}^{\infty }{\frac {(-z)^{k}}{k\;k!}}\qquad (\left|\operatorname {Arg} (z)\right|<\pi )}
where
γ
{\displaystyle \gamma }
is the
Euler–Mascheroni constant . The sum converges for all complex
z
{\displaystyle z}
, and we take the usual value of the
complex logarithm having a
branch cut along the negative real axis.
This formula can be used to compute
E
1
(
x
)
{\displaystyle E_{1}(x)}
with floating point operations for real
x
{\displaystyle x}
between 0 and 2.5. For
x
>
2.5
{\displaystyle x>2.5}
, the result is inaccurate due to
cancellation .
A faster converging series was found by
Ramanujan :
E
i
(
x
)
=
γ
+
ln
x
+
exp
(
x
/
2
)
∑
n
=
1
∞
(
−
1
)
n
−
1
x
n
n
!
2
n
−
1
∑
k
=
0
⌊
(
n
−
1
)
/
2
⌋
1
2
k
+
1
{\displaystyle {\rm {Ei}}(x)=\gamma +\ln x+\exp {(x/2)}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}x^{n}}{n!\,2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}}
Asymptotic (divergent) series
Relative error of the asymptotic approximation for different number
N
{\displaystyle ~N~}
of terms in the truncated sum
Unfortunately, the convergence of the series above is slow for arguments of larger modulus. For example, more than 40 terms are required to get an answer correct to three significant figures for
E
1
(
10
)
{\displaystyle E_{1}(10)}
.
[6] However, for positive values of x, there is a divergent series approximation that can be obtained by integrating
x
e
x
E
1
(
x
)
{\displaystyle xe^{x}E_{1}(x)}
by parts:
[7]
E
1
(
x
)
=
exp
(
−
x
)
x
(
∑
n
=
0
N
−
1
n
!
(
−
x
)
n
+
O
(
N
!
x
−
N
)
)
{\displaystyle E_{1}(x)={\frac {\exp(-x)}{x}}\left(\sum _{n=0}^{N-1}{\frac {n!}{(-x)^{n}}}+O(N!x^{-N})\right)}
The relative error of the approximation above is plotted on the figure to the right for various values of
N
{\displaystyle N}
, the number of terms in the truncated sum (
N
=
1
{\displaystyle N=1}
in red,
N
=
5
{\displaystyle N=5}
in pink).
Asymptotics beyond all orders
Using integration by parts, we can obtain an explicit formula
[8]
Ei
(
z
)
=
e
z
z
(
∑
k
=
0
n
k
!
z
k
+
e
n
(
z
)
)
,
e
n
(
z
)
≡
(
n
+
1
)
!
z
e
−
z
∫
−
∞
z
e
t
t
n
+
2
d
t
{\displaystyle \operatorname {Ei} (z)={\frac {e^{z}}{z}}\left(\sum _{k=0}^{n}{\frac {k!}{z^{k}}}+e_{n}(z)\right),\quad e_{n}(z)\equiv (n+1)!\ ze^{-z}\int _{-\infty }^{z}{\frac {e^{t}}{t^{n+2}}}\,dt}
For any fixed
z
{\displaystyle z}
, the absolute value of the error term
|
e
n
(
z
)
|
{\displaystyle |e_{n}(z)|}
decreases, then increases. The minimum occurs at
n
∼
|
z
|
{\displaystyle n\sim |z|}
, at which point
|
e
n
(
z
)
|
≤
2
π
|
z
|
e
−
|
z
|
{\displaystyle \vert e_{n}(z)\vert \leq {\sqrt {\frac {2\pi }{\vert z\vert }}}e^{-\vert z\vert }}
. This bound is said to be "asymptotics beyond all orders".
Exponential and logarithmic behavior: bracketing
Bracketing of
E
1
{\displaystyle E_{1}}
by elementary functions
From the two series suggested in previous subsections, it follows that
E
1
{\displaystyle E_{1}}
behaves like a negative exponential for large values of the argument and like a logarithm for small values. For positive real values of the argument,
E
1
{\displaystyle E_{1}}
can be bracketed by elementary functions as follows:
[9]
1
2
e
−
x
ln
(
1
+
2
x
)
<
E
1
(
x
)
<
e
−
x
ln
(
1
+
1
x
)
x
>
0
{\displaystyle {\frac {1}{2}}e^{-x}\,\ln \!\left(1+{\frac {2}{x}}\right)<E_{1}(x)<e^{-x}\,\ln \!\left(1+{\frac {1}{x}}\right)\qquad x>0}
The left-hand side of this inequality is shown in the graph to the left in blue; the central part
E
1
(
x
)
{\displaystyle E_{1}(x)}
is shown in black and the right-hand side is shown in red.
Definition by Ein
Both
Ei
{\displaystyle \operatorname {Ei} }
and
E
1
{\displaystyle E_{1}}
can be written more simply using the
entire function
Ein
{\displaystyle \operatorname {Ein} }
[10] defined as
Ein
(
z
)
=
∫
0
z
(
1
−
e
−
t
)
d
t
t
=
∑
k
=
1
∞
(
−
1
)
k
+
1
z
k
k
k
!
{\displaystyle \operatorname {Ein} (z)=\int _{0}^{z}(1-e^{-t}){\frac {dt}{t}}=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}z^{k}}{k\;k!}}}
(note that this is just the alternating series in the above definition of
E
1
{\displaystyle E_{1}}
). Then we have
E
1
(
z
)
=
−
γ
−
ln
z
+
E
i
n
(
z
)
|
Arg
(
z
)
|
<
π
{\displaystyle E_{1}(z)\,=\,-\gamma -\ln z+{\rm {Ein}}(z)\qquad \left|\operatorname {Arg} (z)\right|<\pi }
Ei
(
x
)
=
γ
+
ln
x
−
Ein
(
−
x
)
x
≠
0
{\displaystyle \operatorname {Ei} (x)\,=\,\gamma +\ln {x}-\operatorname {Ein} (-x)\qquad x\neq 0}
Relation with other functions
Kummer's equation
z
d
2
w
d
z
2
+
(
b
−
z
)
d
w
d
z
−
a
w
=
0
{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0}
is usually solved by the
confluent hypergeometric functions
M
(
a
,
b
,
z
)
{\displaystyle M(a,b,z)}
and
U
(
a
,
b
,
z
)
.
{\displaystyle U(a,b,z).}
But when
a
=
0
{\displaystyle a=0}
and
b
=
1
,
{\displaystyle b=1,}
that is,
z
d
2
w
d
z
2
+
(
1
−
z
)
d
w
d
z
=
0
{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(1-z){\frac {dw}{dz}}=0}
we have
M
(
0
,
1
,
z
)
=
U
(
0
,
1
,
z
)
=
1
{\displaystyle M(0,1,z)=U(0,1,z)=1}
for all z . A second solution is then given by E1 (−z ). In fact,
E
1
(
−
z
)
=
−
γ
−
i
π
+
∂
U
(
a
,
1
,
z
)
−
M
(
a
,
1
,
z
)
∂
a
,
0
<
A
r
g
(
z
)
<
2
π
{\displaystyle E_{1}(-z)=-\gamma -i\pi +{\frac {\partial [U(a,1,z)-M(a,1,z)]}{\partial a}},\qquad 0<{\rm {Arg}}(z)<2\pi }
with the derivative evaluated at
a
=
0.
{\displaystyle a=0.}
Another connexion with the confluent hypergeometric functions is that E1 is an exponential times the function U (1,1,z ):
E
1
(
z
)
=
e
−
z
U
(
1
,
1
,
z
)
{\displaystyle E_{1}(z)=e^{-z}U(1,1,z)}
The exponential integral is closely related to the
logarithmic integral function li(x ) by the formula
li
(
e
x
)
=
Ei
(
x
)
{\displaystyle \operatorname {li} (e^{x})=\operatorname {Ei} (x)}
for non-zero real values of
x
{\displaystyle x}
.
Generalization
The exponential integral may also be generalized to
E
n
(
x
)
=
∫
1
∞
e
−
x
t
t
n
d
t
,
{\displaystyle E_{n}(x)=\int _{1}^{\infty }{\frac {e^{-xt}}{t^{n}}}\,dt,}
which can be written as a special case of the upper
incomplete gamma function :
[11]
E
n
(
x
)
=
x
n
−
1
Γ
(
1
−
n
,
x
)
.
{\displaystyle E_{n}(x)=x^{n-1}\Gamma (1-n,x).}
The generalized form is sometimes called the Misra function
[12]
φ
m
(
x
)
{\displaystyle \varphi _{m}(x)}
, defined as
φ
m
(
x
)
=
E
−
m
(
x
)
.
{\displaystyle \varphi _{m}(x)=E_{-m}(x).}
Many properties of this generalized form can be found in the
NIST Digital Library of Mathematical Functions.
Including a logarithm defines the generalized integro-exponential function
[13]
E
s
j
(
z
)
=
1
Γ
(
j
+
1
)
∫
1
∞
(
log
t
)
j
e
−
z
t
t
s
d
t
.
{\displaystyle E_{s}^{j}(z)={\frac {1}{\Gamma (j+1)}}\int _{1}^{\infty }\left(\log t\right)^{j}{\frac {e^{-zt}}{t^{s}}}\,dt.}
The indefinite integral:
Ei
(
a
⋅
b
)
=
∬
e
a
b
d
a
d
b
{\displaystyle \operatorname {Ei} (a\cdot b)=\iint e^{ab}\,da\,db}
is similar in form to the ordinary
generating function for
d
(
n
)
{\displaystyle d(n)}
, the number of
divisors of
n
{\displaystyle n}
:
∑
n
=
1
∞
d
(
n
)
x
n
=
∑
a
=
1
∞
∑
b
=
1
∞
x
a
b
{\displaystyle \sum \limits _{n=1}^{\infty }d(n)x^{n}=\sum \limits _{a=1}^{\infty }\sum \limits _{b=1}^{\infty }x^{ab}}
Derivatives
The derivatives of the generalised functions
E
n
{\displaystyle E_{n}}
can be calculated by means of the formula
[14]
E
n
′
(
z
)
=
−
E
n
−
1
(
z
)
(
n
=
1
,
2
,
3
,
…
)
{\displaystyle E_{n}'(z)=-E_{n-1}(z)\qquad (n=1,2,3,\ldots )}
Note that the function
E
0
{\displaystyle E_{0}}
is easy to evaluate (making this recursion useful), since it is just
e
−
z
/
z
{\displaystyle e^{-z}/z}
.
[15]
Exponential integral of imaginary argument
E
1
(
i
x
)
{\displaystyle E_{1}(ix)}
against
x
{\displaystyle x}
; real part black, imaginary part red.
If
z
{\displaystyle z}
is imaginary, it has a nonnegative real part, so we can use the formula
E
1
(
z
)
=
∫
1
∞
e
−
t
z
t
d
t
{\displaystyle E_{1}(z)=\int _{1}^{\infty }{\frac {e^{-tz}}{t}}\,dt}
to get a relation with the
trigonometric integrals
Si
{\displaystyle \operatorname {Si} }
and
Ci
{\displaystyle \operatorname {Ci} }
:
E
1
(
i
x
)
=
i
−
1
2
π
+
Si
(
x
)
−
Ci
(
x
)
(
x
>
0
)
{\displaystyle E_{1}(ix)=i\left[-{\tfrac {1}{2}}\pi +\operatorname {Si} (x)\right]-\operatorname {Ci} (x)\qquad (x>0)}
The real and imaginary parts of
E
1
(
i
x
)
{\displaystyle \mathrm {E} _{1}(ix)}
are plotted in the figure to the right with black and red curves.
Approximations
There have been a number of approximations for the exponential integral function. These include:
The Swamee and Ohija approximation
[16]
E
1
(
x
)
=
(
A
−
7.7
+
B
)
−
0.13
,
{\displaystyle E_{1}(x)=\left(A^{-7.7}+B\right)^{-0.13},}
where
A
=
ln
(
0.56146
x
+
0.65
)
(
1
+
x
)
B
=
x
4
e
7.7
x
(
2
+
x
)
3.7
{\displaystyle {\begin{aligned}A&=\ln \left[\left({\frac {0.56146}{x}}+0.65\right)(1+x)\right]\\B&=x^{4}e^{7.7x}(2+x)^{3.7}\end{aligned}}}
The Allen and Hastings approximation
[16]
[17]
E
1
(
x
)
=
{
−
ln
x
+
a
T
x
5
,
x
≤
1
e
−
x
x
b
T
x
3
c
T
x
3
,
x
≥
1
{\displaystyle E_{1}(x)={\begin{cases}-\ln x+{\textbf {a}}^{T}{\textbf {x}}_{5},&x\leq 1\\{\frac {e^{-x}}{x}}{\frac {{\textbf {b}}^{T}{\textbf {x}}_{3}}{{\textbf {c}}^{T}{\textbf {x}}_{3}}},&x\geq 1\end{cases}}}
where
a
≜
−
0.57722
,
0.99999
,
−
0.24991
,
0.05519
,
−
0.00976
,
0.00108
T
b
≜
0.26777
,
8.63476
,
18.05902
,
8.57333
T
c
≜
3.95850
,
21.09965
,
25.63296
,
9.57332
T
x
k
≜
x
0
,
x
1
,
…
,
x
k
T
{\displaystyle {\begin{aligned}{\textbf {a}}&\triangleq [-0.57722,0.99999,-0.24991,0.05519,-0.00976,0.00108]^{T}\\{\textbf {b}}&\triangleq [0.26777,8.63476,18.05902,8.57333]^{T}\\{\textbf {c}}&\triangleq [3.95850,21.09965,25.63296,9.57332]^{T}\\{\textbf {x}}_{k}&\triangleq [x^{0},x^{1},\dots ,x^{k}]^{T}\end{aligned}}}
The continued fraction expansion
[17]
E
1
(
x
)
=
e
−
x
x
+
1
1
+
1
x
+
2
1
+
2
x
+
3
⋱
.
{\displaystyle E_{1}(x)={\cfrac {e^{-x}}{x+{\cfrac {1}{1+{\cfrac {1}{x+{\cfrac {2}{1+{\cfrac {2}{x+{\cfrac {3}{\ddots }}}}}}}}}}}}.}
The approximation of Barry et al.
[18]
E
1
(
x
)
=
e
−
x
G
+
(
1
−
G
)
e
−
x
1
−
G
ln
1
+
G
x
−
1
−
G
(
h
+
b
x
)
2
,
{\displaystyle E_{1}(x)={\frac {e^{-x}}{G+(1-G)e^{-{\frac {x}{1-G}}}}}\ln \left[1+{\frac {G}{x}}-{\frac {1-G}{(h+bx)^{2}}}\right],}
where:
h
=
1
1
+
x
x
+
h
∞
q
1
+
q
q
=
20
47
x
31
26
h
∞
=
(
1
−
G
)
(
G
2
−
6
G
+
12
)
3
G
(
2
−
G
)
2
b
b
=
2
(
1
−
G
)
G
(
2
−
G
)
G
=
e
−
γ
{\displaystyle {\begin{aligned}h&={\frac {1}{1+x{\sqrt {x}}}}+{\frac {h_{\infty }q}{1+q}}\\q&={\frac {20}{47}}x^{\sqrt {\frac {31}{26}}}\\h_{\infty }&={\frac {(1-G)(G^{2}-6G+12)}{3G(2-G)^{2}b}}\\b&={\sqrt {\frac {2(1-G)}{G(2-G)}}}\\G&=e^{-\gamma }\end{aligned}}}
with
γ
{\displaystyle \gamma }
being the
Euler–Mascheroni constant .
Inverse function of the Exponential Integral
We can express the
Inverse function of the exponential integral in
power series form:
[19]
∀
|
x
|
<
μ
ln
(
μ
)
,
E
i
−
1
(
x
)
=
∑
n
=
0
∞
x
n
n
!
P
n
(
ln
(
μ
)
)
μ
n
{\displaystyle \forall |x|<{\frac {\mu }{\ln(\mu )}},\quad \mathrm {Ei} ^{-1}(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}{\frac {P_{n}(\ln(\mu ))}{\mu ^{n}}}}
where
μ
{\displaystyle \mu }
is the
Ramanujan–Soldner constant and
(
P
n
)
{\displaystyle (P_{n})}
is
polynomial sequence defined by the following
recurrence relation :
P
0
(
x
)
=
x
,
P
n
+
1
(
x
)
=
x
(
P
n
′
(
x
)
−
n
P
n
(
x
)
)
.
{\displaystyle P_{0}(x)=x,\ P_{n+1}(x)=x(P_{n}'(x)-nP_{n}(x)).}
For
n
>
0
{\displaystyle n>0}
,
deg
P
n
=
n
{\displaystyle \deg P_{n}=n}
and we have the formula :
P
n
(
x
)
=
(
d
d
t
)
n
−
1
(
t
e
x
E
i
(
t
+
x
)
−
E
i
(
x
)
)
n
|
t
=
0
.
{\displaystyle P_{n}(x)=\left.\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)^{n-1}\left({\frac {te^{x}}{\mathrm {Ei} (t+x)-\mathrm {Ei} (x)}}\right)^{n}\right|_{t=0}.}
Applications
Time-dependent
heat transfer
Nonequilibrium
groundwater flow in the
Theis solution (called a well function )
Radiative transfer in stellar and planetary atmospheres
Radial diffusivity equation for transient or unsteady state flow with line sources and sinks
Solutions to the
neutron transport equation in simplified 1-D geometries
[20]
See also
Notes
^ Abramowitz and Stegun, p. 228
^ Abramowitz and Stegun, p. 228, 5.1.1
^ Abramowitz and Stegun, p. 228, 5.1.4 with n = 1
^ Abramowitz and Stegun, p. 228, 5.1.7
^ Abramowitz and Stegun, p. 229, 5.1.11
^ Bleistein and Handelsman, p. 2
^ Bleistein and Handelsman, p. 3
^ O’Malley, Robert E. (2014), O'Malley, Robert E. (ed.),
"Asymptotic Approximations" , Historical Developments in Singular Perturbations , Cham: Springer International Publishing, pp. 27–51,
doi :
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^ Abramowitz and Stegun, p. 229, 5.1.20
^ Abramowitz and Stegun, p. 228, see footnote 3.
^ Abramowitz and Stegun, p. 230, 5.1.45
^ After Misra (1940), p. 178
^ Milgram (1985)
^ Abramowitz and Stegun, p. 230, 5.1.26
^ Abramowitz and Stegun, p. 229, 5.1.24
^
a
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^
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^
"Inverse function of the Exponential Integral Ei-1 (x ) " . Mathematics Stack Exchange . Retrieved 2024-04-24 .
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External links