Plot of the exponential integral function E_n(z) with n=2 for complex z
Plot of the exponential integral function E2(z) for complex z

In mathematics, the exponential integral ⁠Ei {\displaystyle \operatorname {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 of⁠x {\displaystyle x}⁠, the exponential integral ⁠Ei ⁡ ( x ) {\displaystyle \operatorname {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 {\displaystyle \operatorname {Ei} }⁠ is not an elementary function. The definition above can be used for positive values of⁠x {\displaystyle 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 {\displaystyle 0}⁠ and ⁠∞ {\displaystyle \infty }⁠. Instead of ⁠Ei {\displaystyle \operatorname {Ei} }⁠, the following notation is used, E 1 ( z ) = ∫ z ∞ e − t t d t , | Arg ( z ) | < π {\displaystyle E_{1}(z)=\int _{z}^{\infty }{\frac {e^{-t}}{t}}\,dt,\qquad |{\operatorname {Arg} }(z)|<\pi }

Plot of the exponential integral function Ei(z) for complex z
Plot of the exponential integral function Ei(z) for complex z

For positive values of⁠x {\displaystyle 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 {\displaystyle 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 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 {\displaystyle E_{1}}⁠ near the branch cut can be seen by the following relation: 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 E1 function (top) and Ei function (bottom).

For real or complex arguments off the negative real axis, E 1 ( z ) {\displaystyle E_{1}(z)} can be expressed as 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 {\displaystyle 0}⁠ and ⁠2.5 {\displaystyle 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 \mathrm {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 of terms in the truncated sum

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)}⁠. However, for positive values of ⁠x {\displaystyle 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: 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

Normalized exponential integral. The value plotted is ⁠Ei(x)/(exp x)/x⁠. The values of x are written above the corresponding point. The horizontal spacing is according to arctan x. The graph is extended "beyond infinity" a little on both the right and the left to show how the normalized function behaves when 1/x is small. (The horizontal spacing for these points corresponds to angles whose tangent is x.)

Using integration by parts, we can obtain an explicit formula 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 {\begin{aligned}\operatorname {Ei} (z)&={\frac {e^{z}}{z}}\left(\sum _{k=0}^{n}{\frac {k!}{z^{k}}}+e_{n}(z)\right),\\e_{n}(z)&\equiv (n+1)!\ ze^{-z}\int _{-\infty }^{z}{\frac {e^{t}}{t^{n+2}}}\,dt.\end{aligned}}} 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 \vert z\vert }⁠, at which point ⁠| e n ( z ) | ≤ 2 π | z | e − | z | {\displaystyle \textstyle \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 E1 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: 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} }⁠ 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 + Ein ⁡ ( z ) , | Arg ⁡ ( z ) | < π ; Ei ⁡ ( x ) = − γ + ln ⁡ x − E i n ( − x ) , x ≠ 0. {\displaystyle {\begin{aligned}E_{1}(z)&=-\gamma -\ln z+\operatorname {Ein} (z),&&\left|\operatorname {Arg} (z)\right|<\pi;\\\operatorname {Ei} (x)&={\hphantom {-}}\gamma +\ln {x}-\mathrm {Ein} (-x),&&x\neq 0.\end{aligned}}} The function Ein {\displaystyle \operatorname {Ein} } is related to the exponential generating function of the harmonic numbers: Ein ⁡ ( z ) = e − z ∑ n = 1 ∞ z n n ! H n {\displaystyle \operatorname {Ein} (z)=e^{-z}\,\sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}H_{n}}

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 {\displaystyle z}⁠. A second solution is then given by ⁠E 1 ( − z ) {\displaystyle E_{1}(-z)}⁠. In fact, E 1 ( − z ) = − γ − i π + ∂ [ U ( a , 1 , z ) − M ( a , 1 , z ) ] ∂ a , 0 < Arg ⁡ ( z ) < 2 π {\displaystyle E_{1}(-z)=-\gamma -i\pi +{\frac {\partial [U(a,1,z)-M(a,1,z)]}{\partial a}},\qquad 0<\operatorname {Arg} (z)<2\pi } with the derivative evaluated at ⁠a = 0 {\displaystyle a=0}⁠. Another connexion with the confluent hypergeometric functions is that ⁠E 1 {\displaystyle E_{1}}⁠ is an exponential times the function ⁠U ( 1 , 1 , z ) {\displaystyle 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 ⁡ ( e x ) {\displaystyle \operatorname {li} (e^{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}⁠.

The series expansion of the exponential integral immediately gives rise to an expression in terms of the generalized hypergeometric function ⁠2 F 2 {\displaystyle {}_{2}\!F_{2}}⁠: Ei ⁡ ( x ) = x 2 F 2 ( 1 , 1 ; 2 , 2 ; x ) + ln ⁡ x + γ . {\displaystyle \operatorname {Ei} (x)=x{\,}_{2}\!F_{2}(1,1;2,2;x)+\ln x+\gamma .}

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: 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 ⁠φ m ( x ) {\displaystyle \varphi _{m}(x)}⁠, defined as φ m ( x ) = E − m ( x ) . {\displaystyle \varphi _{m}(x)=E_{-m}(x).}

Including a logarithm defines the generalized integro-exponential function 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.}

Derivatives

The derivatives of the generalised functions E n {\displaystyle E_{n}} can be calculated by means of the formula 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 \textstyle {\frac {e^{-z}}{z}}}⁠.

Exponential integral of imaginary argument

E1(ix) against 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 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 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 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}}^{\mathsf {T}}{\textbf {x}}_{5},&x\leq 1\\{\frac {e^{-x}}{x}}{\frac {{\textbf {b}}^{\mathsf {T}}{\textbf {x}}_{3}}{{\textbf {c}}^{\mathsf {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]^{\mathsf {T}}\\{\textbf {b}}&\triangleq [0.26777,8.63476,18.05902,8.57333]^{\mathsf {T}}\\{\textbf {c}}&\triangleq [3.95850,21.09965,25.63296,9.57332]^{\mathsf {T}}\\{\textbf {x}}_{k}&\triangleq [x^{0},x^{1},\dots ,x^{k}]^{\mathsf {T}}\end{aligned}}}
  • The continued fraction expansion 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. 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: ∀ | x | < μ ln ⁡ ( μ ) , Ei − 1 ⁡ ( x ) = ∑ n = 0 ∞ x n n ! P n ( ln ⁡ ( μ ) ) μ n {\displaystyle \forall |x|<{\frac {\mu }{\ln(\mu )}},\quad \operatorname {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 Ei ⁡ ( t + x ) − Ei ⁡ ( x ) ) n | t = 0 . {\displaystyle P_{n}(x)=\left.\left({\frac {d}{dt}}\right)^{n-1}\left({\frac {te^{x}}{\operatorname {Ei} (t+x)-\operatorname {Ei} (x)}}\right)^{n}\right|_{t=0}.}

Applications

See also

Footnotes

Citations

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