The exponential integral function: Ei

6.8.5 The exponential integral function: Ei

The exponential integral Ei is defined for non-zero real numbers x by

Ei(x) =

∫

t=−∞

exp(t)

dt.

For x>0, this integral is improper but the principal value exists. This function satisfies Ei(0) = −∞, Ei(−∞) = 0.

Since

exp(x)

+ 1 +

x²

+ …,

the Ei function can be extended to ℂ − {0} (with a branch cut on the positive real axis) by

Ei(z) = ln(z) + γ + x +

x²

2· 2!

x³

3· 3!

+ …

where γ = 0.57721566490… is Euler’s constant.

The Ei command takes one or two arguments.

With one argument, the Ei command computes the exponential integral.

Ei takes one argument:
z, a complex number.
Ei(z) returns the value of the exponential integral at z.

Examples.

Input:
Ei(1.0)
Output:
1.89511781636
Input:
Ei(-1.0)
Output:
−0.219383934396
Input:
Ei(1.)-Ei(-1.)
Output:
2.11450175075
Input:
int((exp(x)-1)/x,x=-1..1.)
Output:
2.11450175075
The input:
Input:
evalf(Ei(-1)-sum((-1)^n/n/n!,n=1..100))
approximates the Euler’s constant γ
Output:
0.577215664902

Another type of exponential integral is

E₁(x) =

∫

∞

exp(−t)

dt =

∫

∞

exp(−tx)

which satisfies

E₁(x) = −Ei(−x)

This can be generalized to

E_n(x) =

∫

∞

exp(−tx)ⁿ

These functions satisfy

E₁(x)	= −Ei(x)
E₂(x)	= e^−x+ xEi(−x) = e^−x − x*E₁(x)

and, for n ≥ 2,

E_n(x)= (e^−x−x E_n−1(x))/(n−1)

With two arguments, the Ei command computes this version of the exponential integral.

Ei takes two arguments:
- z, a complex number.
- n, a positive integer.
Ei(z,n) returns the value of E_n(z).

Examples.

Input:
Ei(1.0,1)
Output:
0.219383934396
Input:
Ei(3.0,2)
Output:
0.0106419250853