Exponential integral function

7.3.5 Exponential integral function

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

Ei(x)=

∫

−∞

e^t

dt.

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

Since e^x/x=1/x+1+x/2! + x²/3!+⋯, 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 the Euler-Mascheroni constant.

The Ei command takes one or two arguments.

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

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

Examples

Ei(1.0)

1.89511781636

Ei(-1.0)

−0.219383934396

Ei(1.)-Ei(-1.)

2.11450175075

int((exp(x)-1)/x,x=-1..1.)

2.11450175075

The following input approximates the Euler’s constant γ:

evalf(Ei(-1)-sum((-1)^n/n/n!,n=1..100))

0.577215664902

Another type of exponential integral is

E₁(x)=

∫

+∞

e^−t

dt =

∫

+∞

e^−tx

which satisfies E₁(x)=−Ei(−x). This can be generalized to

E_n(x)=

∫

+∞

e^−ntx

dt.

These functions satisfy

E₁(x)	=−Ei(x),
E₂(x)	=e^−x+ x Ei(−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 the above 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

Ei(1.0,1)

0.219383934396

Ei(3.0,2)

0.0106419250853