The exponential integral function: Ei

### 5.8.5 The exponential integral function: Ei

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

Ei(x) = | ∫ |
x
t=−∞
exp(t)
t
| dt.

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

Since

exp(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 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) = | ∫ |
∞
x
exp(−t)
t
| dt = | ∫ |
∞
1
exp(−tx)
t
| dt

which satisfies

E1(x) = −Ei(−x)

This can be generalized to

En(x) = | ∫ |
∞
1
exp(−tx)n
t
| dt

These functions satisfy

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

and, for n ≥ 2,

En(x)= (e−x−x En−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 En(z).

#### Examples.

- Input:
  Ei(1.0,1)
  Output:

  0.219383934396

- Input:
  Ei(3.0,2)
  Output:

  0.0106419250853