7.3.5 Exponential integral function
The exponential integral Ei is defined for non-zero real numbers x
by
For x>0, this integral is improper but the principal value exists.
Also, Ei(0)=−∞, Ei(−∞)=0.
Since ex/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
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
int((exp(x)-1)/x,x=-1..1.) |
The following input approximates the Euler’s constant γ:
evalf(Ei(-1)-sum((-1)^n/n/n!,n=1..100)) |
Another type of exponential integral is
which satisfies E1(x)=−Ei(−x).
This can be generalized to
These functions satisfy
| E1(x) | =−Ei(x), | | | | | | | | | |
E2(x) | =e−x+ x Ei(−x)=e−x−x E1(x),
| | | | | | | | | |
|
and, for n ≥ 2,
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 En(z).
Examples