In mathematics, the exponential integral Ei(x) is defined as

Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value.

The exponential integral arises also in the following sum:

where γ is the Euler gamma constant.

The exponential integral is closely related to the logarithmic integral function li(x),

li(x) = Ei (ln (x))    for all positive real x ≠ 1.

Also closely related is a function which integrates over a different range:

This function may be regarded as extending the exponential integral to the negative reals by Ei( - x) = - E₁(x). We can express both of them in terms of an entire function,

.

Using this function, we then may define, using the logarithm, E₁(x) = - γ - lnx + Ein(x) and Ei(x) = γ + lnx + Ein( - x).

The exponential integral may also be generalized to .

References

• Milton Abromowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions