E_n-Function

The function is defined by the integral

(1)

and is given by the Wolfram Language function ExpIntegralE[n, x]. Defining so that ,

(2)

For integer ,

(3)

Plots in the complex plane are shown above for .

The special case gives

			(4)
			(5)
			(6)
			(7)

where is the exponential integral and is an incomplete gamma function. It is also equal to

(8)

where is the Euler-Mascheroni constant.

			(9)
			(10)

where and are the cosine integral and sine integral.

The function satisfies the recurrence relations

			(11)
			(12)

In general, can be built up from the recurrence

(13)

The series expansions is given by

(14)

and the asymptotic expansion by

(15)