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Generalization

The exponential integral may also be generalized to

which can be written as a special case of the upper incomplete gamma function:

The generalized form is sometimes called the Misra function φ m (x) {\displaystyle \varphi _{m}(x)}, defined as

Many properties of this generalized form can be found in the NIST Digital Library of Mathematical Functions.

Including a logarithm defines the generalized integro-exponential function

The indefinite integral:

is similar in form to the ordinary generating function for d (n) {\displaystyle d(n)}, the number of divisors of n {\displaystyle n}: