The "complete" gamma function can be generalized to the incomplete gamma function such that . This "upper" incomplete gamma function is given by
(1)
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For an integer
(2)
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(3)
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where is the exponential sum function. It is implemented as Gamma[a, z] in the Wolfram Language.
The special case of can be expressed in terms of the subfactorial as
(4)
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The incomplete gamma function has continued fraction
(5)
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(Wall 1948, p. 358).
The lower incomplete gamma function is given by
(6)
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(7)
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(8)
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where is the confluent hypergeometric function of the first kind. For an integer ,
(9)
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(10)
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It is implemented as Gamma[a, 0, z] in the Wolfram Language.
By definition, the lower and upper incomplete gamma functions satisfy
(11)
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The exponential integral is closely related to the incomplete gamma function by
(12)
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Therefore, for real ,
(13)
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