The best-known properties and formulas for exponential integrals

Real values for real arguments

For real values of , the values of the gamma function are real (or infinity). For real values of the parameter and positive arguments , , , the values of the gamma functions , , , , and are real (or infinity).

Simple values at zero

The gamma functions , , , , , , , and have the following values at zero arguments:

Specific values for specialized variables

If the variable is equal to and , the incomplete gamma function coincides with the gamma function and the corresponding regularized gamma function is equal to :

In cases when the parameter equals , the incomplete gamma functions and can be expressed as an exponential function multiplied by a polynomial. In cases when the parameter equals , the incomplete gamma function can be expressed with the exponential integral , exponential, and logarithmic functions, but the regularized incomplete gamma function is equal to . In cases when the parameter equals the incomplete gamma functions and can be expressed through the complementary error function and the exponential function, for example:

These formulas are particular cases of the following general formulas:

If the argument , the log‐gamma function can be evaluated at these points where the gamma function can be evaluated in closed form. The log‐gamma function can also be represented recursively in terms of for :

The generalized incomplete gamma functions and in particular cases can be represented through incomplete gamma functions and and the gamma function :

The inverse of the regularized incomplete gamma functions and for particular values of arguments satisfy the following relations:

Analyticity

The gamma functions , , , , , and are defined for all complex values of their arguments.

The functions and are analytic functions of and over the whole complex ‐ and ‐planes excluding the branch cut on the ‐plane. For fixed , they are entire functions of . The functions and are analytic functions of , , and over the whole complex ‐, ‐, and ‐planes excluding the branch cuts on the ‐ and ‐planes. For fixed and , they are entire functions of .

The function is an analytical function of over the whole complex ‐plane excluding the branch cut.

Poles and essential singularities

For fixed , the functions and have an essential singularity at . At the same time, the point is a branch point for generic . For fixed , the functions and have only one singular point at . It is an essential singularity.

For fixed , the functions and have an essential singularity at (for fixed ) and at (for fixed ). At the same time, the points are branch points for generic . For fixed and , the functions and have only one singular point at . It is an essential singularity.

The function does not have poles or essential singularities.

Branch points and branch cuts

For fixed , not a positive integer, the functions and have two branch points: and .

For fixed , not a positive integer, the functions and are single‐valued functions on the ‐plane cut along the interval , where they are continuous from above:

For fixed , the functions and do not have branch points and branch cuts.

For fixed , or fixed , (with ), the functions and have two branch points with respect to or : , .

For fixed and , the functions and are single‐valued functions on the ‐plane cut along the interval , where they are continuous from above:

For fixed and , the functions and are single‐valued functions on the ‐plane cut along the interval , where they are continuous from above:

For fixed and , the functions and do not have branch points and branch cuts.

The function has two branch points: and .

The function is a single‐valued function on the ‐plane cut along the interval , where it is continuous from above:

Periodicity

The gamma functions , , , , , the log‐gamma function , and their inverses and do not have periodicity.

Parity and symmetry

The gamma functions , , , , and the log‐gamma function have mirror symmetry (except on the branch cut intervals):

Two of the gamma functions have the following permutation symmetry:

Series representations

The gamma functions , , , , the log‐gamma function , and the inverse have the following series expansions:

Asymptotic series expansions

The asymptotic behavior of the gamma functions and , the log‐gamma function , and the inverse can be described by the following formulas (only the main terms of asymptotic expansion are given):

Integral representations

The gamma functions , , , , and the log‐gamma function can also be represented through the following integrals:

Transformations

The argument of the log‐gamma function can be simplified if or :

Multiple arguments

The log‐gamma function with can be represented by a formula that follows from the corresponding multiplication formula for the gamma function :

Identities

The gamma functions , , , , and the log‐gamma function satisfy the following recurrence identities:

The previous formulas can be generalized to the following recurrence identities with a jump of length n:

Representations of derivatives

The derivatives of the gamma functions , , , and with respect to the variables , , and have simple representations in terms of elementary functions:

The derivatives of the log‐gamma function and the inverses of the regularized incomplete gamma functions , and with respect to the variables , , and have more complicated representations by the formulas:

The derivative of the exponential integral by its parameter can be represented in terms of the regularized hypergeometric function :

The derivatives of the gamma functions , , , and , and their inverses and with respect to the parameter can be represented in terms of the regularized hypergeometric function :

The symbolic -order derivatives of all gamma functions , , , , and their inverses , and have the following representations:

Differential equations

The gamma functions , , , and satisfy the following second-order linear differential equations:

where and are arbitrary constants.

The log‐gamma function satisfies the following simple first-order linear differential equation:

The inverses of the regularized incomplete gamma functions and satisfy the following ordinary nonlinear second-order differential equation: