The best-known properties and formulas for exponential integrals

Real values for real arguments

For real values of parameter and positive argument , the values of the exponential integral are real (or infinity). For real values of argument , the values of the exponential integral , the sine integral , and the hyperbolic sine integral are real. For real positive values of argument , the values of the logarithmic integral , the cosine integral , and the hyperbolic cosine integral are real.

Simple values at zero

The exponential integrals have rather simple values for argument :

Specific values for specialized parameter

If the parameter equals , the exponential integral can be expressed through an exponential function multiplied by a simple rational function. If the parameter equals , the exponential integral can be expressed through the exponential integral , and the exponential and logarithmic functions:

The previous formulas are the particular cases of the following general formula:

If the parameter equals , the exponential integral can be expressed through the probability integral , and the exponential and power functions, for example:

The previous formulas can be generalized by the following general representation of this class of particular cases:

Analyticity

The exponential integrals , , , , , , and are defined for all complex values of the parameter and the variable . The function is an analytical functions of and over the whole complex ‐ and ‐planes excluding the branch cut on the ‐plane. For fixed , the exponential integral is an entire function of . The sine integral and the hyperbolic sine integral are entire functions of .

Poles and essential singularities

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

The exponential integral , the cosine integral , and the hyperbolic cosine integral have an essential singularity at .

The function does not have poles and essential singularities.

The sine integral and the hyperbolic sine integral have an essential singularity at .

Branch points and branch cuts

For fixed , the function does not have branch points and branch cuts.

For fixed , not being a nonpositive integer, the function has two branch points and , and branch cuts along the interval . At the same time, the point is an essential singularity for this function.

The exponential integral , the cosine integral, and the hyperbolic cosine integral have two branch points and .

The function has three branch points , , and .

The sine integral and hyperbolic sine integral do not have branch points or branch cuts.

For fixed , not being a nonpositive integer, the function is a single‐valued function on the ‐plane cut along the interval , where it is continuous from above:

The function is a single‐valued function on the ‐plane cut along the interval , where it has discontinuities from both sides:

The function is a single‐valued function on the ‐plane cut along the interval . It is continuous from above along the interval and it has discontinuities from both sides along the interval :

The cosine integral and hyperbolic cosine integral are single‐valued functions on the ‐plane cut along the interval where they are continuous from above:

From below, these functions have discontinuity that are described by the formulas:

Periodicity

The exponential integrals , , , , , , and do not have periodicity.

Parity and symmetry

The exponential integral has mirror symmetry:

The sine integral and the hyperbolic sine integral are odd functions and have mirror symmetry:

The exponential integral , logarithmic integral , cosine integral , and hyperbolic cosine integral have mirror symmetry (except on the branch cut interval (-∞, 0)):

Series representations

The exponential integrals , , , , , , and have the following series expansions through series that converge on the whole ‐plane:

Interestingly, closed‐form expressions for the truncated version of the Taylor series at the origin can be expressed through the generalized hypergeometric function , for example:

Asymptotic series expansions

The asymptotic behavior of the exponential integrals , , , , , , and can be described by the following formulas (only the main terms of the asymptotic expansions are given):

The previous formulas are valid in any direction of approaching point to infinity (z∞). In particular cases, these formulas can be simplified to the following relations:

Integral representations

The exponential integrals , , , and can also be represented through the following equivalent integrals:

The symbol in the second and third integrals means that these integrals evaluate as the Cauchy principal value of the singular integral: .

Transformations

The arguments of the exponential integrals , , , , and that contain square roots can sometimes be simplified:

Identities

The exponential integral satisfies the following recurrence identities:

All of the preceding formulas can be generalized to the following recurrence identities with a jump of length :

Simple representations of derivatives

The derivative of the exponential integral with respect to the variable has a simple representation through itself, but with a different parameter:

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

The derivatives of the other exponential integrals , , , , , and have simple representations through simple elementary functions:

The symbolic -order derivatives with respect to the variable of all exponential integrals , , , , , , and have the following representations:

Differential equations

The exponential integrals , , , , , and satisfy the following linear differential equations of second or third orders:

where , , and are arbitrary constants.

The logarithmic integral satisfies the following ordinary second-order nonlinear differential equation: