The bestknown 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 secondorder nonlinear differential equation:
