You may want to read this overview first: What is an Integral (Entire) Function?

## What is the Exponential Integral Function?

The **exponential integral function** is a special function used in astrophysics, quantum chemistry and many other applied sciences.

It can be defined in two different ways: as a real-valued function or as a complex-valued function.

## 1. Real-Valued Exponential Integral Function

The exponential integral function is defined as a definite integral, with a ratio of the exponential function and its argument:

The function is defined for x in the set of natural numbers. This set excludes zero.

Or, equivalently the function can be defined a little differently through a parity transformation. A *parity transformation* is where the signs are flipped: t→ -t and x→ -x. This gives what Enrico Masina calls a “more suitable” definition:

Note though, that despite the stated suitability of the above form, most authors use the notation E_{1} *only* for the complex-valued version of the function.

## 2. Exponential Integral Function Defined on the Complex Plane

On the complex plane, the function E_{1}is also a definite integral.

The notation is almost the same, with a couple of notable differences:

- The substitution of “z” (to denote a complex number) instead of “x”.
- While the “real” version of the function above is defined for the set of whole numbers, the complex-numbered version is defined for |arg(z) < π)|.*

*This comes from the exponential form of a complex number, z = |z|e^{iθ}, which is just z = x + i y rewritten with exponentials; θ = arg(z) = arctan(y/x).

The complex valued version is not valid when z = 0 or z = ∞, because of a specific kind of discontinuity called a branch point.

## References

Masina, E. (2017). A review on the Exponential-Integral special function and other strictly related special functions. Lectures from a seminar of Mathematical Physics. Retrieved December 10, 2019 from: https://www.researchgate.net/publication/323772322_A_review_on_the_Exponential-Integral_special_function_and_other_strictly_related_special_functions_Lectures_from_a_seminar_of_Mathematical_Physics

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