Calculus How To

Exponential Integral Function

Share on

Types of Functions >

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

exponential integral function graph

Graph of the 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:
exponential integral function

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:
parity transformation

Note though, that despite the stated suitability of the above form, most authors use the notation E1 only for the complex-valued version of the function.

2. Exponential Integral Function Defined on the Complex Plane

On the complex plane, the function E1is also a definite integral.
complex plane exponential function

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, 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.


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:

Stephanie Glen. "Exponential Integral Function" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published. Required fields are marked *