# Definition:Exponential Integral Function

## Definition

The exponential integral function comes in two forms:

### Formulation 1

The exponential integral function is the real function $E_1: \R_{>0} \to \R$ defined as:

$\map {E_1} x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$

### Formulation 2

The exponential integral function is the real function $\Ei: \R_{>0} \to \R$ defined as:

$\map \Ei x = \PV_{t \mathop \to -\infty}^{t \mathop = x} \frac {e^t} t \rd t$

where $\PV$ denotes the Cauchy principal value.

## Also see

• Results about the exponential integral function can be found here.