# Definition:Exponential Integral Function

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## Definition

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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.