# ei

One-argument exponential integral function

## Syntax

``ei(x)``

## Description

example

````ei(x)` returns the one-argument exponential integral defined as$\text{ei}\left(x\right)=\underset{-\text{ }\infty }{\overset{x}{\int }}\frac{{e}^{t}}{t}\text{\hspace{0.17em}}dt.$```

## Examples

### Exponential Integral for Floating-Point and Symbolic Numbers

Compute exponential integrals for numeric inputs. Because these numbers are not symbolic objects, you get floating-point results.

`s = [ei(-2), ei(-1/2), ei(1), ei(sqrt(2))]`
```s = -0.0489 -0.5598 1.8951 3.0485```

Compute exponential integrals for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, `ei` returns unresolved symbolic calls.

`s = [ei(sym(-2)), ei(sym(-1/2)), ei(sym(1)), ei(sqrt(sym(2)))]`
```s = [ ei(-2), ei(-1/2), ei(1), ei(2^(1/2))]```

Use `vpa` to approximate this result with 10-digit accuracy.

`vpa(s, 10)`
```ans = [ -0.04890051071, -0.5597735948, 1.895117816, 3.048462479]```

### Branch Cut at Negative Real Axis

The negative real axis is a branch cut. The exponential integral has a jump of height 2 π i when crossing this cut. Compute the exponential integrals at `-1`, above `-1`, and below `-1` to demonstrate this.

`[ei(-1), ei(-1 + 10^(-10)*i), ei(-1 - 10^(-10)*i)]`
```ans = -0.2194 + 0.0000i -0.2194 + 3.1416i -0.2194 - 3.1416i```

### Derivatives of Exponential Integral

Compute the first, second, and third derivatives of a one-argument exponential integral.

```syms x diff(ei(x), x) diff(ei(x), x, 2) diff(ei(x), x, 3)```
```ans = exp(x)/x ans = exp(x)/x - exp(x)/x^2 ans = exp(x)/x - (2*exp(x))/x^2 + (2*exp(x))/x^3```

### Limits of Exponential Integral

Compute the limits of a one-argument exponential integral.

```syms x limit(ei(2*x^2/(1+x)), x, -Inf) limit(ei(2*x^2/(1+x)), x, 0) limit(ei(2*x^2/(1+x)), x, Inf)```
```ans = 0 ans = -Inf ans = Inf```

## Input Arguments

collapse all

Input specified as a floating-point number or symbolic number, variable, expression, function, vector, or matrix.

## Tips

• The one-argument exponential integral is singular at `x = 0`. The toolbox uses this special value: `ei(0) = -Inf`.

## Algorithms

The relation between `ei` and `expint` is

`ei(x) = -expint(1,-x) + (ln(x)-ln(1/x))/2 - ln(-x)`

Both functions `ei(x)` and `expint(1,x)` have a logarithmic singularity at the origin and a branch cut along the negative real axis. The `ei` function is not continuous when approached from above or below this branch cut.

 Gautschi, W., and W. F. Gahill “Exponential Integral and Related Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.