# Chi

Hyperbolic cosine integral function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```Chi(`x`)
```

## Description

`Chi(x)` represents the hyperbolic cosine integral $\mathrm{EULER}+\mathrm{ln}\left(x\right)+\underset{0}{\overset{x}{\int }}\frac{\mathrm{cosh}\left(t\right)-1}{t}dt$.

If `x` is a floating-point number, then `Chi(x)` returns floating-point results. The special values ```Chi(∞) = ∞```, `Chi(-∞) = ∞ + iπ`, ```Chi(i∞) = iπ/2```, and `Chi(-i∞) = -iπ/2` are implemented. For all other arguments `Chi` returns symbolic function calls.

## Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

Most calls with exact arguments return themselves unevaluated:

`Chi(1), Chi(sqrt(2)), Chi(x + 1), Chi(I*infinity), Chi(-I*infinity)`
``` ```

To approximate exact results with floating-point numbers, use `float`:

`float(Chi(1)), float(Chi(sqrt(2)))`
``` ```

Alternatively, use a floating-point value as an argument:

`Chi(1.0), Chi(2.0 + 10.0*I)`
``` ```

### Example 2

`Chi` is singular at the origin:

`Chi(0)`
```Error: Singularity. [Chi] ```

The negative real axis is a branch cut of `Chi`. A jump of height 2 π i occurs when crossing this cut:

`Chi(-1.0), Chi(-1.0 + 10^(-10)*I), Chi(-1.0 - 10^(-10)*I)`
``` ```

### Example 3

`diff`, `float`, `series`, and other functions handle expressions involving `Chi`:

`diff(Chi(x), x, x, x), float(ln(3 + Chi(sqrt(PI))))`
``` ```
`series(Chi(x), x = 0)`
``` ```
`series(Chi(x), x = infinity, 3);`
``` ```

## Parameters

 `x`

## Return Values

Arithmetical expression.

`x`

## Algorithms

The functions `Ci(x)-ln(x)` and `Chi(x)-ln(x)` are entire functions. Thus, `Ci` and `Chi` have a logarithmic singularity at the origin and a branch cut along the negative real axis. The values on the negative real axis coincide with the limit "from above":

for real x < 0.

`Ci` and `Chi` are related by Ci(x) - ln(x) = Chi(i x) - ln(i x) for all x in the complex plane.

## References

[1] Abramowitz, M. and I. Stegun, "Handbook of Mathematical Functions", Dover Publications Inc., New York (1965).