# Documentation

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# Chi

Hyperbolic cosine integral function

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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);

 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).