Documentation

This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Chi

Hyperbolic cosine integral function

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

Syntax

Chi(x)

Description

Chi(x) represents the hyperbolic cosine integral EULER+ln(x)+0xcosh(t)1tdt.

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

An arithmetical expression

Return Values

Arithmetical expression.

Overloaded By

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).

See Also

MuPAD Functions

Was this topic helpful?