Hyperbolic cosine integral function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.




Chi(x) represents the hyperbolic cosine integral .

If x is a floating-point number, then Chi(x) returns numerical values. The special values Chi(∞) = ∞, Chi(- ∞) = ∞ + i π, , and 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.


Example 1

We demonstrate some calls with exact and symbolic input data:

Ci(1), Ci(sqrt(2)), Ci(x + 1), Ci(infinity), Ci(-infinity)

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

Floating point values are computed for floating-point arguments:

Ci(1.0), Ci(2.0 + 10.0*I)

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

Example 2

Ci and Chi are singular at the origin:

Error: Singularity. [Ci]
Error: Singularity. [Chi]

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

Ci(-1.0), Ci(-1.0 + 10^(-10)*I), Ci(-1.0 - 10^(-10)*I)

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

Example 3

The functions diff, float, and series handle expressions involving Ci and Chi:

diff(Ci(x), x, x, x), float(ln(3 + Ci(sqrt(PI))))

diff(Chi(x), x, x, x), float(ln(3 + Chi(sqrt(PI))))

series(Ci(x), x = 0)

series(Chi(x), x = 0)

series(Ci(x), x = infinity, 5)

series(Chi(x), x = infinity, 3);



An arithmetical expression

Return Values

Arithmetical expression.

Overloaded By



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.


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

See Also

MuPAD Functions

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