Hyperbolic cosine integral function
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Chi(x) represents the hyperbolic cosine integral .
x is a floating-point number, then
floating-point results. The special values
Chi(-∞) = ∞ + iπ,
= iπ/2, and
Chi(-i∞) = -iπ/2 are
implemented. For all other arguments
symbolic function calls.
When called with a floating-point argument, the functions are
sensitive to the environment variable
DIGITS which determines
the numerical working precision.
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
Alternatively, use a floating-point value as an argument:
Chi(1.0), Chi(2.0 + 10.0*I)
Chi is singular at the origin:
Error: Singularity. [Chi]
The negative real axis is a branch cut of
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)
diff(Chi(x), x, x, x), float(ln(3 + Chi(sqrt(PI))))
series(Chi(x), x = 0)
series(Chi(x), x = infinity, 3);
entire functions. Thus,
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.
related by Ci(x)
- ln(x) = Chi(i x)
- ln(i x) for
all x in
the complex plane.
 Abramowitz, M. and I. Stegun, “Handbook of Mathematical Functions”, Dover Publications Inc., New York (1965).