Hyperbolic cosine integral function
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Chi(x
)
Chi(x)
represents the hyperbolic cosine integral $$\mathrm{EULER}+\mathrm{ln}\left(x\right)+{\displaystyle \underset{0}{\overset{x}{\int}}\frac{\mathrm{cosh}\left(t\right)1}{t}dt}$$.
If x
is a floatingpoint number, then Chi(x)
returns
floatingpoint 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.
When called with a floatingpoint 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 floatingpoint numbers, use float
:
float(Chi(1)), float(Chi(sqrt(2)))
Alternatively, use a floatingpoint value as an argument:
Chi(1.0), Chi(2.0 + 10.0*I)
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)
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);

Arithmetical expression.
x
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).