Hyperbolic cosine integral function
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.
When called with a floating-point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.
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)
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)
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);
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.
 Abramowitz, M. and I. Stegun, "Handbook of Mathematical Functions", Dover Publications Inc., New York (1965).