Comment #4 on issue 564 by [email protected]: series expansion of acosh
and acoth
http://code.google.com/p/sympy/issues/detail?id=564
acosh(x) = (1/2)*pi*I - I*x - (1/6)*I*x**3 - (3/40)*I*x**5 + O(x**6)
(SYMPY)
acosh(x) = -(1/2)*pi*I + I*x + (1/6)*I*x**3 + (3/40)*I*x**5 + O(x**6)
(SAGE)
interestingly, both evaluate acosh(0) to (1/2)*pi*I.
The problem here is that cosh(x) is one to many. For real valued x, cosh(x)
is symmetric about the y axis (one to two) and kind of looks like f(x) =
x**2. In this case it's pretty natural to take the principal value of
acosh(x) to be the positive number. In the complex plane while cosh(z) =
cosh(-z) it is also the case that cosh(z) = cosh(z + I*2k*pi).
The standard definition for acosh(x) seems to be log(x + Sqrt(x**2 - 1))
which is multiple valued in the complex plane in exactly the way we expect.
The standard solution is to put a "branch cut" in the plane along (-oo, 0]
making the domain simply connected. This is how Stein and Shakarchi handle
it (Princeton Lectures in Analysis II, 2003). I've evaluated the
derivatives by hand and they come out in agreement with Sympy for the
principal values of log(z).
The situation is similar for acoth(x).
I'll add a note to the documentation and submit a patch.
Note: I'm not sure where the expansions Sympy uses came from but if they
were borrowed from mathworld, they adopt the conventions that Mathematica
uses for branch cuts. I think these are just branch cuts that correspond to
(-oo, 0] (the principal branch) for log(z) since each inverse hyperbolic
function has some logarithm of some F(z).
function name branch cut(s)
inverse hyperbolic cosecant (-I, I)
inverse hyperbolic cosine (-oo, 1)
inverse hyperbolic cotangent [-1, 1]
inverse hyperbolic secant (-oo, 0] and (1, oo)
inverse hyperbolic sine (-I* oo, -I) and (I, I* oo)
inverse hyperbolic tangent (-oo, 1] and [1, oo)
see,
http://mathworld.wolfram.com/InverseHyperbolicFunctions.html
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