arctanh

[Albert Chan]

(02-26-2024 07:17 AM)hp-zl Wrote:  Enter e.g. tanh( ln(3)/2 - i*pi/2*5) and press Shift-Enter - you will get 2.

atanh(z) only return principal value, above does not imply its reverse is true.
(same reason (-1)^2 = 1 does not imply sqrt(1) = -1)

With signed zero, atanh argument real/imag part sign matched its result.

atanh(2 ± 0i) = ln(3)/2 ± (pi/2)*i

We get 2 answers because HP Prime does not have signed zero.
There is a discontinuity with "true" 0 imag part, its limit does not exist.

Cas> atanh(2+1e-10*i)      → 0.549306144334+1.57079632676*i
Cas> atanh(2−1e-10*i)      → 0.549306144334−1.57079632676*i

We may arbitrarily pick a side for this slit, as long as it is applied consistently.
The side picked at the slit depends on formula used.

atanh(z) = 1/2 * ln( (1+z)/(1-z) )         // XCas atanh
atanh(2) = 1/2 * ln(3/-1) = 1/2 * (ln(3) + pi*i) = ln(3)/2 + pi/2*i

atanh(z) = 1/2 * (ln(1+z) - ln(1-z))      // mpmath mpc_atanh
atanh(2) = 1/2 * (ln(3) - ln(-1)) = 1/2 * (ln(3) - pi*i) = ln(3)/2 - pi/2*i

Again, without signed zero, there is no right answer, it is just an arbitrary pick.