Trac #20098: doctest fix for: Re/Im(tanh) wrong formula

In analogy to #19791:
{{{
sage: tanh(1+2*I).n()
1.16673625724092 - 0.243458201185725*I
sage: tanh(1+2*I).real().n()
0.202053272732553
sage: tanh(1+2*I).imag().n()
-0.579697798322816
sage: tanh(x).real()
tanh(real_part(x))/(tan(imag_part(x))^2*tanh(real_part(x))^2 + 1)
sage: tanh(x).imag()
tan(imag_part(x))/(tan(imag_part(x))^2*tanh(real_part(x))^2 + 1)
}}}
See for example http://dlmf.nist.gov/4.35.E36

Will be fixed in Pynac-0.6.3 and should be doctested here.

URL: http://trac.sagemath.org/20098
Reported by: rws
Ticket author(s): Ralf Stephan
Reviewer(s): Marc Mezzarobba

src/sage/functions/hyperbolic.py

@@ -189,6 +189,19 @@ def __init__(self):
 
             sage: latex(tanh(x))
             \tanh\left(x\right)
+
+        Check that real/imaginary parts are correct (:trac:`20098`)::
+
+            sage: tanh(1+2*I).n()
+            1.16673625724092 - 0.243458201185725*I
+            sage: tanh(1+2*I).real().n()
+            1.16673625724092
+            sage: tanh(1+2*I).imag().n()
+            -0.243458201185725
+            sage: tanh(x).real()
+            sinh(2*real_part(x))/(cos(2*imag_part(x)) + cosh(2*real_part(x)))
+            sage: tanh(x).imag()
+            sin(2*imag_part(x))/(cos(2*imag_part(x)) + cosh(2*real_part(x)))
         """
         GinacFunction.__init__(self, "tanh", latex_name=r"\tanh")