# Update

This bug has been fixed in v11.3.0. And as @Carl Woll has pointed out, the imaginary part of Sin and negative sign of Cos are due to the selections of different branch cuts.

# Original Post

v11.2.0.0 on Windows 10, 64bit, check this:

InverseFunction /@ {f[#] &, f[2 #] &, f[#/2] &, f[# + 1] &}
/. {{f -> Sinh}, {f -> Cosh}, {f -> Tanh}, {f -> Coth}} // TableForm


The result is only correct for Coth case; Sinh picks up an imaginary part, Cosh adds an negative sign, Tanh entangles with ArcCoth instead of ArcTanh.

This is really a disturbing error. Can someone explain this behavior?

The results for Sinh and Cosh are not incorrect. All of your functions have multi-valued inverses, so you're seeing Mathematica choose different branches than you were expecting. For example:

x = Sinh[2 3.]
y = 1/2 (I Pi - ArcSinh[#])&[x]
Sinh[2 y]


201.713

-3. + 1.5708 I

201.713 + 2.4703*10^-14 I

If you want to restrict to real inverses, you could use ConditionalExpression:

if = InverseFunction[ConditionalExpression[Sinh[2 #], # ∈ Reals]&];


Then:

if[Sinh[2 3.]]


3.

On the other hand, I do think the results for Tanh are incorrect. It would be worth reporting the issue to support. In particular, I think the following demonstrates the issue:

if = InverseFunction[Tanh[2#]&]

x = Tanh[2 .130]
y = if[x]
Tanh[2 y]


InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses.

ArcCoth[#1]/2 &

0.197375320224904000738157318811

0.100000000000000000000000000000 - 0.785398163397448309615660845820 I

5.0664895634394727136317877883 + 0.*10^-30 I

The output of Tanh[2 y] should be the same as x. Again, you can work around the issue by including a ConditionalExpression wrapper:

if = InverseFunction[ConditionalExpression[Tanh[2#], # ∈ Reals]&]

x = Tanh[2 .130]
y = if[x]
Tanh[2 y]


ConditionalExpression[ArcTanh[#1]/2, -1 < #1 < 1] &

0.197375320224904000738157318811

0.100000000000000000000000000000

0.197375320224904000738157318811

• Thanks. But what about Tanh etc? Jan 8, 2018 at 18:29
• @luyuwuli See update. I do think you found an issue with Tanh. Jan 8, 2018 at 18:48
• Thank you very much. I've already report this issue to the official feedback. Jan 8, 2018 at 20:34