# atanh

hyperbolic tangent inverse

### Syntax

t = atanh(x)

### Arguments

- x
a real or complex vector/matrix.

- t
a real or complex vector/matrix.

### Description

The components of vector `t`

are the hyperbolic
tangent inverse of the corresponding entries of vector
`x`

. Definition domain is `[-1,1]`

for
the real function (see Remark).

### Remark

In Scilab (as in some others numerical software) when you try to
evaluate an elementary mathematical function outside its definition domain
in the real case, then the complex extension is used (with a complex
result). The most famous example being the `sqrt`

function (try
`sqrt(-1)`

!). This approach have some drawbacks when you
evaluate the function at a singular point which may led to different
results when the point is considered as real or complex. For the
`atanh`

this occurs for `-1`

and
`1`

because the at these points the imaginary part do not
converge and so `atanh(1) = +Inf + i NaN`

while
`atanh(1) = +Inf`

for the real case (as lim `x->1`

of
`atanh(x)`

). So when you evaluate this function on the vector `[1 2]`

then like `2`

is outside the definition
domain, the complex extension is used for all the vector and you get
`atanh(1) = +Inf + i NaN`

while you get ```
atanh(1)
= +Inf
```

with `[1 0.5]`

for instance.

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