YouTipRef Math Atanh
## Python math.atanh() Method
The `math.atanh()` method is a built-in function in Python's standard `math` module. It returns the inverse hyperbolic tangent of a given number $x$.
Mathematically, the inverse hyperbolic tangent is defined as:
$$\operatorname{atanh}(x) = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right)$$
This method is widely used in scientific computing, physics, and advanced mathematical modeling.
---
### Syntax
To use the `math.atanh()` method, you must first import the `math` module:
```python
import math
math.atanh(x)
```
### Parameters
| Parameter | Type | Description |
| :--- | :--- | :--- |
| `x` | `float` or `int` | **Required.** A numeric value representing the value whose inverse hyperbolic tangent you want to calculate. The value of `x` must be strictly between **-1.0 and 1.0** (excluding -1.0 and 1.0). |
### Return Value
* **Type:** `float`
* **Description:** Returns the inverse hyperbolic tangent of the specified number `x`.
---
### Code Examples
#### Example 1: Basic Usage
The following example demonstrates how to calculate the inverse hyperbolic tangent for positive and negative values within the valid range.
```python
# Import the math module
import math
# Calculate and print the inverse hyperbolic tangent
print(math.atanh(0.59))
print(math.atanh(-0.12))
```
**Output:**
```text
0.6776660677579618
-0.12058102840844402
```
#### Example 2: Boundary Values
As $x$ approaches $1$ or $-1$, the result approaches positive or negative infinity.
```python
import math
# Values close to the boundaries
print(math.atanh(0.9999))
print(math.atanh(-0.9999))
```
**Output:**
```text
4.951710188619111
-4.951710188619111
```
---
### Exceptions and Considerations
When working with `math.atanh()`, you must ensure that the input values fall strictly within the mathematical domain of the function: $-1 < x < 1$.
#### 1. ValueError (Out of Domain)
If you pass a value $x \ge 1$ or $x \le -1$, Python will raise a `ValueError: math domain error`.
```python
import math
try:
# This will raise a ValueError because 1.0 is out of the valid domain
print(math.atanh(1.0))
except ValueError as e:
print(f"Error: {e}")
```
**Output:**
```text
Error: math domain error
```
#### 2. TypeError (Non-numeric Input)
If you pass a non-numeric value (such as a string or a list), Python will raise a `TypeError`.
```python
import math
try:
# This will raise a TypeError
print(math.atanh("0.5"))
except TypeError as e:
print(f"Error: {e}")
```
**Output:**
```text
Error: must be real number, not str
```
### Version History
* **Introduced in:** Python 2.6