mech-math 0.3.3

Math library for the Mech language
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66

math/sinh
===============================================================================

%% Hyperbolic sine of argument

1. Usage
-------------------------------------------------------------------------------

```mech:disabled
Y := math/sinh(X)
```

2. Description
-------------------------------------------------------------------------------

Computes the hyperbolic sine of each element of `X`. The input `X` is interpreted as a real or complex number. The result `Y` has the same shape as the input `X`.

3. Input
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `X`      | `float`, `[float]`       | Input value(s), real or complex. If `X` is complex, sinh returns complex results. |

4. Output
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Hyperbolic sine of the input values. For real `X`, the output is always real. For complex `X`, the output may have both real and imaginary parts. The shape of `Y` matches the shape of `X`. |

5. Examples  
-------------------------------------------------------------------------------

(a) Find the hyperbolic sine of a number 

```mech:ex1
y := math/sinh(1.0)
```

(b) Find the hyperbolic sine for a vector of numbers

```mech:ex2
x := [0, 1, 2]
y := math/sinh(x)
```

(c) Find the hyperbolic sine for a matrix of numbers

```mech:ex3
x := [0, 1; 2, 3]
y := math/sinh(x)
```

6. Details
-------------------------------------------------------------------------------

The hyperbolic sine function is defined as:

$$ sinh(x) = \frac{e^x - e^{-x}}{2}

It is one of the basic hyperbolic functions, closely related to the exponential function.

Unlike the trigonometric sine, which oscillates between -1 and 1, the hyperbolic sine grows rapidly for large values of $$x$$ and is unbounded in both positive and negative directions.

For complex numbers, the definition using exponentials extends naturally, making the function applicable to both real and complex domains.