Pure functions in Wolfram language: Definitions, examples, and use cases

[Splendid Digital Solutions]

Disclaimer: This article was created with the assistance of an AI language model and is intended for informational purposes only. Please verify any technical details before implementation.

 

In Wolfram Language (Mathematica), a **pure function** is a function defined without giving it an explicit name. It operates directly on its inputs, which are represented by placeholders. Pure functions are useful for quick, temporary functions—especially when you want to apply an operation to a list, manipulate data, or simplify expressions without defining a full function.

### Syntax of Pure Functions

The syntax for defining a pure function is:

– `#` represents the argument(s) of the function.
– `&` indicates the end of the pure function.

For example:
“`mathematica
#^2 &
“`
This pure function takes an input and squares it. To use it, you would apply it to a number like this:

“`mathematica
(#^2 &)[5]
“`
**Output**: `25`

 

Alternatively, if you want to apply the pure function to each element in a list, you could use the `Map` function:
“`mathematica
#^2 & /@ {1, 2, 3, 4, 5}
“`
**Output**: `{1, 4, 9, 16, 25}`

### Examples of Pure Functions

1. **Square each number in a list**:
“`mathematica
#^2 & /@ {2, 4, 6}
“`
**Output**: `{4, 16, 36}`

2. **Add two numbers**:
“`mathematica
(#1 + #2) & [3, 5]
“`
**Output**: `8`

Here, `#1` and `#2` represent the first and second arguments, respectively.

3. **Apply a more complex operation (e.g., square the number and add 1)**:
“`mathematica
(#^2 + 1) & /@ {1, 2, 3, 4}
“`
**Output**: `{2, 5, 10, 17}`

4. **Filter a list to find even numbers**:
“`mathematica
Select[{1, 2, 3, 4, 5, 6}, EvenQ[#] &]
“`
**Output**: `{2, 4, 6}`

### Differences Between Pure Functions and Named Functions

1. **Definition Style**:
– **Pure Functions**: Defined inline, using `#` for arguments and `&` at the end. They don’t need a name, which makes them concise.
– **Named Functions**: Created using `FunctionName[x_] := expression`, where the function is explicitly named and can be reused across the code.

**Example**:
“`mathematica
(* Pure function *)
# + 1 &

(* Named function *)
IncrementByOne[x_] := x + 1
“`

2. **Usage and Reusability**:
– **Pure Functions**: Used for quick, one-off operations or when mapping over lists.
– **Named Functions**: Useful when a function will be called multiple times or in various places.

3. **Scope and Readability**:
– **Pure Functions**: Ideal for small operations but can become hard to read if they’re too complex.
– **Named Functions**: Better for complex functions as they allow you to give descriptive names to functions, improving code readability.

4. **Performance**: Generally, both pure and named functions perform similarly. However, in functional programming tasks (like mapping over lists), pure functions offer a more concise syntax, making them suitable for functional-style code.

### Use Cases for Pure Functions

– **Quick Calculations**: When you need a simple function for a single task, like calculating squares or adding numbers.
– **Mapping**: When applying a transformation to each element in a list.
– **Filtering**: When selecting or rejecting elements in a list based on a condition.

### Example Comparisons

1. **Square Each Element of a List**

Using a pure function:
“`mathematica
#^2 & /@ {1, 2, 3, 4}
“`

Using a named function:
“`mathematica
Square[x_] := x^2
Square /@ {1, 2, 3, 4}
“`

2. **Filter Even Numbers from a List**

Using a pure function:
“`mathematica
Select[{1, 2, 3, 4, 5, 6}, EvenQ[#] &]
“`

Using a named function:
“`mathematica
EvenFilter[x_] := EvenQ[x]
Select[{1, 2, 3, 4, 5, 6}, EvenFilter]
“`

In summary, pure functions are a powerful, concise feature in the Wolfram Language, especially useful for quick tasks. Named functions, on the other hand, are preferable for more complex or frequently used operations, as they provide better readability and reusability.

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