Wolfram: Function

By Xah Lee. Date: 2022-04-18. Last updated: 2025-02-04.

Define Function

Function can be defined in 2 ways: Pure Function and pattern matching.

Define Function as Pure Function by 「Function」

Using Function. This is fast and direct. Function defined this way is called Pure Function.

Pure function is similar to lambda in other languages, e.g. • Elisp: Lambda Function • JS: Arrow Function

Define Function by Pattern Matching

Create a global rule of pattern matching.

This is slower but more powerful. It allows you to define optional parameters, default values, rest parameters, parameters of any tree structure, definition of special cases, recursive sequences. In general, it defines rules for transformation of expressions. (which includes the so-called polymorphism) 〔see Wolfram: Define Function by Pattern〕

Define Pure Function via Function

Function[x, body]

same as

Function[{x}, body]

Function[{x}, body]

represent a function with one parameter.

🛑 WARNING: you cannot assign values to function's parameter variable in function body.

(* a function that adds one. applied to 3 *)
Function[ {x}, x + 1 ][3]

(* 4 *)

(* give the function a name *)
f = Function[ {x}, x + 1 ];
f[3]

(* 4 *)

(* in a function, you cannot change the parameter value *)
(* this is error *)
Function[{x}, x = 3][4]

(* if you really want to change param value, create local var *)
Function[{x}, Module[{y = 3}, y] ][4]

Function[{x1, x2, etc}, body]

represent a function with formal parameters x1, x2 etc.
When the function is called, body is evaluated, with formal parameters replaced by arguments.

Function

f = Function[ {x,y}, x + y ];
f[3, 4]

(* 7 *)

Named Parameters, Default Values, Polymorphism