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The names "lambda abstraction", "lambda function", and "lambda expression" refer to the notation of function abstraction in lambda calculus, where the usual function f (x) = M would be written (λx. M), and where M is an expression that uses x. Compare to the Python syntax of lambda x: M.
In Python, functions are first-class objects that can be created and passed around dynamically. Python's limited support for anonymous functions is the lambda construct. An example is the anonymous function which squares its input, called with the argument of 5:
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The term closure is often used as a synonym for anonymous function, though strictly, an anonymous function is a function literal without a name, while a closure is an instance of a function, a value, whose non-local variables have been bound either to values or to storage locations (depending on the language; see the lexical environment section below).
The function that accepts a callback may be designed to store the callback so that it can be called back after returning which is known as asynchronous, non-blocking or deferred. Programming languages support callbacks in different ways such as function pointers, lambda expressions and blocks.
This definition recognizes a lambda abstraction with an actual parameter as defining a function. Only lambda abstractions without an application are treated as anonymous functions. lambda-named A named function. An expression like (.) where M is lambda free and N is lambda free or an anonymous function.
A function's identity is based on its implementation. A lambda calculus function (or term) is an implementation of a mathematical function. In the lambda calculus there are a number of combinators (implementations) that satisfy the mathematical definition of a fixed-point combinator.
Dirichlet lambda function, λ(s) = (1 – 2 −s)ζ(s) where ζ is the Riemann zeta function; Liouville function, λ(n) = (–1) Ω(n) Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p; Modular lambda function, λ(τ), a highly symmetric holomorphic function on the complex upper half-plane