In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every total recursive function is a primitive recursive function—the most famous example is the Ackermann function.
Other equivalent classes of functions are the functions of lambda calculus and the functions that can be computed by Markov algorithms.
The subset of all total recursive functions with values in {0,1} is known in computational complexity theory as the complexity class R.