An implicit function is a function specified by a relation between the input and output that is left unsolved for the output. An equation defines on a domain if it determines a unique allowed value of for each . The choice of allowed values can select a branch. For example, the circle defines the function on when restricted to , and the function on the same domain when restricted to .

An example in which a single equation already determines a unique output is Kepler's equation. For a body orbiting on an ellipse of fixed eccentricity , the mean anomaly is known from the elapsed time, and the eccentric anomaly is found from

The right-hand side has derivative and tends to as . Although the equation gives explicitly in terms of , it defines implicitly as a function of , with exactly one real for each real .

See also

Function, Implicit Differentiation, Implicit Function Theorem

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Cite this as:

Weisstein, Eric W. "Implicit Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ImplicitFunction.html

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