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implicit function (plural implicit functions)

1. (mathematical analysis, algebraic geometry) A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially where said equation is such that the value is not directly calculable from the other variables.

   An important class of implicit functions comprises those defined by equations of the form ${\displaystyle x-f(y)=0}$. Choosing ${\displaystyle y}$ as the value means that the implicit function is the inverse of ${\displaystyle f}$. This type of implicit function is of particular interest when that inverse is not expressible as a closed-form expression, and thus is difficult to study directly.

   • 1871, William G. Peck, Practical Treatise on the Differential and Integral Calculus, A. S. Barnes & Company, page 50:

     Thus, in the equation,
     ${\displaystyle \quad \quad \quad ay^{2}+bxy+cx^{2}+d=0}$;
     ${\displaystyle y}$ is an implicit function of ${\displaystyle x}$. Implicit functions are generally connected with their variables by one or more equations. When these equations are solved the implicit equation becomes explicit.
     […] The differential of an implicit function may be found without first finding the function itself.

   • 1875, J. Minot Rice, W. Woolsey Johnson, The Elements of the Differential Calculus, Part 2, Washington: Government Printing Office, page 71:

     The process of evaluating the derivatives in the case of implicit functions is exemplified below.

   • 2002, Michael W. Klein, Mathematical Methods for Economics, Addison-Wesley, page 245:

     In this section we discuss the analysis of implicit functions for addressing questions like this.

使用する際の注意点

• Definition notes:
  • An implicit equation is a relation expressed in the form ${\displaystyle R(x_{1},\dots ,x_{n})=0}$, where ${\displaystyle R}$ is a function of several variables (often a polynomial). More broadly, the relation can be a set of simultaneous equations.
  • The expression not directly calculable here simply means that the implicit equation must be manipulated in order to solve for the chosen value. Such manipulation is not always possible, and, even if it is, the result may be an expression that is not computable in a finite number of "standard operations" (an ambiguous term). (See also Closed-form expression on Wikipedia.Wikipedia )
• Notes concerning related terms:
  • Given an arbitrary implicit equation ${\displaystyle R(x_{1},\dots ,x_{n-1},y)=0}$, the selection of ${\displaystyle y}$ as the value (and thus the ${\displaystyle x_{i}}$ as arguments) does not by any means entail that the map ${\displaystyle f:(x_{1},\dots ,x_{n-1})\rightarrow y}$ is single-valued (and thus a function). The nonstandard concept of multivalued function (or multifunction) is sometimes invoked. Additionally, the terms implicit curve and implicit surface may be used. In general, the solution space of the equation is some subset of ${\displaystyle \mathbb {R} ^{n-1}}$. The implicit function theorem deals, in part, with what constraints on the equation ensure that ${\displaystyle f}$ is a function.
    • For example, solving for ${\displaystyle y}$ in the implicit equation ${\displaystyle x^{2}+y^{2}-1=0}$ yields the formula ${\displaystyle \textstyle y=\pm {\sqrt {1-x^{2}}}}$. This formula produces real number values only over the interval ${\displaystyle x\in [-1,1]}$, and across that interval has two branches corresponding to the cases ${\displaystyle y\geq 0}$ and ${\displaystyle y\leq 0}$. Depending on the equation, the number of branches can be infinite.
  • The technique of implicit differentiation enables the conversion of an implicit equation directly into a differential equation, which may reveal properties of an implicit function without its needing to be explicitly formulated.

関連する語

• explicit function
• implicit curve
• implicit equation
• implicit differentiation
• implicit function theorem

参考

• algebraic function
• functional equation

Further reading

• Implicit curve on Wikipedia.Wikipedia
• Implicit function theorem on Wikipedia.Wikipedia
• Algebraic variety on Wikipedia.Wikipedia
• Functional equation on Wikipedia.Wikipedia