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名詞

implicit function (複数形 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, Government Printing Office, Washington, 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.