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Difference between revisions of "Implicitly defined function"

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An '''implicit function''' is a function that isn't explicit. That is, it doesn't express <math>y</math> in terms of <math>x</math>. It is often difficult or inconvenient to rewrite an implicit function in explicit form.
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An '''implicitly defined function''' is a [[function]] that is presented as the solution of some [[equation]] or system of equations, rather than being given by an explicit formula.
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Equations defining functions implicitly can sometimes be solved to give the function explicitly.  However, this is not generally true, and it can be difficult or impossible to express simple implicitly defined functions in explicit form.
  
 
==Examples==
 
==Examples==
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<math>x^2(x^2 + y^2) = y^2</math>
 
  
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<math>x^2 + y^2 = (1 + tan^{-1}\frac{y}{x})^2</math>
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* The equation <math>y + \tan^{-1} y = x</math> implicitly defines <math>y</math> as a function of <math>x</math>.  This equation cannot be solved for <math>y</math> as an [[elementary function]] in terms of <math>x</math>.
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* The equation <math>y^3 + 2y^2 + 2y + x^2 + x = 0</math> implicitly defines <math>y</math> as a function of <math>x</math> over the [[real number]]s.  This equation can be solved for <math>y</math> in terms of <math>x</math> by using the [[cubic formula]], but the resulting expression is very unpleasant to work with.
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* The equation <math>w \cdot e^w = x</math> for <math>x \geq -\frac{1}{e}</math> and <math>w \geq -1</math> implicitly defines <math>w</math> as a function of <math>x</math>.  This function is known as the [[Lambert W function]].  This equation cannot be solved for <math>w</math> as an elementary function in <math>x</math>, but this function is of sufficient importance that it has been given its own name.
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== See also ==
  
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<math>x^2 + y^2 = x^2 y^2</math>
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* [[Implicit Function Theorem]]

Latest revision as of 18:48, 8 September 2009

An implicitly defined function is a function that is presented as the solution of some equation or system of equations, rather than being given by an explicit formula.

Equations defining functions implicitly can sometimes be solved to give the function explicitly. However, this is not generally true, and it can be difficult or impossible to express simple implicitly defined functions in explicit form.

Examples

  • The equation $y + \tan^{-1} y = x$ implicitly defines $y$ as a function of $x$. This equation cannot be solved for $y$ as an elementary function in terms of $x$.
  • The equation $y^3 + 2y^2 + 2y + x^2 + x = 0$ implicitly defines $y$ as a function of $x$ over the real numbers. This equation can be solved for $y$ in terms of $x$ by using the cubic formula, but the resulting expression is very unpleasant to work with.
  • The equation $w \cdot e^w = x$ for $x \geq -\frac{1}{e}$ and $w \geq -1$ implicitly defines $w$ as a function of $x$. This function is known as the Lambert W function. This equation cannot be solved for $w$ as an elementary function in $x$, but this function is of sufficient importance that it has been given its own name.

See also

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