Parametric representation

of a function

The specification of a function $ f $, say defined on $ [ a, b] $, by means of a pair of functions $ \phi , \psi $, say on $ [ \alpha , \beta ] $, for which $ \phi : [ \alpha , \beta ] \rightarrow [ a, b] $ has a single-valued inverse $ \phi ^ {- 1} : [ a, b] \rightarrow [ \alpha , \beta ] $ such that $ f = \psi \circ \phi ^ {- 1} $, that is, for any $ x \in [ a, b] $,

$$ f( x) = \psi [ \phi ^ {- 1} ( x)]. $$

Example. The pair of functions $ x = \cos t $, $ y = \sin t $, $ 0 \leq t \leq \pi $, is a parametric representation of the function $ y = \sqrt {1- x ^ {2} } $, $ - 1 \leq x \leq 1 $.

If at a point $ t _ {0} \in [ \alpha , \beta ] $ a parametric representation of $ f $ is differentiable, that is, $ \phi $ and $ \psi $ are differentiable, and if $ \phi ^ \prime ( t _ {0} ) \neq 0 $, then $ f $ is differentiable at $ x _ {0} = \phi ( t _ {0} ) $ and $ f ^ { \prime } ( x _ {0} ) = \psi ^ \prime ( t _ {0} )/ \phi ^ \prime ( t _ {0} ) $. Furthermore, if $ \phi $ and $ \psi $ have at $ t _ {0} $ derivatives of order $ n $, $ n = 2, 3, \dots $ then $ f $ has a derivative of order $ n $ at $ x _ {0} $, which is a fractional-rational function of the derivatives of $ \phi $ and $ \psi $ of orders $ k $, $ k = 1, \dots, n $, where in the denominator there stands the $ ( 2n- 1) $-th power of $ \phi ^ \prime ( t _ {0} ) $; for example,

$$ f ^ { \prime\prime } ( x _ {0} ) = \frac{\psi ^ {\prime\prime} ( t _ {0} ) \phi ^ \prime ( t _ {0} ) - \psi ^ \prime ( t _ {0} ) \phi ^ {\prime\prime} ( t _ {0} ) }{[ \phi ^ \prime ( t _ {0} )] ^ {3} } . $$

Comments

The functions need not be real, the same as above holds for complex functions (i.e. $ f: D \rightarrow \mathbf C $, $ D \subset \mathbf C $).

References

[a1]	T.M. Apostol, "Calculus" , 1–2 , Blaisdell (1967)