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)