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In mathematics, inverse mapping theorem may refer to: the inverse function theorem on the existence of local inverses for functions with non-singular derivatives the bounded inverse theorem on the boundedness of the inverse for invertible bounded linear operators on Banach spaces
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.}
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
The open mapping theorem forces the inverse function (defined on the image of ) to be holomorphic. Thus, under this definition, a map is conformal if and only if it is biholomorphic. The two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative.
Non-linear inverse problems constitute an inherently more difficult family of inverse problems. Here the forward map is a non-linear operator. Modeling of physical phenomena often relies on the solution of a partial differential equation (see table above except for gravity law): although these partial differential equations are often linear ...
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The inverse of a K-quasiconformal homeomorphism is K-quasiconformal. The set of 1-quasiconformal maps forms a group under composition. The space of K-quasiconformal mappings from the complex plane to itself mapping three distinct points to three given points is compact.