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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 ...
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
The composition of two homotheties with centers S 1, S 2 and ratios k 1, k 2 = 0.3 mapping P i &rarrow; Q i &rarrow; R i is a homothety again with its center S 3 on line S 1 S 2 with ratio k ⋅ l = 0.6.
a symbol for psychology; the wave function in the Schrödinger equation of quantum mechanics; represents: the J/psi mesons in particle physics; the stream function in fluid dynamics; the reciprocal Fibonacci constant; the second Chebyshev function in number theory; the polygamma function in mathematics; the supergolden ratio [8]
For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture). The arcsine is a partial inverse of the sine function. These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since
According to this weaker definition, a conformal map need not be biholomorphic, even though it is locally biholomorphic, for example, by the inverse function theorem. For example, if f: U → U is defined by f(z) = z 2 with U = C–{0}, then f is conformal on U, since its derivative f’(z) = 2z ≠ 0, but it is not biholomorphic, since it is 2-1.
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From the definition, it follows that any isomorphism : will map the identity element of to the identity element of , =, that it will map inverses to inverses, = (), and more generally, th powers to th powers, = (), and that the inverse map : is also a group isomorphism.