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The graph of an involution (on the real numbers) is symmetric across the line y = x. This is due to the fact that the inverse of any general function will be its reflection over the line y = x. This can be seen by "swapping" x with y. If, in particular, the function is an involution, then its graph is its own reflection.
In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called the inverse. If g {\displaystyle g} is a left inverse and h {\displaystyle h} a right inverse of f {\displaystyle f} , for all y ∈ Y {\displaystyle y\in Y} , g ( y ) = g ( f ( h ( y ) ) = h ( y ) {\displaystyle g(y)=g(f ...
For example, there is a well known proof relating the Riemann zeta function to the prime zeta function that uses the series-based form of Möbius inversion in the previous equation when =. Namely, by the Euler product representation of ζ ( s ) {\displaystyle \zeta (s)} for ℜ ( s ) > 1 {\displaystyle \Re (s)>1}
As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed-point theorem (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations ...
In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k 2. The inverse of the curve C is then the locus of P as Q runs over C.
This reflection operation turns the gradient of any line into its reciprocal. [1] Assuming that has an inverse in a neighbourhood of and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at and have a derivative given by the above formula.