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The Schwarz function of a curve in the complex plane is an analytic function which maps the points of the curve to their complex conjugates.It can be used to generalize the Schwarz reflection principle to reflection across arbitrary analytic curves, not just across the real axis.
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation.It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane.
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
The function D(z) is sometimes called the Bloch-Wigner function. [1] Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. [2]
For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from A n in this way, and the corresponding Coxeter group is the affine Weyl group of A n (the affine symmetric group). For n = 2, this can be pictured as a subgroup of the symmetry group of the standard tiling of the plane by equilateral triangles.
Suppose X and Y in the Lie algebra are given. Their exponentials, exp(X) and exp(Y), are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some Z in the Lie algebra, exp(Z) = exp(X) exp(Y), and one may tentatively write = (,), for C some expression in X and Y.
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A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.