harmonic conjugates การใช้

• The harmonic conjugate of is the unique real function on such that is holomorphic.
• The purely geometric approach of von Staudt was based on the complete quadrilateral to express the relation of projective harmonic conjugates.
• The projective harmonic conjugate of such a " midpoint " with respect to the two endpoints is the point at infinity.
• He also made discoveries about projective harmonic conjugates; relating these to the poles and polar lines associated with conic sections.
• There is an additional occurrence of the term "'harmonic conjugate "'in mathematics, and more specifically in geometry.
• The conformal mapping property of analytic functions ( at points where the derivative is not zero ) gives rise to a geometric property of harmonic conjugates.
• The second set of fixed points is This situation is what is classically called the " " ", and arises in projective harmonic conjugates.
• For instance, the polar line can be viewed as the set of projective harmonic conjugates of a given point, the pole, with respect to a conic.
• If a variable line through the point is a secant of the conic, the harmonic conjugates of with respect to the two points of on the secant all lie on the " polar " of.
• Observe that if the functions related to " u " and " v " were interchanged, the functions would not be harmonic conjugates, since the minus sign in the Cauchy Riemann equations makes the relationship asymmetric.
• Since  " " u " is the harmonic conjugate of " v ", the process of taking harmonic conjugates is one-one, so the multiplicity of  " ? as an eigenvalue is the same as that of ?.
• Since  " " u " is the harmonic conjugate of " v ", the process of taking harmonic conjugates is one-one, so the multiplicity of  " ? as an eigenvalue is the same as that of ?.
• For instance, given a line containing the points and, the "'midpoint "'of line segment is defined as the point which is the projective harmonic conjugate of the point of intersection of and the absolute line, with respect to and.
• Karl von Staudt reformed mathematical foundations in 1847 with the complete quadrangle when he noted that a " harmonic property " could be based on concomitants of the quadrangle : When each pair of opposite sides of the quadrangle intersect on a line, then the diagonals intersect the line at projective harmonic conjugate positions.
• Two points A and B are said to be harmonic conjugates of each other with respect to another pair of points C, D if ( ABCD ) = & minus; 1, where ( ABCD ) is the cross-ratio of points A, B, C, D ( See Projective harmonic conjugates .)
• Two points A and B are said to be harmonic conjugates of each other with respect to another pair of points C, D if ( ABCD ) = & minus; 1, where ( ABCD ) is the cross-ratio of points A, B, C, D ( See Projective harmonic conjugates .)
• There is an operator taking a harmonic function " u " on a simply connected region in "'R "'2 to its harmonic conjugate " v " ( putting e . g . " v " ( x 0 ) = 0 on a given x 0 in order to fix the indeterminacy of the conjugate up to constants ).