The Schwarz Function and Its Applications |
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Page 16
... unit circle . The clinant of Oz1 is 21/21 1/212 . The equation of the tangent to the circle at = z1 is , by ( 3.7 ) , = - 1 ( 2 212 - 21 ) + 21 = 2 2 + 212 21 • Solving this equation with the equation of the tangent at 22 we arrive at ...
... unit circle . The clinant of Oz1 is 21/21 1/212 . The equation of the tangent to the circle at = z1 is , by ( 3.7 ) , = - 1 ( 2 212 - 21 ) + 21 = 2 2 + 212 21 • Solving this equation with the equation of the tangent at 22 we arrive at ...
Page 91
... unit circle with S ( z ) 1/2 , as we already know . We may therefore say that the 45 ° spiral of Bernoulli passing through z = 1 is " the ith power of the unit circle . " ω If w is any complex number on the unit circle , then the " wth ...
... unit circle with S ( z ) 1/2 , as we already know . We may therefore say that the 45 ° spiral of Bernoulli passing through z = 1 is " the ith power of the unit circle . " ω If w is any complex number on the unit circle , then the " wth ...
Page 92
Philip J. Davis. → z1 , we have the sequence of figures : unit circle →→ 45 ° spiral real axis → 135 ° spiral → unit circle , under the ith powering of the Schwarz Function . The cycle can also be interpreted geometrically in terms ...
Philip J. Davis. → z1 , we have the sequence of figures : unit circle →→ 45 ° spiral real axis → 135 ° spiral → unit circle , under the ith powering of the Schwarz Function . The cycle can also be interpreted geometrically in terms ...
Contents
CHAPTER PAGE 1 Prologue 15739 | 1 |
Conjugate Coordinates in the Plane | 2 |
Elementary Geometrical Facts | 3 |
Copyright | |
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A₁ algebra analytic arc analytic continuation analytic curve analytic function anti-analytic assume b₁ bilinear bisector Cauchy Chapter clinant coefficients complex number complex variables conformal invariant conformal mapping conjugate coordinates convergence curvilinear angle defined derivative designate differential equations dxdy Example F₂ fixed point follows formal power series formula func Function for f functional composition functional equation geometry harmonic function Hence horn angle identity integral interior intersect inverse inversive geometry iteration LEMMA linear Math mathematics matrix meromorphic Möbius Möbius transformation neighborhood nine-point circle obtain orthocenter orthogonal plane polynomial problem Proof real axis schlicht Schroeder Function Schwarz Function S(z Schwarzian reflection simply connected simply connected region single-valued solution spiral straight line Suppose symmetric with respect tangent theorem theory tion transformation triangle unit circle write x-axis z₁ дв