Schwarz function explained

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.

\Gamma in the complex plane, there is an open neighborhood \Omega of \Gamma and a unique analytic function S on \Omega such that S(z) = \overline{z} for every z \in \Gamma .

The "Schwarz function" was named by Philip J. Davis and Henry O. Pollak (1958) in honor of Hermann Schwarz,^[1] who introduced the Schwarz reflection principle for analytic curves in 1870.^[2] However, the Schwarz function does not explicitly appear in Schwarz's works.

Examples

The unit circle is described by the equation

|z| ^{2} = 1 , or \overline{z} = 1 /z . Thus, the Schwarz function of the unit circle is S(z) = 1 /z .

A more complicated example is an ellipse defined by

(x/a) ^{2} + (y/b) ^{2} = 1 . The Schwarz function can be found by substituting style x = (z + \overline{z)/() } and style y = (z - \overline{z)/() } and solving for \overline{z} . The result is:S(z) = (1)/(a ^{2-b} ^{2}) \left( (a ^{2+b} ^{2)z} - 2 ab\sqrt{z ^{2+b} ^{2-a} ^{2}} \right) .This is analytic on the complex plane minus a branch cut along the line segment between the foci \pm \sqrt{a ^{2-b} ^{2}} .

References

• Book: Davis, Philip J. . The Schwarz function and its applications . Philip J. Davis . . 1974 . . 978-0-883-85017-6 . 912405492 .
• Book: Needham, Tristan . Tristan Needham . 1997 . Visual Complex Analysis . Clarendon Press . 978-0-19-853447-1.
• Book: Shapiro, Harold S.. The Schwarz Function and Its Generalization to Higher Dimensions. 1992-03-18. John Wiley & Sons. 978-0-471-57127-8. en. 924755133. Harold S. Shapiro.

Notes and References

1. Davis . Philip J. Davis . Phillip . Pollak . Henry . Henry O. Pollak . On the Analytic Continuation of Mapping Functions . Transactions of the American Mathematical Society . 87 . 198–225 . January 1958 . 1 . 10.2307/1993097 . 1993097. free .
2. Schwarz . H.A. . Hermann Schwarz . Ueber die Integration der paritellen Differentialgleichung style (\partial ^{2} u)/(\partial x ^{2}) + (\partial ^{2} u)/(\partial y ^{2}) = 0 unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen . Monatsberichte der Königlichen Preussische Akademie des Wissenschaften zu Berlin . 1870 . 767–795 . Reprinted in: Book: Schwarz, H.A. . Hermann Schwarz . Gesammelte Mathematische Abhandlungen . 1890 . II . 144–171 . Berlin J. Springer .