Schwarz function

Not to be confused with Schwartz function or Schwarz triangle function.

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.

The Schwarz function exists for analytic curves. More precisely, for every non-singular, analytic Jordan arc ${\displaystyle \Gamma }$ in the complex plane, there is an open neighborhood ${\displaystyle \Omega }$ of ${\displaystyle \Gamma }$ and a unique analytic function ${\displaystyle S}$ on ${\displaystyle \Omega }$ such that ${\displaystyle S(z)={\overline {z}}}$ for every ${\displaystyle z\in \Gamma }$.^[1]

The "Schwarz function" was named by Philip J. Davis and Henry O. Pollak (1958) in honor of Hermann Schwarz,^[2][3] who introduced the Schwarz reflection principle for analytic curves in 1870.^[4] However, the Schwarz function does not explicitly appear in Schwarz's works.^[5]

Examples

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

A more complicated example is an ellipse defined by ${\displaystyle (x/a)^{2}+(y/b)^{2}=1}$. The Schwarz function can be found by substituting ${\displaystyle \textstyle x={\frac {z+{\overline {z}}}{2}}}$ and ${\displaystyle \textstyle y={\frac {z-{\overline {z}}}{2i}}}$ and solving for ${\displaystyle {\overline {z}}}$. The result is:^[6]

${\displaystyle S(z)={\frac {1}{a^{2}-b^{2}}}\left((a^{2}+b^{2})z-2ab{\sqrt {z^{2}+b^{2}-a^{2}}}\right)}$.

This is analytic on the complex plane minus a branch cut along the line segment between the foci ${\displaystyle \pm {\sqrt {a^{2}-b^{2}}}}$.

References

1. Shapiro 1992, p. 3
2. Davis, Phillip; Pollak, Henry (January 1958). "On the Analytic Continuation of Mapping Functions" (PDF). Transactions of the American Mathematical Society. 87 (1): 198–225. doi:10.2307/1993097. JSTOR 1993097.
3. Needham 1997, p. 255
4. Schwarz, H.A. (1870). "Ueber die Integration der paritellen Differentialgleichung ${\displaystyle \textstyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}$ unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen". Monatsberichte der Königlichen Preussische Akademie des Wissenschaften zu Berlin: 767–795. Reprinted in: Schwarz, H.A. (1890). Gesammelte Mathematische Abhandlungen. Vol. II. pp. 144–171.
5. Shapiro 1992, p. 2
6. Needham 1997, p. 256

• Davis, Philip J. (1974). The Schwarz function and its applications. Carus Monographs 17. Mathematical Association of America. ISBN 978-0-883-85017-6. OCLC 912405492.
• Needham, Tristan (1997). Visual Complex Analysis. Clarendon Press. ISBN 978-0-19-853447-1.
• Shapiro, Harold S. (1992-03-18). The Schwarz Function and Its Generalization to Higher Dimensions. John Wiley & Sons. ISBN 978-0-471-57127-8. OCLC 924755133.