TOPICS
Search

Köbe Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
KoebeFunction
KoebeReImAbs
Min Max
Re
Im Powered by webMathematica

The function

f_theta(z)=z/((1+e^(itheta)z)^2)
(1)

defined on the unit disk |z|<1. For theta in [0,2pi), the Köbe function is a schlicht function

f(z)=z+sum_(j=2)^inftya_jz^j
(2)

with |a_j|=j for all j (Krantz 1999, p. 149). For theta=0,

f_0(z)=z/((z-1)^2),
(3)

illustrated above.

See also

Köbe's One-Fourth Theorem, Schlicht Function

Explore with Wolfram|Alpha

References

Bombieri, E. "On the Local Maximum of the Koebe Function." Invent. Math. 4, 26-67, 1967.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 149, 1999.Pederson, R. and Schiffer, M. "A Proof of the Bieberbach Conjecture for the Fifth Coefficient." Arch. Rational Mech. Anal. 45, 161-193, 1972.Stewart, I. From Here to Infinity: A Guide to Today's Mathematics. Oxford, England: Oxford University Press, pp. 164-165, 1996.

Referenced on Wolfram|Alpha

Köbe Function

Cite this as:

Weisstein, Eric W. "Köbe Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KoebeFunction.html

Subject classifications