Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation at Göttingen in 1851.

Consider the function from the complex plane to itself given by
w=f(z)=z^{n}, where n is at least 2. The z-plane may be divided
into n sectors given by arg z lying between (k-1)(2pi/n) and k(2pi/n) for
k=1,...,n. There is a one-to-one correspondence between each sector and
the whole w-plane, except for the positive real axis. The image of
each sector is obtained by performing a cut along the positive real axis;
this cut has an upper and a lower edge. Corresponding to the n sectors
in the z-plane, take n identical copies of the w-plane with the cut. These
will be the *sheets* of the Riemann surface and are distinguished
by a label k which serves to identify the corresponding sector. For k=1,...,n-1
attach the lower edge of the sheet labeled k with the upper edge of the
sheet labeled k+1. To complete the cycle, attach the lower edge of the
sheet labeled n to the upper edge of the sheet labeled 1. In a physical
sense, this is not possible without self-intersection but the idealized
model shall be free of this discrepancy. The result of the construction
is a *Riemann surface* whose points are in one-to-one correspondence
with the points of the z-plane. This correspondence is continuous in the
following sense. When z moves in its plane the corresponding point w is
free to move on the Riemann surface. The point w=0 connects all the sheets
and is called the *branch point*. A curve must wind n times around
the branch point before it closes. Now consider the n-valued relation
z=n^{th}root(w). To each nonzero w, there correspond n values of z.
If the w-plane is replaced by the Riemann surface just constructed, then each
complex nonzero w is represented by n points of the Riemann surface at
superposed positions. Let the point on the uppermost sheet represent the
principal value and the other n-1 points represent the other values. Then
z=n^{th}root(w) becomes a single-valued, continuous, one-to-one
correspondence of the points of the Riemann surface with the points of the
z-plane. The Riemann surface is orientable, since every orientation of a
sheet is carried over to the sheet next to it.

Model produced with: JavaView v.2.21

Keywords | Riemann Surface; Algebraic Function | |

MSC-2000 Classification | 30F99 | |

Zentralblatt No. | 05264891 |

- B. Riemann: Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, Inauguraldissertation, Göttingen (1851), http://www.emis.de/classics/Riemann/.
- D. Hilbert and S. Cohn-Vossen: Anschauliche Geometrie, English Translation by Chelsea Publishing Company (1990) (1932), .

- Master File: Riemann-Surface_Master.jvx
- Master File: Riemann-3-Surface_Master.jvx
- Applet File: Riemann-Surface_Master.jvx
- Applet File: Riemann-3-Surface_Master.jvx
- Preview: Riemann-Surface_Preview.gif
- Preview: Riemann-3-Surface_Preview.gif

Submitted: Sun May 19 09:03:19 CEST 2002.

Revised: Fri Jan 10 14:19:31 GMT 2003.

Accepted: Fri Feb 14 18:02:36 CET 2003.

Ansal Institute of Technology

H-501 Palam Vihar

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Haryana 122017

India

ashay@dharwadker.org

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