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Introduction

This is the second part of a (long-term) five-part series on the visualization of Riemann surfaces. The first part [1] dealt with algebraic functions: [Graphics:../Images/index_gr_1.gif], [Graphics:../Images/index_gr_2.gif].  In this case [Graphics:../Images/index_gr_3.gif] describes an [Graphics:../Images/index_gr_4.gif]-valued (or [Graphics:../Images/index_gr_5.gif]-sheeted) function, locally described by radicals, or Root objects for the generic case[Graphics:../Images/index_gr_6.gif].

Here we will deal with compositions of radicals and elementary transcendental functions. In addition to the single-valued functions Exp, Sin, Cos, Tan, Cot, Sec, Csc, Sinh, Cosh, Tanh, Coth, Sech, Csch, the multivalued functions Power[_,Rational[p, q]] (which has q many sheets), Log, ArcSin, ArcCos, ArcTan, ArcCot, ArcSec, ArcCsc, ArcSinh, ArcCosh, ArcTanh, ArcCoth, ArcSech, ArcCsch, and Power[[Graphics:../Images/index_gr_7.gif], [Graphics:../Images/index_gr_8.gif]] will be taken into account. All of the last set of functions have infinitely many sheets. We will also allow Root objects to occur in the functions to be visualized. In the following we will assume that the function is already given in the form [Graphics:../Images/index_gr_9.gif]. (Solve can generate such an explicit  form in many cases from a function given in the form [Graphics:../Images/index_gr_10.gif].)

The unifying feature of the functions listed above is that all of their branch points are immediately recognized as logarithmic or algebraic branch points. This property makes dealing with this class of functions much easier than investigating general functions, such as implicitly defined functions. An example of a more general class of functions is [Graphics:../Images/index_gr_11.gif] defined implicitly through [Graphics:../Images/index_gr_12.gif]. Such functions will be discussed in Part III of this series. The classical special functions, such as the Gauss function [Graphics:../Images/index_gr_13.gif] and elliptic integrals, will be visualized in Part IV. Part V of this series will deal with the graphical representation of Riemann surfaces as double tori, triple tori, and spheres with handles.

Graphing the real or imaginary part (or a combination of them) of the function [Graphics:../Images/index_gr_14.gif] is often a faithful representation of the Riemann surface of [Graphics:../Images/index_gr_15.gif]. But in distinction to the case of algebraic functions, now some of the sheets might coincide globally for the real or imaginary part. In such cases, one has to decide from function to function what represents a faithful representation of the Riemann surface. Functions like Abs, (Re[#]+Im[#])& will typically be a faithful representation.

A generalized semi-numerical algorithm for the construction of the Riemann surface follows. It will be used to visualize the Riemann surfaces for a variety of  compositions of transcendental functions.

In this article we start with some generic remarks about the use of multivalued functions. Before we deal with the general case we will--as a warm up--construct some simple Riemann surfaces. In the next installment we will treat Weierstrass's method of analytic continuation for the function [Graphics:../Images/index_gr_16.gif] in detail, and implement a generic method for calculating the sheets of the Riemann surfaces of functions built from compositions of elementary transcendental functions.


Converted by Mathematica      May 9, 2000

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