Phase Plots of Complex Functions:a Journey in Illustration

Elias Wegert, TU Freiberg

July 14, 2010

Introduction

This work was inspired by the recent article“Mobius Transformations Revealed” by Dou-glas Arnold and Jonathan Rogness [3]. Therethe authors write:

“Among the most insightful tools that mathe-matics has developed is the representation of afunction of a real variable by its graph. . . . Thesituation is quite different for a function of acomplex variable. The graph is then a surface infour dimensional space, and not so easily drawn.Many texts in complex analysis are without asingle depiction of a function. Nor it is unusualfor average students to complete a course in thesubject with little idea of what even simple func-tions, say trigonometric functions, ‘look like’.”

In the printed literature there are a few laud-able exceptions to this rule, such as the prize-winning “Visual Complex Analysis” by Tris-tan Needham [27], Steven Krantz’ textbook [20]with a chapter on computer packages for study-ing complex variables, and the Maple based(German) introduction to complex function the-ory [15] by Wilhelm Forst and Dieter Hoffmann.

But looking behind the curtain, one encoun-ters a different situation which is evolving veryquickly. Many of us have developed our owntechniques for visualizing complex functions inteaching and research, and one can find many

beautiful illustrations of complex functions onthe internet.This paper is devoted to “phase plots”, a spe-cial tool for visualizing and exploring analyticfunctions. Figure 1 shows such a fingerprint ofa function in the complex unit disk.

Figure 1: The phase plot of an analyticfunction in the unit disk

The explanation of this illustration is deferredto a later section where it is investigated in de-tail.Phase plots have been invented independentlyby a number of people and it is impossible togive credit to someone for being the first. Origi-

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nally, they were mainly used in teaching as sim-ple and effective methods for visualizing com-plex functions. Over the years, and in partic-ular during the process of writing and rewrit-ing this manuscript, the topic developed its owndynamics and gradually these innocent illustra-tions transmuted to sharp tools for dissectingcomplex functions.

So the main purpose of this paper is not onlyto present nice pictures which allow one to rec-ognize complex functions by their individualface, but also to develop the mathematical back-ground and demonstrate the utility and creativeuses of phase plots. That they sometimes alsofacilitate a new view on known results and mayopen up new perspectives is illustrated by a uni-versality property of the Riemann Zeta functionwhich, in the setting of phase plots, can be ex-plained to (almost) anyone.

The final section is somewhat special. It re-sulted from a self-experiment carried out todemonstrate that phase plots are sources of in-spiration which can help to establish new re-sults. The main finding is that any meromor-phic function is associated with a dynamicalsystem which generates a phase flow on its do-main and converts the phase plot into a phasediagram. These diagrams will be useful toolsfor exploring complex functions, especially forthose who prefer thinking geometrically.

Visualization of Functions

The graph of a function f : D ⊂ C→ C lives infour real dimensions, and since our imaginationis trained in three dimensional space, most ofus have difficulties in “seeing” such an object.1

Some old books on complex function theoryhave nice illustrations of analytic functions.These figures show the analytic landscape of afunction, which is the graph of its modulus.

Figure 2: The analytic landscape off(z) = (z − 1)/(z2 + z + 1)

The concept was not introduced by JohannJensen in 1912 as sometimes claimed, but prob-ably earlier by Edmond Maillet [26] in 1903 (seealso Otto Reimerdes’ paper [31] of 1911).Differential geometric properties of analyticlandscapes have been studied in quite a numberof early papers (see Ernst Ullrich [34] and thereferences therein). Jensen [19] and others alsoconsidered the graph of |f |2, which is a smoothsurface. The second edition of “The Jahnke-Emde” [18] made analytic landscapes popularin applied mathematics.Analytic landscapes involve only one part of thefunction f , its modulus |f |; the argument arg fis lost. In the era of black-and-white illustra-tions our predecessors sometimes compensatedthis shortcoming by complementing the analyticlandscape with lines of constant argument. To-day we can achieve this much better using col-ors. Since coloring is an essential ingredient ofphase plots we consider it in some detail.Recall that the argument arg z of a complexnumber z is unique up to an additive multi-ple of 2π. In order to make the argument well-defined its values are often restricted to the in-terval (−π, π], or, even worse, to [0, 2π). Thisambiguity disappears if we replace arg z withthe phase z/|z| of z. Though one usually doesnot distinguish between the notions of “argu-ment” and “phase”, it is essential here to keepthese concepts apart.The phase lives on the complex unit circle T,and points on a circle can naturally be encodedby colors. We thus let color serve as the lacking

1One exception is Thomas Banchoff who visualized four dimensional graphs of complex functions at [6].

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fourth dimension when representing graphs ofcomplex-valued functions.

Figure 3: The color circle and the color-codedphase of points close to the origin

The colored analytic landscape is the graphof |f | colored according to the phase of f .Since the modulus of analytic functions typi-cally varies over a wide range one better usesa logarithmic scaling of the vertical axis. Thisrepresentation is also more natural since log |f |and arg f are conjugate harmonic functions.

Figure 4: The colored logarithmic analyticlandscape of f(z) = (z − 1)/(z2 + z + 1)

Colored analytic landscapes came to life witheasy access to computer graphics and by nowquite a number of people have developed soft-ware for their visualization. Andrew Bennett[7] has an easy-to-use Java implementation,and an executable Windows program can bedownloaded from Donald Marshall’s web site[25]. We further refer to Chapter 12 of StevenKrantz’ book [20], as well as to the web sitesrun by Hans Lundmark [24] and Tristan Need-ham [28]. Very beautiful pictures of (uncolored)analytic landscapes can be found on the “TheWolfram Special Function Site” [39].2

Though color printing is still expensive, coloredanalytic landscapes meanwhile also appear inthe printed literature (see, for example, the out-standing mathematics textbook [1] by Arens etal. for engineering students).With colored analytic landscapes the problemof visualizing complex functions could be con-sidered solved. However, there is yet anotherapproach which is not only simpler but evenmore general.Instead of drawing a graph, one can depict afunction directly on its domain by color-codingits values, thus converting it to an image. Suchcolor graphs of functions f live in the productof the domain of f with a color space.Coloring techniques for visualizing functionshave been customary for many decades, forexample in depicting altitudes on maps, butmostly they represent real valued functions us-ing a one dimensional color scheme. It is re-ported that two dimensional color schemes forvisualizing complex valued functions have beenin use for more than twenty years by now (LarryCrone [9], see Hans Lundmark [24]), but theybecame popular only with Frank Farris’ review[13] of Needham’s book and its complement [14].Farris also coined the name “domain coloring”.Domain coloring is a natural and universal sub-stitute for the graph of a function. Moreover iteasily extends to functions on Riemann surfacesor on surfaces embedded R3 (see KonstantinPoelke and Konrad Polthier [30], for instance).It is worth mentioning that we human beingsare somewhat limited with respect to the avail-able color spaces. Since our visual system hasthree different color receptors, we can only rec-ognize colors from a three–dimensional space.Mathematicians of the species gonodactylus oer-stedii3 could use domain coloring techniques toeven visualize functions with values in a twelve–dimensional space (Welsch and Liebmann [37]p. 268, for details see Cronin and King [10]).Indeed many people are not aware that natu-ral colors in fact provide us with an infinitedimensional space - at least theoretically. In

2As of June 2009 Wolfram’s tool visualizes the analytic landscape and the argument, but not phase.3 This is a species of shrimps which have 12 different photo receptors.

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reality “color” always needs a carrier. “Col-ored light” is an electromagnetic wave which isa mixture of monochromatic components withdifferent wavelengths and intensities. A sim-ple prismatic piece of glass reveals how light iscomposed from its spectral components. Read-ers interested in further information are recom-mended to visit the fascinating internet site ofDieter Zawischa [40].

Figure 5: A typical spectrum of polar light

The wavelengths of visible light fill an intervalbetween 375 nm and 750 nm approximately, andhence color spectra form an infinite-dimensionalspace.How many color dimensions are distinguishablein reality depends on the resolution of the mea-suring device. A simple model of the humaneye, which can be traced back to Thomas Youngin 1802, assumes that our color recognition isbased on three types of receptors which are sen-sitive to red, green, and blue, respectively.Since, according to this assumption, our visualcolor space has dimension three, different spec-tra of light induce the same visual impression.Interestingly, a mathematical theory of this ef-fect was developed as early as in 1853 by Her-mann Grassmann, the ingenious author of the“Ausdehnungslehre”, who found three funda-mental laws of this so-called metamerism [16](see Welsch and Liebmann [37]).Bearing in mind that the world of real colors isinfinite dimensional, it becomes obvious that itscompression to at most three dimensions cannotlead to completely satisfying results, which ex-plains the variety of color schemes in use fordifferent purposes. The two most popular colorsystems in our computer dominated world arethe RGB (CYM) and HSV schemes.In contrast to domain colorings which color-code the complete values f(z) by a two dimen-sional color scheme, phase plots display only

f(z)/|f(z)| and thus require just a one dimen-sional color space with a circular topology. Aswill be shown in the next section, they nev-ertheless contain almost all relevant informa-tion about the depicted analytic or meromor-phic function.In the figure below the Riemann sphere C (withthe point at infinity on top) is colored using twotypical schemes for phase plots (left) and do-main coloring (right), respectively.

Figure 6: Color schemes for phase plots anddomain coloring on the Riemann sphere

Somewhat surprisingly, the number of peopleusing phase plots seems to be quite small. Theweb site of Francois Labelle [23] has a nicegallery of nontrivial pictures4, including Euler’sGamma and Riemann’s Zeta function.Since the phase of a function occupies only onedimension of the color space, there is plenty ofroom for depicting additional information. Itis recommended to encode this information bya gray scale, since color (HUE) and brightnessare visually orthogonal. Figure 7 shows two suchcolor schemes on the Riemann w-sphere.

Figure 7: Two color schemes involvingsawtooth functions of gray

The left scheme is a combination of phase plotsand standard domain coloring. Here the valueg of gray does not depend directly on log |w|,

4Labelle justifies the sole use of phase by reasons of clarity and aesthetics.

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but is a sawtooth function thereof, g(w) =log |w| − blog |w|c. This coloring works equallywell, no matter in which range the values of thefunction are located.In the right scheme the gray value is the productof two sawtooth functions depending on log |w|and w/|w|, respectively. The discontinuities ofthis shading generate a logarithmically scaledpolar grid. Pulling back the coloring from thew-sphere to the z-domain of f by the map-ping w = f(z) resembles a conformal grid map-ping, another well-known technique for depict-ing complex functions (see Douglas Arnold [2]).Note that pulling back a grid instead of pushingit forward avoids multiple coverings. Of courseall coloring schemes can also be applied to func-tions on Riemann surfaces.For comparison, the figure below shows the fourrepresentations of f(z) := (z−1)/(z2 +z+1) inthe square |Re z| ≤ 2, |Im z| ≤ 2 correspondingto the color schemes of Figure 6 and Figure 7,respectively.

Figure 8: Four representations of the functionf(z) = (z − 1)/(z2 + z + 1)

Though these pictures (in particular the uppertwo) look quite similar, which makes it simpleto use them in parallel, the philosophy and themathematics behind them is quite different. Weshall comment on this issue in the final section.

The Phase Plot

The phase of a complex function f : D → Cis defined on D0 := z ∈ D : f(z) ∈ C×,where C× denotes the complex plane puncturedat the origin. Nevertheless we shall speak ofphase plots P : D → T, z 7→ f(z)/|f(z)| onD, considering those points where the phase isundefined as singularities. Recall that T standsfor the (colored) unit circle.

To begin with we remark that meromorphicfunctions are characterized almost uniquely bytheir phase plot.

Theorem 1. If two non–zero meromorphicfunctions f and g on a connected domain Dhave the same phase, then f is a positive scalarmultiple of g.

Proof. Removing from D all zeros and poles off and g we get a connected domain D0. Since,by assumption, f(z)/|f(z)| = g(z)/|g(z)| for allz ∈ D0, the function f/g is holomorphic andreal-valued in D0, and so it must be a (posi-tive) constant.

It is obvious that the result extends to the casewhere the phases of f and g coincide merely onan open subset of D.

In order to check if two functions f and g withthe same phase are equal, it suffices to comparetheir values at a single point which is neither azero nor a pole. For purists there is also an in-trinsic test which works with phases alone: As-sume that the non-constant meromorphic func-tions f and g have the same phase plot. Thenit follows from the open mapping principle thatf 6= g if and only if the phase plots of f + cand g+ c are different for one, and then for all,complex constants c 6= 0.

Zeros and Poles

Since the phases of zero and infinity are unde-fined, zeros and poles of a function are singular-ities of its phase plot. What does the plot looklike in a neighborhood of such points?

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If a meromorphic function f has a zero of degreen at z0 it can be represented as

f(z) = (z − z0)n g(z),

where g is meromorphic and g(z0) ∈ C×. It fol-lows that the phase plot of f close to z0 lookslike the phase plot of zn at 0, rotated by the an-gle arg g(z0). The same reasoning, with a nega-tive integer n, applies to poles.

Figure 9: A function with a simple zero, adouble zero, and a triple pole

Note that the colors are arranged in oppositeorders for zeros and poles. It is now clear thatthe phase plot does not only show the locationof zeros and poles but also reveals their multi-plicity.A useful tool for locating zeros is the argumentprinciple. In order to formulate it in the con-text of phase plots we translate the definitionof winding number into the language of colors:Let γ : T → D0 be a closed oriented path inthe domain D0 of a phase plot P : D0 → T.Then the usual winding number of the mappingP γ : T → T is called the chromatic numberof γ with respect to the phase plot P and isdenoted by chromPγ or simply by chrom γ.Less formally, the chromatic number counts howmany times the color of the point γ(t) movesaround the complete color circle when γ(t) tra-verses γ once in positive direction.Now the argument principle can be rephrasedas follows: Let D be a Jordan domain with pos-itively oriented boundary ∂D and assume thatf is meromorphic in a neighborhood of D. If fhas n zeros and p poles in D (counted with mul-tiplicity), and none of them lies on ∂D, then

n− p = chrom ∂D.

Figure 10: This function has no poles. Howmany zeros are in the displayed rectangle?

Looking at Figure 10 in search of zeros imme-diately brings forth new questions, for example:Where do the isochromatic lines end up? Canthey connect two zeros? If so, do these lineshave a special meaning? What about “basins ofattraction”? Is there always a natural (cyclic)ordering of zeros? What can be said about theglobal structure of phase plots? We shall returnto these issues later.

The Logarithmic Derivative

Along the isochromatic lines of a phase plotthe argument of f is constant. The Cauchy-Riemann equations for any continuous branchof the logarithm log f = ln |f | + i arg f implythat these lines are orthogonal to the level linesof |f |, i.e. the isochromatic lines are parallel tothe gradient of |f |. According to the chosencolor scheme, we have red on the right and greenon the left when walking on a yellow line in as-cending direction.To go a little beyond this qualitative result, wedenote by s the unit vector parallel to the gradi-ent of |f | and set n := is. With ϕ := arg f andψ := log |f | the Cauchy-Riemann equations forlog f imply that the directional derivatives of ϕand ψ satisfy

∂sψ = ∂nϕ > 0, ∂nψ = −∂sϕ = 0,

at all points z of the phase plot where f(z) 6= 0and f ′(z) 6= 0. Since the absolute value of ∂nϕmeasures the density of the isochromatic lines,we can visually estimate the growth of log |f |along these lines from their density. Because

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the phase plot delivers no information on theabsolute value, this does not say much aboutthe growth of |f |. But taking into account thesecond Cauchy-Riemann equation and

|(log f)′|2 = (∂nϕ)2 + (∂sϕ)2,

we obtain the correct interpretation of the den-sity ∂nϕ: it is the modulus of the logarithmicderivative,

∂nϕ = |f ′/f | . (1)

So, finally, we need not worry about branchesof the logarithm. It is worth mentioning that∂nϕ(z) behaves asymptotically like k/|z− z0| ifz approaches a zero or pole of order k at z0.

But this is not yet the end of the story. Whatabout zeros of f ′ ? Equation (1) indicatesthat something should be visible in the phaseplot. Indeed, points z0 where f ′(z0) = 0 andf(z0) 6= 0 are “color saddles”, i.e. intersectionsof isochromatic lines.

If f ′ has a zero of order k at z0, then z 7→f(z) − f(z0) has a zero of order k + 1 at z0.Consequently f can be represented as

f(z) = f(z0) + (z − z0)k+1 g(z)

where g(z0) 6= 0. It follows that f(z) travelsk + 1 times around f(z0) when z moves oncearound z0 along a small circle. In conjunctionwith f(z0) 6= 0 this can be used to show thatthere are exactly 2k + 2 isochromatic lines em-anating from z0 where the phase of f is equalto the phase of f(z0). Alternatively, one canalso think of k + 1 smooth isochromatic linesintersecting each other at z0.

Figure 11: Zeros of f ′ are color saddles

Color saddles appear as diffuse spots like in theleft picture of Figure 11. To locate them pre-cisely it is helpful to modify the color schemeby superimposing a gray component which hasa jump at some point t of the unit circle. Ift := f(z0)/|f(z0)| is chosen, then the phase plotshows a sharp saddle at the zero z0 of f ′ as inthe right picture.

Essential Singularities

Have you ever seen an essential singularity?Here is the picture which usually illustrates thissituation.

Figure 12: The analytic landscape off(z) = e1/z

Despite the massive tower this is not veryimpressive, and with regard to the Casorati-Weierstrass Theorem or the Great Picard The-orem one would expect something much wilder.Why does the analytic landscape not reflect thisbehavior? For the example the answer is easy:the function has a tame modulus, every contourline is a single circle through the origin. Nowlook at the phase plot in Figure 13:

Figure 13: A phase plot depicting the essentialsingularity of f(z) = e1/z

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But must there not be a symmetry betweenmodulus and phase? In fact not. There is sucha symmetry of modulus and argument (for non-vanishing functions), but phase plots depict thephase and not the argument – and this makes adifference.

So much for the example, but what about thegeneral case? Perhaps there are also functionswhich conceal their essential singularities in thephase plot?

In order to show that this cannot happen, weassume that f : D → C is analytic and has anessential singularity at z0.

By the Great Picard Theorem, there exists acolor c ∈ T such that any punctured neighbor-hood U of z0 contains infinitely many pointszk ∈ U with f(zk) = c. Moreover, the set ofzeros of f ′ in D is at most countable, and hencewe can choose c such that c 6= f(z)/|f(z)| forall zeros z of f ′.

Since |c| = 1, all isochromatic lines of the phaseplot through the points zk have the color c. Aswas shown in the preceding section, the modu-lus of f is strictly monotone along these lines.Now it is not hard to see that two distinct pointszk cannot lie on the same isochromatic line, be-cause these lines can meet each other only ata zero of f ′, which has been excluded by thespecial choice of c.

Consequently any neighborhood of an essentialsingularity contains a countable set of pairwisedisjoint isochromatic lines with color c. Com-bining this observation with the characteriza-tion of phase plots near poles and removablesingularities we obtain the following result.

Theorem 2. An isolated singularity z0 of ananalytic function f is an essential singularityif and only if any neighborhood of z0 intersectsinfinitely many isochromatic lines of the phaseplot with one and the same color.

Note that a related result does not hold forthe argument, since then, in general, the val-ues of arg f(zk) are different. For example, anytwo isochromatic lines of the function f(z) =exp(1/z) have a different argument.

Periodic Functions

Obviously, the phase of a periodic function isperiodic, but what about the converse?Though there are only two classes (simply anddoubly periodic) of nonconstant periodic mero-morphic functions on C, we can observe threedifferent types of periodic phase plots.

Figure 14: Phase plot of f(z) = ez

Figure 15: Phase plot of f(z) = sin z

Figure 16: Phase plot of a Weierstrass℘-function

“Striped” phase plots like in Figure 14 alwaysdepict exponential functions f(z) = eaz+b witha 6= 0. Functions with simply p–periodic phaseneed not be periodic, but have the more gen-eral form eαz/p g(z) with α ∈ R and a p–periodic

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function g. Somewhat surprisingly, doubly peri-odic phase plots indeed always represent ellipticfunctions.

The first result basically follows from the factthat the function arg f is harmonic and has par-allel straight contour lines, which implies thatarg f(x + iy) = αx + βy + γ. Since log |f | isconjugate harmonic to arg f , it necessarily hasthe form log |f(x+ iy)| = −αy + βx+ δ.

If the phase of f is p-periodic, then we have

h(z) :=f(z + p)

f(z)=|f(z + p)||f(z)|

∈ R+,

and since h is meromorphic on C, it must be apositive constant eα. Now it follows easily thatg(z) := f(z) · e−αz/p is periodic with period p.

Finally, if p1 and p2 are periods of f/|f | withp1/p2 /∈ R, then there exist α1, α2 ∈ R such thatf(z+pj) = eαj f(z). The meromorphic functiong defined by g(z) := f ′(z)/f(z) has only sim-ple poles and zeros. Integration of g = (log f)′

along a (straight) line from z0 to z0 + pj whichcontains no pole of g yields that

αj =

∫ z0+pj

z0

g(z) dz.

Evaluating now the area integral∫∫

Ωg dx dy

over the parallelogram Ω with vertices at0, p1, p2, p1 + p2 by two different iterated inte-grals, we obtain α2 p1 = α1 p2. Since α1, α2 ∈ Rand p1/p2 /∈ R this implies that αj = 0.

Partial Sums of Power Series

Figure 17 shows a strange image which, in sim-ilar form, occurred in an experiment. Since itlooks so special, one could attribute it to a pro-gramming error. A moment’s thought revealswhat is going on here, at least at an intuitivelevel. This example demonstrates again thatlooking at phase plots can immediately provokenew questions.

Figure 17: A Taylor polynomial off(z) = 1/(1− z)

Indeed the figure illustrates a rigorous result(see Titchmarsh [33], Section 7.8) which wasproven by Robert Jentzsch in 1914:

If a power series a0 +a1z+a2z2 + . . . has a pos-

itive finite convergence radius R, then the zerosof its partial sums cluster at every point z with|z| = R.

The reader interested in the life and personalityof Robert Jentzsch is referred to the recent pa-per [11] by Peter Duren, Anne-Katrin Herbig,and Dmitry Khavinson.

Boundary Value Problems

Experimenting with phase plots raises a numberof new questions. One such problem is to finda criterion for deciding which color images areanalytic phase plots, i.e., phase plots of analyticfunctions.Since phase plots are painted with the restrictedpalette of saturated colors from the color cir-cle, Leonardo’s Mona Lisa will certainly neverappear. But for analytic phase plots there aremuch stronger restrictions: By the uniquenesstheorem for harmonic functions an arbitrarilysmall open piece determines the plot entirely.So let us pose the question a little differently:What are appropriate data which can be pre-scribed to construct an analytic phase plot, say,in a Jordan domain D ? Can we start, for in-stance, with given colors on the boundary ∂D?

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If so, can the boundary colors be prescribed ar-bitrarily or are they subject to constraints?In order to state these questions more preciselywe introduce the concept of a colored set KC ,which is a subset K of the complex plane to-gether with a mapping C : K → T. Any suchmapping is referred to as a coloring of K.For simplicity we consider here only the fol-lowing setting of boundary value problems forphase plots with continuous colorings:

Let D be a Jordan domain and let B be a con-tinuous coloring of its boundary ∂D. Find allcontinuous colorings C of D such that the re-striction of C to ∂D coincides with B and therestriction of C to D is the phase plot of ananalytic function f in D.

If such a coloring C exists, we say that the col-oring B admits a continuous analytic extensionto D.The restriction to continuous colorings auto-matically excludes zeros of f in D. It does,however, not imply that f must extend contin-uously onto D – and in fact it is essential notto require the continuity of f on D in order toget a nice result.

Theorem 3. Let D be a Jordan domain witha continuous coloring B of its boundary ∂D.Then B admits a continuous analytic extensionto D if and only if the chromatic number of Bis zero. If such an extension exists, then it isunique.

Proof. If C : D → T is a continuous coloring,then a simple homotopy argument (contract ∂Dinside D to a point) shows that the chromaticnumber of its restriction to ∂D must vanish.Conversely, any continuous coloring B of ∂Dwith chromatic number zero can be representedas B = eiϕ with a continuous function ϕ : ∂D →R. This function admits a unique continuousharmonic extension Φ to D. If Ψ denotes a har-monic conjugate of Φ, then f = eiΦ−Ψ is analyticin D. Its phase C := eiΦ is continuous on D andcoincides with B on ∂D.

Theorem 3 parametrizes analytic phase plotswhich extend continuously on D by their

boundary colorings. This result can be gener-alized to phase plots which are continuous onD with the exception of finitely many singular-ities of zero or pole type in D. Admitting nowboundary colorings B with arbitrary color indexwe get the following result:For any finite collection of given zeros with or-ders n1, . . . , nj and poles of orders p1, . . . , pk theboundary value problem for meromorphic phaseplots with prescribed singularities has a uniquesolution if and only if the (continuous) bound-ary coloring B satisfies

chromB = n1 + . . .+ nj − p1 − . . .− pk.

The Riemann Zeta Function

After these preparations we are ready to paya visit to “Zeta”, the mother of all analyticfunctions. Here is a phase plot in the square−40 ≤ Re z ≤ 10, −2 ≤ Im z ≤ 48.

Figure 18: The Riemann Zeta function

We see the pole at z = 1, the trivial zeros atthe points −2,−4,−6, . . . and several zeros onthe critical line Re z = 1/2. Also we observethat the isochromatic lines are quite regularlydistributed in the left half plane.Saying that Zeta is the mother of all functionsalludes to its universality. Our starting point

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is the following strong version of Voronin’s Uni-versality Theorem due to Bagchi [4] (see alsoKaratsuba and Voronin [21], Steuding [32]):

Let D be a Jordan domain such that D is con-tained in the strip

R := z ∈ C : 1/2 < Re z < 1,

and let f be any function which is analytic inD, continuous on D, and has no zeros in D.Then f can be uniformly approximated on D byvertical shifts of Zeta, ζt(z) := ζ(z + it) witht ∈ R.

Recall that a continuously colored Jordan curveJC is a continuous mapping C : J → T from asimple closed curve J into the color circle T.

Figure 19: Arepresentative of a

string

A string S is an equiva-lence class of all such col-ored curves with respectto rigid motions of theplane. Like colored Jor-dan curves, strings fallinto different classes ac-cording to their chro-matic number. Figure 19depicts a representativeof a string with chro-matic number one.

We say that a string S lives in a domain D ifit can be represented by a colored Jordan curveJC with J ⊂ D. A string can hide itself in aphase plot P : D → T, if, for every ε > 0, it hasa representative JC such that J ⊂ D and

maxz∈J|C(z)− P (z)| < ε.

In less technical terms, a string can hide itselfif it can move to a place where it is invisiblesince it blends in almost perfectly with the back-ground.In conjunction with Theorem 3 the followinguniversality result for the phase plot of the Rie-mann Zeta function can easily be derived fromVoronin’s theorem.

Theorem 4. Let S be a string which lives inthe strip R. Then S can hide itself in the phase

plot of the Riemann Zeta function on R if it haschromatic number zero.

In view of the extreme richness of Jordan curvesand colorings this result is a real miracle. Thethree pictures below show phase plots of Zetain the critical strip. The regions with saturatedcolors belong to R. The rightmost figure depictsthe domain considered on p. 342 of Conrey’s pa-per [8].

Figure 20: The Riemann Zeta function atIm z = 171, 8230 and 121415

What about the converse of Theorem 4 ? Ifthere existed strings with nonzero chromaticnumber which can hide themselves in the stripR, their potential hiding-places must be Jor-dan curves with non-vanishing chromatic num-ber in the phase plot. By the argument prin-ciple, this would imply that Zeta has zeros inR. If we assume this, for a moment, then suchstrings indeed exist: They are perfectly hiddenand wind themselves once around such a zero.So the converse of Theorem 4 holds if and onlyif R contains no zeros of Zeta, which is knownto be equivalent to the Riemann hypothesis (seeConrey [8], Edwards [12]).

Phase Flow and Diagrams

Mathematical creativity is based on the inter-play of problem posing and problem solving,and it is my belief that the former is even more

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important than the latter: often the key to solv-ing a problem lies in asking the right questions.Illustrations have a high density of informa-tion and stimulate imagination. Looking at pic-tures helps in getting an intuitive understandingof mathematical objects and finding interestingquestions, which then can be investigated usingrigorous mathematical techniques.This section intends to demonstrate how phaseplots can produce novel ideas. The mate-rial presented here is the protocol of a self-experiment which has been carried out by theauthor in order to check the creative potentialof phase plots.Let us start by looking at Figure 1 again. Itdepicts the phase of a finite Blaschke product,which is a function of the form

f(z) = cn∏k=1

z − zk1− zkz

, z ∈ D,

with |zk| < 1 and |c| = 1. Blaschke products arefundamental building blocks of analytic func-tions in the unit disc and have the property|f(z)| = 1 for all z ∈ T. The function shown inFigure 1 has 81 zeros zk in the unit disk.Looking at Figure 1 for a while leaves the im-pression of a cyclic ordering of the zeros. Let ustest this with another example having only fivezeros (Figure 21, left).

Figure 21: The phase plot of a Blaschkeproduct with five zeros

The left picture seems to confirm the expecta-tion: if we focus attention to the yellow color,any of these lines connects a zero with a certainpoint on the boundary, thus inducing a cyclicordering.However, looking only at one specific color ismisleading. Choosing another, for instance

blue, can result in a different ordering. So whatis going on here? More precisely: What is theglobal structure of the phase plot of a Blaschkeproduct? This could be a good question.An appropriate mathematical framework to de-velop this idea is the theory of dynamical sys-tems. We here only sketch the basic facts; fordetails see [36].With any meromorphic function f in a domainD we associate the dynamical system

z = g(z) :=f(z) f ′(z)

|f(z)|2 + |f ′(z)|2. (2)

The function g on the right-hand side of (2) ex-tends from D0 to a smooth function on D. Thissystem induces a flow Φ on D, which we desig-nate as the phase flow of f .The fixed points of (2) are the zeros of f (re-pelling), the poles of f (attracting), and the ze-ros of f ′ (saddles). The remaining orbits arethe components of the isochromatic lines of thephase plot of f when the fixed points are re-moved. Thus the orbits of the phase flow endowthe phase plot with an additional structure andconvert it into a phase diagram.Intuitively, the phase flow Φ transports a col-ored substance (“phase”) from the zeros to thepoles and to the boundary of the domain.

Figure 22: Phase transport to the boundary,zeros of f and f ′ with invariant manifolds

The left part of Figure 23 illustrates how“phase” of measure 2π emerging from the zeroin the highlighted domain is transported alongthe orbits of Φ until it is finally deposited along(parts of) the boundary.

In the general case, where f : D → C is mero-morphic on a domain D and G ⊂ D is a Jordandomain with boundary J in D0, the phase flow

12

of any zero (pole) of f in G generates a (signed)measure on J . The result is a quantitative ver-sion of the argument principle which tells us inwhich way the phase of the zeros (poles) is dis-tributed along J (see Figure 23, right).

Figure 23: The phase flow and the argumentprinciple

The question about the structure of phase plotsof Blaschke products can now be rephrased inthe setting of dynamical systems: What are thebasins of attraction of the zeros of f with respectto the (reversed) phase flow?The key for solving this problem is given bythe invariant manifolds of the saddle points,i.e., the points aj ∈ D where f ′(aj) = 0 andf(aj) 6= 0.

Figure 24: Invariant manifolds of the saddlesand basins of attraction of the zeros

Removing all unstable manifolds of the pointsaj from D results in an open set B, which isthe union of connected components Bj. Anycomponent Bj contains exactly one zero bj off , where multiple zeros are counted only once.The intersection of every set Bj with T is notempty and consists of a finite number of arcsAji. The complete set of these arcs covers theunit circle and two arcs are either disjoint ortheir intersection is a singleton. These sep-arating points are the endpoints of unstable

manifolds which originate from saddle points.For later use we renumber the arcs Aji asA1, A2, . . . , As in counter-clockwise direction.

It is obvious that the number s of separatingpoints cannot be less than the number of dis-tinct zeros of f . In order to get an upperbound of s we assume that f has m distinctzeros with multiplicities β1, . . . , βm and k sad-dle points where f ′ has zeros of multiplicitiesα1, . . . , αk, respectively. Then we have

α1 + . . .+αk = m−1, β1 + . . .+βm = n−1.

The first equation follows from the well-knownfact that the derivative of a Blaschke product oforder n has exactly n − 1 zeros in D, this timecounting multiplicity. From any saddle point ajexactly αj + 1 rays emerge which belong to theunstable manifold of aj. Since any separatingpoint must be the endpoint of one such line,the total number s of separating points cannotbe greater than k + β1 + . . .+ βk = k +m− 1.Thus we finally get

m ≤ s ≤ m+ k − 1.

Examples show that both estimates are sharp.

It turns out that the global topological struc-ture of the phase plot is completely character-ized by the sequence S of integers, which asso-ciates with any of the arcs A1, . . . , As (in con-secutive order) the number of the correspond-ing zero. This sequence depends on the spe-cific numbering of the zeros and the arcs, butan appropriate normalization makes it unique.For example, the Blaschke product depictedin Figure 24 is represented by the sequenceS = (1, 2, 3, 2, 4, 5, 4, 2).

Let us now return to Figure 1 again. Picturingonce more that “phase” is a substance emergingfrom sources at the zeros which can exit the do-main only at its boundary, is it then not quitenatural that phase plots of Blaschke productsmust look like they do?

And if you are asking yourself what “natural”means, then this is already another question.

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Concluding Remarks

Phase plots result from splitting the informa-tion about the function f into two parts (phaseand modulus), and one may ask why we do notseparate f into its real and imaginary part. Onereason is that often zeros are of special interest;their presence can easily be detected and char-acterized using the phase, but there is no wayto find these from the real or imaginary partalone.

And what is the advantage of using f/|f | in-stead of ln |f | ? Of course, zeros and poles canbe seen in the analytic landscape, but they aremuch better represented in the phase plot. Infact there is a subtle asymmetry between mod-ulus and argument (respectively, phase). Forexample, Theorem 3 has no counterpart for themodulus of a function.5

Since phase plots and standard domain coloringproduce similar pictures, it is worth mentioningthat they are based on different concepts andhave a distinct mathematical background.

Recall that standard domain coloring methodsuse the complete values of an analytic function,while phase plots depict only its phase. Takinginto account that phase can be considered as aperiodization of the argument, which is (locally)a harmonic function, reveals the philosophy be-hind phase plots: Analytic functions are con-sidered as harmonic functions, endowed with aset of singularities having a special structure.Algebraically, phase plots forget about the lin-ear structure of analytic functions, while theirmultiplicative structure is preserved.

This approach has at least two advantages. Thefirst one is almost trivial: phase has a smallrange, the unit circle, which allows one visual-izing all functions with one and the same colorscheme. Moreover, a one–dimensional colorspace admits a better resolution of singulari-ties. Mathematically more important is the ex-istence of a simple parametrization of analyticand meromorphic phase plots by their bound-ary values and their singularities (Theorem 3).

There is no such result for domain colorings ofanalytic functions.

The following potential fields of applicationsdemonstrate that phase plots may be a usefultool for anyone working with complex–valuedfunctions.

1. A trivial but useful application is visual in-spection of functions. If, for example, it is notknown which branch of a function is used in acertain software, a glance at the phase plot mayhelp. In particular, if several functions are com-posed, software implementations with differentbranch cuts can lead to completely different re-sults. You may try this with the Mathemat-ica functions Log (Gamma) and LogGamma. An-other useful exercise in teaching is to comparethe phase plots of exp(log z) and log(exp z).

2. A promising field of application is visualanalysis and synthesis of transfer functions insystems theory and filter design. Since here themodulus (gain) is often more important thanphase, it is recommended to use the left colorscheme of Figure 7.

3. Further potential applications lie in the areaof Laplace and complex Fourier transforms, inparticular to the method of steepest descent (orstationary phase).

4. Phase plots also allow to guess the asymp-totic behavior of functions (compare, for exam-ple, the phase plots of exp z and sin z), and tofind functional relations. A truly challengingtask is to rediscover the functional equation ofthe Riemann Zeta–function from phase plots ofζ and Γ.

5. Complex dynamical systems, in the sense ofiterated functions, have been investigated by byFelix Huang [17] and Martin Pergler [29] usingdomain coloring methods. The problem of scal-ing the modulus disappears when using phaseplots; see the pictures of Francois Labelle [23]and Donald Marshall [25].

6. The utility of phase plots is not restrictedto analytic functions. Figure 25 visualizes the

5There is such a result for outer functions, but it is impossible to see if a function is outer using only the boundaryvalues of its modulus.

14

function

h(z) := Im(e−

iπ4 zn

)+ i Im

(e

iπ4 (z − 1)n

),

with n = 4. This is Wilmshurst’s example [38]of a harmonic polynomial of degree n having themaximal possible number of n2 zeros. For back-ground information we recommend the paper ongravitational lenses by Dmitry Khavinson andGenevra Neumann [22].

Figure 25: A modified phase plot ofWilmshurst’s example for n = 4

To understand the construction of the depictedfunction it is important to keep track of the ze-ros of its real and imaginary parts. In the figurethese (straight) lines are visualized using a mod-ified color scheme which has jumps at the points1, i,−1 and −i on the unit circle.

Besides these and other concrete applicationsone important feature of phase plots is theirpotential to bring up interesting questions andproduce novel ideas. If you would like to tryout phase plots on your own problems, you maystart with the following self-explaining Matlabcode:6

xmin=-0.5; xmax=0.5; ymin=-0.5; ymax=0.5;

xres = 400; yres = 400;

x = linspace(xmin,xmax,xres);

y = linspace(ymin,ymax,yres);

[x,y] = meshgrid(x,y); z = x+i*y;

f = exp(1./z);

p = surf(real(z),imag(z),0*f,angle(-f));

set(p,’EdgeColor’,’none’);

caxis([-pi,pi]), colormap hsv(600)

view(0,90), axis equal, axis off

Though the phase of a function is at least of thesame importance as its modulus, it has not yetbeen studied to the same extent as the latter.It is my conviction that phase plots are problemfactories, which have the potential to changethis situation.

Technical Remark. All images of this articlewere created using Mathematica and Mat-lab.

Acknowledgement. I would like to thank allpeople who supported me in writing this pa-per: Gunter Semmler read several versions ofthe manuscript, his valuable ideas and construc-tive criticism led to a number of significant im-provements. The project profited much fromseveral discussions with Albrecht Bottcher whoalso generously supported the production of afirst version. Richard Varga and George Csor-das encouraged me during a difficult period.Jorn Steuding and Peter Meier kindly advisedme in computing the Riemann Zeta function.Oliver Ernst tuned the language and eliminatedsome errors.The rewriting of the paper would not have beenpossible without the valuable comments andcritical remarks of several referees.Last but not least I would like to thank StevenKrantz and Marie Taris for their kind, construc-tive, and professional collaboration and an in-spiring exchange of ideas.

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Author’s address: Elias Wegert, Institute of Applied Mathematics, Technical University BergakademieFreiberg, D-09596 Freiberg, Germany. Email: wegert@math.tu-freiberg.de

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