Visualization of Complex Functions

– Plea for the Phase Plot –

Elias Wegert, Technical University Freiberg

May 25, 2009

This work has been inspired by the recent article“Mobius Transformations Revealed” by Dou-glas N. Arnold and Jonathan Rogness [2] in theNotices. The 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. Not 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’.”

There are praiseworthy exceptions from thisrule, like the textbooks by Tristan Needham [10]with its beautiful illustrations, and by StevenKrantz [7], with a chapter on computer packagesfor studying complex variables”. And certainlysome of us have invented their own techniquesto visualize complex functions in teaching andresearch.

The objective of this paper is to promote onesuch method which is equally simple and pow-erful. It does not only allow to depict complexfunctions, but may also serve as a tool for theirvisual exploration. Sometimes it also providesa new view on known results and opens up newperspectives, as is demonstrated in this paperfor a universality property of Riemann’s Zetafunction.

The Analytic Landscape

The visualization of functions is based on andlimited by our intuitive understanding of geo-metric objects. The graph of a complex functionf : D ⊂ C → C lives in four real dimensions,and since our imagination is trained in three di-mensional space, most of us have difficulties to“see” such an object.

Of course one can separate complex-valuedfunctions into their real and imaginary parts.This, however, destroys their mutual relation,and exactly this interplay, controlled by theCauchy-Riemann equations, is essential for an-alytic functions.

Some old books on function theory used to havenice illustrations of complex functions. Thesefigures show the analytic landscape of a func-tion f , which is the graph of its modulus |f |.

Figure 1: Analytic landscape of the complexGamma function

Taking into account how much easier it is to-day to produce such illustrations, it seems to

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be strange that we do not find them more oftenin contemporary textbooks.But in fact this may happen for good reasons.Analytic landscapes must be generated withcare, otherwise discretization effects may resultin pictures which rather hide what they shoulddemonstrate.

Figure 2: Some “poles” of the complexGamma function

In order to understand what a pole is we neednot really see the volcanoes of an analytic land-scape, and students looking at Figure 2 mayeven be in doubt about the validity of the max-imum principle.Also, the perspective view of the surface embed-ded in three dimensional space does not allowto see things very precisely. For instance, it isalmost impossible to read off location and de-gree of a zero from the analytic landscape. Sowhat else can we do?

Colored Analytic Landscape

The analytic landscape involves only one part ofthe function f , namely its modulus |f |. The sec-ond part, its argument arg f , is lost. How canwe incorporate this missing information appro-priately?Recall that the argument of a complex num-ber is only defined up to an additive multiple of2π. In order to get a well-defined function, wefrequently restrict it artificially to the interval(−π, π], or, even worse, to [0, 2π). This draw-back vanishes if we replace ϕ with the phaseeiϕ, which lives on the complex unit circle T.

And points on a circle can naturally be encodedby colors. So we let color serve as the lack-ing fourth dimension in representing graphs ofcomplex-valued functions.

Figure 3: The color circle and the colorencoded phase of points close to the origin

Though one usually does not distinguish be-tween the notions of “argument” and “phase”,this difference is essential for our purposes.Since the phase of f(z) is just the quotientf(z)/|f(z)|, one need not worry about the an-noying multi–valuedness of the argument.

The colored analytic landscape is the graph of|f | colored according to the phase of f .

Figure 4: Colored analytic landscape of thecomplex Gamma function

Typically analytic functions grow very fast atinfinity, at singularities or at the boundary oftheir (natural) domains. In such cases it is con-venient to use the graph of ln |f | instead of |f |for constructing the surface of the analytic land-scape. Since ln |f | and arg f are harmonic con-jugate functions the logarithmic representationis also more natural.

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The Phase Plot

One drawback of the colored analytic land-scapes is their three–dimensionality. The per-spective view makes it difficult to locate pointson the surface, and essential details may be in-visible. Also, there is no standard view, whichwould allow to recognize functions visually bytheir canonical appearance, as we do with thereal sine function, for instance.But indeed there is such a distinguished per-spective, namely the view “straight from thetop”. What we see then is a flat color im-age which just displays the phase. As will beshown in a moment, this phase plot alone gives(almost) all relevant information about the de-picted analytic function.

Figure 5: Phase plot of the Gamma functionwith logarithmically spaced level lines

And if one would not like to miss the infor-mation about |f |, it can easily be incorporatedinto the phase plot as (preferably logarithmi-cally spaced) level lines.

How to Read It

Though the phase of a function f with domainD lives on the set

D0 := z ∈ D : f(z) ∈ C \ 0

we shall nevertheless speak of phase plots onD, considering those points where the phase isundefined as singularities.

If f is analytic or, more generally, meromorphicin D, then its phase (plot) encodes the full in-formation about the function up to a positivescaling factor.

Theorem 1. If two (non–zero) meromorphicfunctions f and g on a connected domain Dhave the same phase plot, then f is a positivescalar multiple 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 clear that the result extends to the casewhere the phases of f and g coincide on an opensubset 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.There is also an intrinsic test which avoids val-ues and works with phases alone: If f and gare not constant (in which case the phase plotis isochromatic), it follows from the open map-ping principle that f = g if the phases of f + ciand g+ ci coincide for two distinct constants c1and c2.

Zeros and Poles

Since the phases of zero and infinity are unde-fined, zeros and poles of f are singularities ofits phase plot. How does the plot look like in aneighborhood of such points?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) 6= 0. 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.

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Figure 6: Phase plots in a neighborhood of azero (left) and a pole (right) of third order

Note that the colors are arranged in differentorders for zeros and poles. It is now clear thatthe phase plot does not only show the locationof zeros and poles but also reveals their degrees.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 numbers into the language of colors:Let γ : T → D0 be a closed oriented path inthe domain D0 of a phase plot P : D0 → D.Then the usual winding number (or index) ofthe mapping P γ : T → T is called the colorindex of γ with respect to the phase plot P andis denoted by cind γ.Less formally, the color index counts how manytimes the color of the point γ(t) moves aroundthe color circle when γ(t) traverses γ once inpositive direction.Now the argument principle can be rephrased asfollows: 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 = cind ∂D.

Figure 7: This function has no poles. Howmany zeros are in the displayed rectangle?

Looking at this picture in search for zeros imme-diately brings forth new questions, for example:Where do the isochromatic lines end up? Canthese lines connect two zeros? If so, do theyhave 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?

Essential Singularities

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

Figure 8: Analytic landscape of f(z) = e1/z

Despite of the massive tower this is not veryimpressive, and with regard to the Casorati-Weierstrass theorem, or even Picard’s great the-orem one would expect something much wilder.Now look at the colored logarithmic analyticlandscape and the phase plot:

Figure 9: Colored logarithmic analyticlandscape of f(z) = e1/z

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Figure 10: Phase plot of the essentialsingularity of f(z) = e1/z

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 im-ply that these lines are orthogonal to the levellines of |f |, i.e. the isochromatic lines are par-allel to the gradient of |f |. According to thechosen color scheme, we have red on the rightand green on the left when walking on a yellowline in ascending direction.To go a little beyond this qualitative result, wedenote by s the unit vector parallel to the gra-dient of |f | and set n := is. With ϕ := arg fand ψ := ln |f | the Cauchy-Riemann equationsfor log f yield that the directional derivatives ofϕ and ψ satisfy

∂sψ = ∂nϕ > 0, ∂nψ = −∂sϕ = 0,

at all regular points of the phase plot. Sincethe absolute value of ∂nϕ measures the densityof the isochromatic lines we can visually esti-mate the growth of ln |f | along these lines fromtheir density. Because the phase plot deliversno information on the absolute value, this doesnot say much about the growth of |f |. But tak-ing into account the second Cauchy-Riemannequation and

|(log f)′|2 = (∂nϕ)2 + (∂sϕ)2,

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

derivative,

∂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) indicates thatsomething should be visible in the phase plot.Indeed points z0 where f ′(z0) = 0 and f(z0) 6= 0are “color saddles”, i.e. intersections of isochro-matic lines. If f ′ has a zero of order k at z0 then2k + 2 such lines of a specific color emanatefrom z0, which is equivalent to saying that k+1smooth isochromatic lines intersect at z0.

Figure 11: Zeros of f ′ are color saddles

Color saddles appear as diffuse spots and itneeds some training to detect them. Here ithelps to use a color scheme which has a jumpat some point t of the unit circle. If we chooset := f(z0)/|f(z0)|, then the phase plot depictsa sharp saddle at the zero z0 of f ′. When t ro-tates through the full (color) circle, every saddleshows up clearly at some moment.

Periodic Functions

Obviously, the phase of a periodic function isperiodic, but what about the converse? If, forexample, a phase plot is doubly periodic, canwe then be sure that it represents an ellipticfunction?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.

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Figure 12: Phase plot of f(z) = ez

Figure 13: Phase plot of f(z) = sin z

Figure 14: Phase plot of a Weierstrass℘-function

Motivated by these pictures we give the follow-ing definition: A nonconstant map Φ is said tobe

(i) striped if there exists p0 6= 0 such that forall p = αp0 with α ∈ R

Φ(z + p) = Φ(z) for all z, (2)

(ii) simply periodic if there exists p1 6= 0 suchthat (2) holds if and only if p = k p1 forall k ∈ Z,

(iii) doubly periodic if there exist p1, p2 6= 0with p1/p2 /∈ R such that (2) holds if andonly if p = k1p1 + k2p2 for all k1, k2 ∈ Z.

As usual the numbers p1, p2 in (ii) and (iii) arereferred to as fundamental periods of Φ.

Theorem 2. Let f be a nonconstant meromor-phic function on C with phase Φ := f/|f |.

(i) The phase Φ is striped if and only if thereexist a, b ∈ C with a 6= 0 such that

f(z) = exp(az + b).

(ii) The phase Φ is simply periodic with fun-damental period p if and only if there exista simply periodic function g : C → C withperiod p and a real number α such that

f(z) = exp(αz/p) · g(z).

(iii) The phase Φ is doubly periodic if and onlyif f is doubly periodic.

Proof. The “if-direction” of all statements iseasy to verify.(i) After rotating the z-plane we may assumethat p0 is real. Then Φ(z) only depends on theimaginary part y of z. Writing Φ (locally) asΦ = eiϕ we get a harmonic function ϕ whichdepends only on y. Hence ϕ(z) = αy + β. Thefunction − ln |f | is (locally) a harmonic conju-gate of ϕ, thus ln |f(z)| = αx + γ. Finallyf(z) = eαz+b with b = γ + iβ. The case α = 0corresponds to a constant function with isochro-matic phase plot, which was excluded.(ii) If (2) holds for the phase of f we have

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

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

∈ R+.

Since h is meromorphic on C it must be apositive constant eα. It is easy to check thatg(z) := f(z) · e−αz/p defines a simply periodicfunction g with fundamental period p.

(iii) If p1 and p2 are fundamental periods of Φ,then it follows from the proof of (ii) that thereexist α1, α2 ∈ R such that f(z + pj) = eαj f(z).The meromorphic function g defined by g(z) :=f ′(z)/f(z) has only simple poles and zeros. In-tegration of g = (log f)′ along a (straight) linefrom z0 to z0 + pj which contains no pole of gyields that

αj =

∫ z0+pj

z0

g(z) dz.

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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 get α2 p1 = α1 p2. Since α1, α2 ∈ Rand p1/p2 /∈ R this implies that αj = 0.

Partial Sums of Power Series

Here is a strange image which, in similar form,occurred in an experiment. Since it looks sospecial, one could attribute it to a programmingerror.

Figure 15: A Taylor polynomial off(z) = 1/(1− z)

A moment’s thought shows what is going onhere, at least at an intuitive level. This exam-ple demonstrates that looking at phase plots canimmediately bring forth new questions whichwould perhaps not have been posed otherwise.But in the case at hand the question has notonly been posed, it has also been answered muchearlier (see [12], Section 7.8) – and probablywithout looking at pictures.

Theorem 3 (Robert Jentzsch, 1914). If apower series a0 + a1z + a2z

2 + . . . has a posi-tive finite convergence radius R, then the zerosof its partial sums cluster at every point z with|z| = R.

The reader interested in life and personal-ity of Robert Jentzsch is referred to the re-cent paper [5] by P.Duren, A.-K.Herbig, andD.Khavinson.

Riemann Surfaces

Any landscape invites for exploration. An ap-propriate vehicle for a journey through analyticlandscapes and phase plots is Weierstrass’ discchain method (‘Kreiskettenverfahren’) for ana-lytic continuation of a function along a path γ.Phase plots are appropriate to visualize thismethod not only in its abstract setting withempty discs, but using concrete functions, likethe square root or the logarithm. Such exper-iments are not only helpful to develop an in-tuitive understanding of Riemann surfaces, butcan even guide students to the discovery of suchobjects.

Figure 16: Square root and logarithm on theirRiemann surfaces.

It is an advantage of the phase plot that it al-lows to visualize analytic function directly ontheir Riemann surface.

Boundary Value Problems

Experimenting with phase plots generates anumber of new questions. One such problemis to find a criterion for deciding which colorimages are analytic phase plots, i.e. phase plotsof analytic functions.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

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in a Jordan domain D ? Can we start, for in-stance, with given colors on the boundary ∂D?If so, can the boundary colors be prescribed ar-bitrarily or must they be subject to certain con-ditions?In order to state these questions more precisewe 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 the sake of simplicity we here consider onlyboundary value problems for phase plots withcontinuous 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 B admitsa continuous extension to an analytic phase plotin 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 this continuity in order to get a niceresult.

Theorem 4. Let D be a Jordan domain witha continuous coloring B of its boundary ∂D.Then B admits a (unique) continuous extensionto an analytic phase plot in D if and only if thecolor index of B is zero.

Proof. If C : D → T is a continuous coloring,then a simple homotopy argument (contract ∂Dinside D to a point) shows that the color indexof its restriction to ∂D must vanish.Conversely, any continuous coloring B of ∂Dwith color index zero can be represented as B =eiϕ with a continuous function ϕ : ∂D → R.This function admits a (unique) continuous har-monic extension Φ to D. If Ψ denotes a har-monic conjugate of Φ, then f = eiΦ−Ψ is ana-lytic in D. Its phase C := eiΦ is continuous onD and coincides with B on ∂D.

Though the function f need not be continuous,it belongs to the Smirnov spaces Ep(D) for allp <∞.The figure below shows the phase plots off1(z) = −1, f2(z) = (z + 1)2/(z − 1)2, andf3(z) = (z2 + 5

2z + 1)/(z2 − 5

2z + 1) which all

have constant phase −1 almost everywhere onT. Though f1 and f2 are analytic and have con-tinuous phase in D, only f1 is a solution of theboundary value problem in the sense of Theo-rem 4.

Figure 17: Three functions with phase −1almost everywhere on T

The theorem can be rephrased in yet anotherintuitive form, which avoids the color index, asfollows.

Let C : C → T be any continuous coloring of theplane. Draw an arbitrary Jordan curve J anderase the color in the interior of J . Then theempty space can be filled in a unique way withthe phase plot of an analytic function such thatthe resulting coloring of the plane is continuousagain.Theorem 4 parametrizes analytic phase plotswhich extend continuously on D by theirboundary colorings. This result can be ex-tended to phase plots which are continuous onD with the exception of finitely many singular-ities of zero or pole type in D. If we then admitboundary 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 analytic phase plotswith prescribed singularities has a (unique) so-lution if and only if the (continuous) boundarycoloring B satisfies

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

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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: Riemann’s 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.The regular behavior of the phase of Zeta onthe imaginary axis provides a basis for count-ing its zeros in the rectangle 0 < Re z < 1,0 < Im z < T by the argument principle. Fur-ther, the striking visual similarity between thephase plots of ζ and Γ in the left half plane,together with the location of poles of Gammaand zeros of Zeta, suggests to look at the prod-uct ζ(z) Γ(z/2). Observing then that the phaseof this function is changing very slowly alongthe imaginary axis, we get a clue of the famousRiemann-Siegel formula (see [6]).The saying that Zeta is the mother of all func-tions alludes to its universality. The objectiveof this section is to present a version of thisresult which can be communicated to (almost)everybody.1.

Our starting point is the following variant ofVoronin’s universality theorem (Karatsuba andVoronin [8], see also Steuding [11]):

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. Astring S is the equivalence class of all such col-ored curves with respect to rigid motions of theplane.

Strings can be classifiedby their chromatic num-ber, which is the wind-ing number of the colormap of any representa-tive. The figure depictsa string with chromaticnumber 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 itself ifit can move to a place where it is invisible sinceit looks almost like the background.In conjunction with Theorem 4 the followinguniversality result for the phase plot of the Rie-mann Zeta function can easily be derived fromVoronin’s theorem.

Theorem 5. Let S be a string which lives inthe strip R. Then S can hide itself in the phaseplot of the Riemann Zeta function on R if it haschromatic number zero.

1“Man hat eine Sache erst dann verstanden, wenn man sie seiner Großmutter erklaren kann.” (Albert Einstein)

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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 [4].

Figure 19: Riemann’s Zeta function atIm z = 171, 8230 and 121415

What about the converse of Theorem 5 ? 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 color index inthe phase plot. By the argument principle thiswould imply that Zeta has zeros in R. If we as-sume this, for a moment, then such strings in-deed exist: They are perfectly hidden and windthemselves once around such a zero. So the con-verse of Theorem 5 holds if and only if R con-tains no zeros of Zeta, which is known to beequivalent to the Riemann hypothesis (see [4],[6]).

The Phase Plot as a Tool

Phase plots may be a useful tool for everybodywho is working with complex–valued functions.Here are a few examples which also demonstratesome additional tricks.

If it is not clear which branch of a functionis used in a certain software implementationof a special function a glimpse of the phaseplot may help. In particular, if several func-tions are composed, implementations with dif-ferent branch cuts can lead to completely dif-ferent results. Figure 20 illustrates the dif-ference between the Mathematica functionsLog (Gamma) and LogGamma.

Figure 20: The Mathematicaimplementations of LogGamma and Log(Gamma)

Another promising field of application is the vi-sual analysis of transfer functions in systemstheory and filter design.

Figure 22: The transfer function of aButterworth filter

The left part of the figure depicts the phase plotof a Butterworth filter. The frequency responseis on the white line (imaginary axis). In theright figure the color scheme is a modified by ablack component, which is a sawtooth functionof ln |f |. This allows to read off the gain of thefilter directly from the plot. Numerically this

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is much more efficient than computing contourlines of the modulus.Phase plots also allow to guess the asymptoticbehavior of functions and to find functional re-lations. For example you may wish to comparethe phase plots of exp z and sin z or, more chal-lenging, reinvent Riemann’s ξ-function and dis-cover its functional relation from the phase plotsof Zeta and Gamma.Further potential applications are in the fieldof Laplace and complex Fourier transforms, inparticular to the method of steepest descent (orstationary phase).Of course, the utility of phase plots is notrestricted to analytic functions. Figure 23shows the phase plot of a harmonic polynomial(Wilmshurst’s example [13], see also the paperon gravitational lenses by Khavinson and Neu-mann [9]).

Figure 23: 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.Finally, it should be mentioned that phase plotscan easily be animated. For example, a spin-ning color wheel with a discontinuity is helpfulto identify zeros of f ′ as color saddles.

In the phase plot of f − c, with a complex con-stant c, the c-values of f appear as singulari-ties. Now let c move and observe what hap-pens. Good examples are Blaschke products2

or functions with an essential singularity.

Figure 24: A finite Blaschke product withrandomly distributed zeros

Can you predict how the phase plot in Figure 24will morph when c travels along the real axisfrom 0 to 1 ?Even more interesting are animated phase plotsof functions involving parameters. For example,varying the module of elliptic functions allowsto visualize their metamorphosis from simplyperiodic to doubly periodic functions.

If you meet an interesting complex function inyour own work, I recommend that you look atits phase plot. There is a good chance to dis-cover new aspects of the subject. Here is a self-explaining Matlab code3:

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);

z = ones(yres,1)*x + i*y’*ones(1,xres);

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

2For readers interested in other color representations of Blaschke products the paper [3] is a must.3This is a tribute to N.Trefethen’s proposal to communicate ideas by exchanging ten-lines computer code

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Conclusion

Color encoding of function values has been cus-tomary for many decades (think about altitudeson maps or temperature and air pressure distri-bution in weather forecast). Also the idea torepresent the argument of a complex functionby a circular color scheme (its phase) is quitenatural.

Since color printing is not yet standard in math-ematical publications, analytic functions areusually either depicted by their analytic land-scape or by images of grids (see Section 12 of[7]). The latter approach is perfect for con-formal mappings, but the visualization of non-univalent function is problematic.

To the best of my knowledge the colored an-alytic landscape appears for the first time inprinted form in the outstanding mathematicstextbook [1] for engineering students.

In the internet, the page on special functionsat Wolfram Research is a well-structuredsource for all kinds of information, includ-ing tools for graphical visualization of complexfunctions. There are a few colored analyticlandscapes, but at the time of this writing phaseplots play only a marginal role. A good sourcefor aesthetic “phase plot like” pictures (see thecomments below) is the Wikipedia page of JanHomann.

Though I have been working with phase plotsin education and research for several years, ittook me some time to realize that they are muchmore than just one of several options to visual-ize analytic functions, and that forgetting aboutthe modulus may even be an advantage.

The concept of phase plots starts with split-ting the information about the function into twoparts, such that one (phase f/|f |) can be easilyrepresented, while the second one (ln |f |) can bereconstructed from the same picture.

So why not separate f into its real and imagi-nary part? One reason is that often zeros areof special interest, which can easily be detectedand characterized using the phase, but there isno way to find them from the real or imaginarypart alone.

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 modu-lus and argument (respectively phase). For ex-ample, Theorem 4 has no counterpart for themodulus of a function.Another advantage of phase is its small range,the unit circle. So one standardized colorscheme is appropriate for all functions.Reducing the range by replacing the argumentarg f with the phase f/|f | has yet another inter-pretation: we periodize the values of a function.The same idea was applied to ln |f | in Figure 22(Butterworth filter), where the gray scale waschosen according to the fractional part of ln |f |.If a sawtooth function with high frequency isapplied to both, the phase and the module, theresulting plots show the preimages of a polargrid, which yields a conformal representation ofthe function.

Figure 25: A conformal representation of thesine function

Finally, why do we not use a gray scale in gen-eral to represent the modulus and show the fullinformation about f in a “value plot”?The interested reader can find such pictures(where large values of |f | are associated withdark colors and small values are bright) on theWikipedia page of Jan Homann.There is no doubt that value plots outperformphase plots in some applications. However, foranalytic functions their main drawback is thelack of a simple characterization among all pos-sible colorings of the domain. In particular,

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there is no result about the boundary coloringcorresponding to Theorem 4.As a consequence the universality of Riemann’sZeta function for value plots needs the con-cept of “analytic color patterns”. For a non-mathematician this does not say much, in par-ticular it is not clear if this class is large or small.For some applications value plots have too manycolors.

Technical Remark. All images of this arti-cle have been created using Mathematica andMatlab.

Acknowledgement. I would like to thankGunter Semmler for many stimulating discus-sions and ideas as well as for his constructivecriticism, and Albrecht Bottcher for his inter-est, support and encouragement. Jorn Steud-ing and Peter Meier kindly advised me how tocompute the Riemann Zeta function.

References

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[4] J.B. Conrey, The Riemann Hypothesis. No-tices of the AMS, 50 (2003) 341–353.

[5] P.Duren, A.-K.Herbig, and D.Khavinson,Robert Jentzsch – Mathematiker und Poet.Mitteilungen DMV 16 (2008) 233–240.

[6] H.M.Edwards, Riemann’s Zeta Function.Academic Press, New York 1974.

[7] S.G.Krantz, Complex Variables. A Physi-cal Approach with Applications and Mat-lab. Chapman and Hall/CRC 2008.

[8] A.A.Karatsuba, and S.M. Voronin.The Riemann Zeta-Function. Walter deGruyter, 1992

[9] D.Khavinson, and G. Neumann, From theFundamental Theorem of Algebra to As-trophysics: A “Harmonious” Path. Noticesof the AMS 55 (2008) 666–675.

[10] T.Needham, Visual Complex Analysis.Clarendon Press, Oxford University Press,1997.

[11] J. Steuding, Value-Distribution of L-Func-tions. Springer, 2007.

[12] E.C. Titchmarsh, The Theory of Func-tions, Second Edition. Oxford UniversityPress, 1939.

[13] A.S.Wilmshurst, The valence of harmonicpolynomials. Proc. Am. Math. Soc. 126(1998) 2077-2081.

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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