Category Archives: Science and Math

Writing a paper in E-prime

Many top scientists communicate clearly, sometimes seemingly effortlessly. The papers by Einstein flow elegantly in clear and logical steps, almost as if choreographed, from one idea to the next. Some articles even have qualities more commonly seen in great works of art, for example, Dirac’s seminal book on quantum mechanics or Shannon’s paper introducing his celebrated entropy. What a pleasure to read! Most physicists similarly recognize Feynman as a master of clear communication.

Before I became a grad student, I had underestimated the importance of good and effective communication. My former PhD advisor, an excellent communicator, taught me the crucial role played by communication in scientific discourse and debate.

Let me explain this point in greater detail. As an illustrative example, imagine if Einstein had not written clearly. Then it may very well have taken much longer for his ideas to percolate and gain acceptance throughout the scientific community. Indeed, Boltzmann, in contrast to Einstein, wrote lengthy and admittedly difficult-to-read texts. Some of his critics perhaps  failed to grasp his seminal ideas. Disappointed and possibly depressed, he eventually committed suicide while still in his prime. Today, the top prize in the field  of statistical physics honors his name— the Boltzmann Medal. Nevertheless, it took many years and the efforts of other scientists (e.g. Gibbs) for the physics community to recognize the full extent of Boltzmann’s contributions.    Clear exposition can make a big difference.

In this blog post, I do not give tips or advice about how to write clearly. Good tips on how to write clearly abound.  Instead, I want to draw your attention to how this article does not contain a single instance of the verb “to be” or any of its conjugations or derived words, such as “being,” “was,” “is,” and so forth — excepting this sentence, obviously. The subset of the English language that remains after the removal of these words goes by the name E-prime, often written E’. In other words, E’ equals English minus all words derived from the above-mentioned verb.

Writing in E’ usually forces a person to think more carefully. Scientists need to communicate not only clearly, but with a slightly higher degree of precision than your typical non-scientist. I have found that fluency in E’ helps me to spot certain kinds of errors of reasoning. The key error of reasoning attenuated by the use of E’ relates to identification.   Too often, the referents of the grammatical subject and object become identified in standard English, where in fact no such identification exists in the real world.  E’ helps to reduce this improper identification, or at least to call attention to it.  The topic of E’, and of related subjects, such as  its ultimate historical origins in general semantics, the study of errors of reasoning, the nature of beliefs, cognitive biases, etc., would require too broad a digression for me to discuss here, so I recommend that interested readers research such topics on their own.

In my early 30s, soon after I obtained tenure in my first faculty position, I decided to write a full article entirely in E’.  What a wonderful and interesting exercise!  Of course, I did not find it easy to write in E’, but with few exceptions, the finished paper contained only E’ sentences.  Forcing myself to think and write in E’ helped me to give a better description of what we, as scientists, really did.  I would cautiously claim that writing in E’ benefited our paper, at least as far as concerns clarity and precision.  No longer do I publish papers in E’, but I learned a lot about how to write (and think) a little bit more clearly.

That paper, about an empirical approach to music, appeared in print in 2004 in the statistical physics journal  Physica A. It eventually ended up cited very well: 33 citations according to  Thomson Reuters’  Web of Science and 60 citations on Google Scholar, as of May 2016.  Most incredibly, it even briefly shot up to the top headline at (click here to see)!  We had never expected this.

In that paper, my co-authors and I proposed a method for studying rhythmic complexity. The collaboration team included as first author Heather Jennings, a professor of music (and also my spouse). We took an empirical approach for comparing the rhythmic structures of Javanese Gamelan, Jazz, Hindustani music, Western European classical music, Brazilian popular music (MPB), techno (dance), New Age music, the northeastern Brazilian folk music known as Forró and last but not least: Rock’n Roll. Excepting a few sentences, the paper consists entirely of E’ sentences.

You can read the paper by clicking here for the PDF. A fun exercise: as you read the paper, (1) try to imagine how you would normally rephrase the E’ sentences in ordinary English; (2) try to spot the subtle difference in meaning between the English and E’ sentences.


Colóquio na USP sobre movimento de animais

Aqui está o link para o video de um Colóquio que proferi na USP em 09/04/2015. A palestra está em português, embora o título esteja em inglês.   Esse assunto representa o “feijão com arroz” das minhas pesquisas na área de física estatística aplicada.

Vale a pena também destacar que o professor que me apresenta no início do video é o professor titular Mario de Oliveira, autor do livro sobre termodinâmica que virou referência no Brasil. Seu livro é frequentemente usado como texto principal junto a disciplinas de termodinâmica nos cursos de graduação em física.

Scale invariance, random walks and complex networks

Here is the link to a youtube video of a talk I gave at the International Institute of Physics (IIP) at UFRN, in Natal, Brazil.  It is one of many talks given by invited lecturers at the school on Physics and Neuroscience, which was held at the IIP during 11 to 17 of August 2014.

This talk touches on the bread and butter of my research activities.  It should be completely or almost completely understandable to anyone at least midway through an undergraduate degree in the sciences. Since the participants in the conference came from diverse backgrounds, I had made a special effort to avoid the use of jargon and to speak in as clear a language as I could. (It is probably the longest talk I have given about my research.)

An explanation about the initial statement regarding elves and hobbits, etc.:  These comments  refer to a running “inside joke” at the school, contrasting the distinct scientific cultures of the participants, for example biologists vs. applied mathematicians and physicists etc.


Fermionization of the 2-D Ising model: The method of Schultz, Mattis and Lieb

F. A da Costa, R. T. G. de Oliveira, G. M. Viswanathan

This blog post was written in co-authorship with my physics department colleague Professor Francisco “Xico” Alexandre da Costa and Professor Roberto Teodoro Gurgel de Oliveira, of the UFRN mathematics department. Xico obtained his doctorate under Professor Sílvio Salinas at the University of São Paulo. Roberto was a student of Xico many years ago, but left physics to study mathematics at IMPA in Rio de Janeiro in 2010. During 2006–2007, Roberto and Xico had written up a short text in Portuguese that included the exact solution of the Ising model on the infinite square lattice using the method of fermion operators developed by Schultz, Mattis and Lieb. With the aim of learning this method, I adapted their text and expanded many of the calculations for my own convenience. I decided to post it on this blog since others might also find it interesting. I have previously written an introduction to the 2-D Ising model here, where I review a combinatorial method of solution.

1. Introduction

The spins in the Ising model can only take on two values, {\pm 1}. This behavior is not unlike how the occupation number {n} for some single particle state for fermions can only take on two values, {n=0,1}. It thus makes sense to try to solve the Ising model via fermionization. This is what Schultz, Mattis and Lieb accomplished in their well-known paper of 1964. In turn, their method of solution is a simplified version of Bruria Kaufman’s spinor analysis method, which is in turn a simplification of Onsager’s original method.

We will proceed as follows. First we will set up the transfer matrix. Next we will reformulate it in terms of Pauli’s spin matrices for spin-{\tfrac 1 2} particles. Recall that in quantum field theory boson creation and annihilation operators satisfy the well-known commutation relations of the quantum harmonic oscillator, whereas fermion operators satisfy analogous anticommutation relations. The spin annihilation and creation operators {\sigma_j^\pm } do not anticommute at distinct sites {j} but instead commute, whereas fermion operators must anticommute at different sites. This problem of mixed commutation and anticommutation relations can be solved using a method known as the Jordan-Wigner transformation. This step completes the fermionic reformulation of the 2-D Ising model. To obtain the partition function in the thermodynamic limit, which is the largest eigenvalue of the transfer matrix, one diagonalizes the fermionized transfer matrix using appropriate canonical transformations.

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Are science and religion compatible?

This blog post explores whether or not science and religion are compatible. I use the term religion in the usual sense, to mean a system of faith, worship and sacred rituals or duties. Religions typically consist of an organized code or collection of beliefs related to the origins and purpose of humanity (or subgroups thereof), together with a set of practices  based on those beliefs. Can such belief systems be compatible with science?

Since this topic is controversial, I only reluctantly decided to write about it.  Being a physics professor and a research scientist, I decided not to flee debate on this issue (which is like the third rail of science). Instead, here I detail my thoughts in writing.

Actually, I spent decades trying to reconcile science and (organized) religion, however I made little or no significant progress. Eventually, after much hesitation and discomfort, I was forced to conclude that full reconciliation between science and organized religion may not be possible, even in principle.   Although this realization was initially surprising (and unpleasant) to me, I soon discovered new and more fulfilling ways of approaching issues such as ethics, morals and the purpose or meaning of life, which religion has traditionally monopolized.

1. Short answer: science and religion are incompatible

`Religion is a culture of faith and science is a culture of doubt.’  This statement is usually attributed to Richard Feynman.   Faith and doubt are indeed antagonistic, like water and fire. How can it be possible to fully reconcile religious views, which are based on faith, with the systematic doubt and the skeptical questioning that are intrinsic to the scientific method? Like many scientists, I too have concluded that full reconciliation of science and religion is not possible.

One caveat: obviously, if one removes the element of dogma and faith from religion, then reconciliation might be possible. But religion without dogma is more like a social club than a traditional religion. What would become of Christianity without faith in Jesus Christ? Can you imagine Islam without faith in the Koran?  So, by religion I always mean organized religion, with a set of teachings or dogmas.

Nevertheless, this caveat actually points to a possible way forward  at reconciliation of science and religion: if religions do become more like social clubs and less dogmatic, then disagreement with science can be minimized or even avoided. I see some movement in this direction. There is growing realization, even among people of faith,  that the arbitrary divisions of race, ethnicity and religion, for example, do not have a clear and well-established scientific foundation. In this context, it is admirable that the  new pope of the Catholic Church,  Pope Francis, has defended interfaith dialogue.  He has said, for example, that even atheists can be redeemed. This concession is a major advance, compared to the old threats about burning in hell in eternal damnation!  Moreover, by claiming that he would baptize even Martians, he has (perhaps inadvertently) signaled an openness to the possibility of extraterrestrial intelligence (i.e., aliens). Similarly, his emphasis on raising awareness of climate change is also most encouraging. These are all welcome developments. Other religions have also responded positively to the challenges brought on by science. The Dalai Lama is good example: a Buddhist religious leader who has shown interest in and kept an open mind about science. He has stated that “…if scientific analysis were conclusively to demonstrate certain claims in Buddhism to be false, then we must accept the findings of science and abandon those claims.” Pope Francis, the Dalai Lama and many others like them have contributed positively towards the reconciliation of science and religion. They are forward-thinking and broad-minded religious leaders. Maybe, in some sense, they have more in common with liberal social and political leaders than with the dogmatic defenders of religious orthodoxy and closed-minded conservatism. So I do see a ray of hope. While I welcome the positive change in attitudes brought by religious leaders like Pope Francis and the Dalai Lama, on the other hand the fact remains that their religions are still based on dogma. Religions still have too much dogma, too much superstition and too much bigotry. So, even considering the above caveat, overall I still feel  generally pessimistic about science and religion being compatible.

Below I explore these issues in some detail.

2. Dirac and Feynman on religion

In the list of the all-time greatest physicists, Newton and Einstein invariably take the top positions. Paul A. M. Dirac, of Dirac equation fame, is considered to be an intellectual giant, ranking just a few notches below Einstein or Newton. And Feynman, who usually ranks just below or comparable to Dirac, has rock star status in the physics community.

Feynman had the following to say about religion:

It doesn’t seem to me that this fantastically marvelous universe, this tremendous range of time and space and different kinds of animals, and all the different planets, and all these atoms with all their motions, and so on, all this complicated thing can merely be a stage so that God can watch human beings struggle for good and evil — which is the view that religion has. The stage is too big for the drama.

Dirac had the following to say about religion:

If we are honest — and scientists have to be — we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can’t for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards — in heaven if not on earth — all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.

I do not accept arguments from authority. But it is nevertheless interesting to read about what these eminent physicists had to say.

3. Scientists abandon God and religion

Most scientists are non-religious. Many are atheist. A Pew survey from 2009 found that while over 80% of Americans believed in God, less than 50% of scientists believed in God. The percentage was actually 33% in that particular survey. These numbers are typical. For instance, among the members of the US National Academy of Sciences, more than 60% of biological scientists had disbelief in God (i.e., were what most people call `atheists’) according to a study from 1998. In the physical sciences, 79% had disbelief in God.

This issue is relevant in society because most politicians and people in leadership positions are, at least outwardly, sympathetic to religion if not actively religious. So there is at least this one important difference between the majority of scientists and the rest of society. Whereas most people are religious, most scientists are non-believers.

More worrying is that many politicians actively campaign against science and science education. We have all heard about attempts by the religious to eliminate (or water down) the teaching of Darwinian evolution in schools. At least in the West, these attempts have largely failed.

Fortunately, the voting population does not particularly crave a return to the dark ages. It  is easy to understand why. The experience of the last few centuries has shown that social and economic development is only possible when there is political support and commitment to science research and education. Science is responsible for the invention of the Internet, cell phones, radio, TV, cars, trains, airplanes, X-rays, MRIs, the eradication of smallpox, etc.   Rich and socially developed countries are precisely those in which science education and research are well funded. Economic pressures have thus led to investment in science and in science education.

At the same time, science has led to unintended consequences. The more a person is exposed to science, the less religious they are likely to become. (Possibly as a consequence, wealth is also negatively correlated with religiosity. In other words, on average the richer you are, the less religious you are likely to be.)

Especially among those with less science education, there is a fear that exposure to science and to “subversive” ideas such as Darwinian evolution will infect the minds of young people and turn them into “Godless infidels.”  In fact, fear is a constant theme in religion: fear of God, fear of divine punishment, fear of hell, fear of forbidden knowledge, etc.  Science education dispels such fears, and replaces it with the cultivation of curiosity, wonder, questioning, doubt and awe. Since fear is often used as an instrument of control and power, the loss of fear can be a setback for the power structures of organized religion. In this sense, science and science education sometimes directly threaten some religious movements.

Consider, as an example, suicide bombing as a form of jihad by Islamic militant organizations. It is perfectly fair for us to ask: is it even remotely plausible that these hapless suicide bombers correctly understood the scientific method? This is a rhetorical question, of course. A genuinely curious and scientifically literate potential candidate for suicide bombing would immediately ask questions, especially when faced with death by suicide. Is life after death a sure thing? Will Allah really reward a suicide bomber? How it is possible for the big-breasted and hot Houri girls and women to recover their sexual virginity every morning? The young man may then go on to ask: is there a remote possibility, perhaps, that such ridiculous claims are not a sign of pulling the wool over the eyes of the naïve young men in their sexual prime who crave sex and intimacy with women, but who are forbidden by religion to engage in casual sex? And why is recreational and free sex allowed in Paradise but not on earth? A scientifically literate young man would probably say `Thanks but no thanks, I’ll let you go first to set the example!’  Hence the fear and loathing of doubt, curiosity and questioning. Indeed, scientific illiteracy makes people gullible and easier to manipulate.

There is no denying the statistics: exposure to science is correlated with loss of religious faith. This raises two questions: (i) why does this happen and (ii) is this good or bad? I am mostly concerned here with question (i) and only briefly touch upon point (ii).

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Onsager’s solution of the 2-D Ising model: The combinatorial method

1. Introduction

In this blog post, I derive, step by step, the exact partition function for the ferromagnetic Ising model on the square lattice. The result is celebrated as “Onsager’s solution” of the 2-D Ising model. It was originally derived by Lars Onsager in 1942 and published in 1944 in Physical Review [1]. That paper revolutionized the study of phase transitions and what we now call critical phenomena [2].

Somewhat ironically, I first heard about the Ising model when I was working in industry. I was 20 and held a summer job at what was then known as British Telecom Research Labs (BTRL), near Ipswich in the UK. This was before I had ever seen a cell phone or heard of the Internet (although I knew about BITNET and JANET). I worked there in the summer of 1990 and again for a month or so around April 1991. My job at BT involved writing C implementations of multilayer perceptrons and Hopfield neural nets. In those days, BT was interested in implementing hardware neural networks and my boss mentioned casually to me that certain kinds of neural nets are basically just special cases of the Ising model. (Indeed, the Hopfield network is closely related to the Ising spin glass.) Thus began my fascination with the Ising model. Later, in 1994 in Boston, I took a course given by Bill Klein at BU on statistical mechanics, where we went through the solution of the 1-D ferromagnetic Ising model. Still, I never had the chance to study properly the 2-D Ising model. As a PhD student, I would almost daily pass by a poster with a background photo of Lars Onsager (with a cigarette in his hand), hung near the office door of my advisor Gene Stanley, so I was regularly reminded of the 2-D Ising model. I kept telling myself that one day I would eventually learn how Onsager managed to do what seemed to me, at the time, an “impossible calculation.” That was 1994 and I am writing this in 2015!

In what follows, I solve the Ising model on the infinite square lattice, but I do not actually follow Onsager’s original argument. There are in fact several different ways of arriving at Onsager’s expression [3–9]. The method I use below is known as the combinatorial method and was developed by van der Waerden, Kac and Ward among others and relies essentially on counting certain kinds of closed graphs (see refs. [3,10–13]). I more or less follow Feynman [3] and I have also relied on the initial portions of ref. [13].

2. The 2-D Ising model

Consider a two dimensional lattice {\Bbb Z^2} where at each point of the lattice is located a (somewhat idealized) spin-{\tfrac{1}{2}} particle. Consider a finite subset of this lattice, of size {L\times L} and let {i,j=1,2,\ldots L^2}. Let {\cal N} denote the set of pairs of integers {(i,j)} such that spins {\sigma_i} and {\sigma_j} are nearest neighbors. In the ferromagnetic 2-D Ising model with nearest neighbor interactions, spins {i} and {j} interact if and only if {(i,j)\in \cal N}. Each spin {\sigma_i} can assume only 2 values: {\sigma=\pm 1}.

Consider a system of {N=L^2} spins. The Hamiltonian for a spin configuration {\sigma=(\sigma_1,\sigma_2,\ldots \sigma_N)} is given by

\displaystyle H(\sigma)= -J \displaystyle\sum_{i,j\in \cal N} \sigma_i \sigma_j ~. \ \ \ \ \ (1)

The sum over the nearest neighbors should avoid double counting, so that {(i,j)} and {(j,i)} are not counted separately. Without loss of generality, we will assume {J=1} for simplicity.

3. The canonical partition function

In the theory of equilibrium statistical mechanics, the canonical partition function contains all the information needed to recover the thermodynamic properties of a system with fixed number of particles, immersed in a heat bath, details of which can be found in any textbook on statistical mechanics [3-6,14].

I prefer to define the partition function as the two-sided Laplace transform of the degeneracy {\Omega(E)} of the energy level {E}. But traditionally, the partition function is defined as a sum or integral over all possible states of the system:

\displaystyle Z(\beta) = \displaystyle\sum _{\sigma} e^{-\beta H(\sigma)} ~. \ \ \ \ \ (2)

The two ways of thinking are equivalent. The Laplace transform variable {\beta} is related to the thermodynamic temperature {T} via {\beta =1/k_B T}, where {k_B} is the Boltzmann constant. What follows is the exact calculation of {Z(\beta)}.

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Domain coloring for visualizing complex functions



The above figure shows a domain color plot for the function {z\mapsto {1/(1 - i z )} -(1 + i z)} .

I have been relearning complex analysis and decided to have some fun plotting complex functions. It is easy to visualize graphically a real function {f:\Bbb R\rightarrow \Bbb R} of a real variable. To visualize {f(x)}, we can plot {x} on the horizontal axis and {y=f(x)} on the vertical axis, as we learn in high school mathematics. It is more difficult, however, to visualize the function {g:\Bbb C\rightarrow \Bbb C}, because {g}  is a  complex function of a complex variable. A complex number {z} can be expressed in terms of real numbers x, ~y as {z=x+ i y}. To visualize the complex domain of a function, one thus requires 2 real dimensions. The complex function {g(z)} must then be plotted on the remaining dimensions. If we lived in 4 dimensional space, it would be easy to plot a complex function of a complex variable. The problem is that we live in three dimensions, not four. Domain coloring is a method to overcome this limitation. The basic idea is to use colors and shades etc. as extra dimensions for visualizing functions.

I first heard about domain coloring when I came across Hans Lundmark’s complex analysis pages. For readers unfamiliar with domain coloring, I recommend reading up on it first. There is also a wikipedia article on the subject.


The pictures below were generated using Wolfram’s Mathematica software (many thanks to my department colleague Professor Marcio Assolin). I adapted the code from the discussion on stackexchange here, and also here.

The first figure below shows the identity function {f(z)=z}:


The horizontal axis is the real axis and the vertical the imaginary axis of the domain {z}. The 29×29 grid is not deformed in this case because the identity function {f(z)=z} does “nothing.” But I decided to include this example to illustrate how the colors and shades code information. First of all, notice that as we go around the origin {z=0}, the colors go from red on the positive real line, to green on the positive imaginary axis, to cyan on the negative real axis, to purple on the negative imaginary axis and then back to red again.

The color represents the argument of the function. In this case, since we can write {z=r e^{i\theta}} the argument {\theta} wraps back around every time we turn {2\pi} radians around the origin in the complex plane.

Note that in addition to color, there is shading. Notice that at {|z|=1/2}, {|z|=1} and {|z|=2} there are discontinuities in shading. As we increase the absolute value of {z}, the shading gets darker and discontinuously gets brighter and this process repeats itself. Indeed, the shading has been used to plot the absolute value of the function. Every time the absolute value doubles, the shading goes through one cycle, becoming discontinuously brighter.


Let us now look at more complicated functions. The pictures below show the functions {z\mapsto z^2} and {z\mapsto z^3}.

domain-coloring-z2 domain-coloring-z3

Now, things look more interesting! The first thing to note is that the 29×29 grid is distorted. The grid shown is the inverse image of the grid of the identity map. So for the function defined by {f(z)=z^2}, the procedure is as follows. Take the points of the grid and put them into some set {S}. Then calculate the inverse image {f^{-1}(S)}, i.e. the set of points that map to {S}. This inverse image is the grid shown above.

Notice how the distorted grid seems to preserve right angles, except at the origin. Indeed, holomorphic functions are typically also conformal (i.e., angle preserving) in many instances, and we will return to this topic further below. At the origin, the monomial functions above are clearly not conformal.

Take a look at the colors. Instead of cyclying through the rainbow colors going once around the origin, the colors cycle 2 and 3 times, respectively, for {z^2} and {z^3}. This is easy to understand if we write {z=re^{i \theta}} (for real {r} and {\theta}), so that {z^2=r^2 e^{2i\theta}} and {z^3=r^3 e^{3i\theta}}. So {z^2} and {z^3} circle the origin 2 and 3 times every time {z} goes around the origin once.

Finally, notice that the shading cycles through more quickly. Indeed, {|z^2|} and {|z^3|} double more quickly than {|z|}.


The monomial functions above have zeroes, but no poles. What do poles look like? The image below shows the function {z\mapsto 1/z}.


Notice how the colors cycle around “backwards.” Indeed, the argument of {z} and {z^{-1}} are negative of each other. Notice also how the shading now has discontinuities in the “opposite” sense (compare with {z\mapsto z} in the first image above).

It is also worthwhile to look at the grid. The original grid of lines has transformed into a patchwork of circles. To show this more clearly, I modifed the image to show only a few grid lines, corresponding to real and imaginary lines at {\pm 1,2,3,4,5}:


In the image above the grid lines are now clearly seen to have mapped into circles. In higher order poles, the colors cycle around (backwards) a number equal to the order of the pole. Here are poles of order 2 and 3, shown with only some grid lines for greater clarity:



In addition to poles, there are other kinds of singularities. Removable singularities are not too interesting, because basically a “point” is missing. If we “manually add” the point, the singularity is “removed” — hence the name.

In addition to removable singularities and poles, there are also what are known as essential singularities. Essential singularities can be thought of, loosely speaking, as poles of infinite order. Further below, we will take a look at essential singularities.

Now consider a function with a zero as well as a pole:


The function shown above is {z\mapsto (z-1)+ 1/(z+1)}, which has a zero at {z=0} and simple pole at {z=-1}.

Non-holomorphic functions

Having looked at examples of holomorphic and meromorphic functions, let us look at more complicated non-holomorphic function. The figure below shows {|z|}, the absolute value of {z}.


The color is red because {|z|} is always non-negative real. The grid is gone and we have circles instead: we do not have conformality. The Cauchy-Riemann equations are impossible to satisfy because {|z|} is always real, the imaginary part being identically zero everywhere.

The complex conjugate function {z\mapsto \bar z} also is not analytic. Here is {z\mapsto \bar z}:


Notice that it looks just like the identity map, but reflected along the real axis. The imaginary axis is “backwards”.  Still, angles are preserved, so why is this function not holomorphic? The answer is that the angle orientations are reversed, i.e. the function is antiholomorphic rather than holomorphic. The colors cycle around “backwards”  in this case because the complex conjugate of  r e^{i \theta} is r e ^{- i \theta}.  Conformal maps preserve oriented angles, rather than just angles. Indeed, the complex conjugate function is neither holomorphic nor conformal.

The exponential function and its Taylor polynomials

Let us now look at a transcendental holomorphic function: the exponential function {z\mapsto \exp(z)}. The images below show the exponential function at two scales.

domain-coloring-exp-z-large domain-coloring-exp-z

Notice that the colors now cycle through going up and down vertically. The reason for this is as follows. If we write {z=x + i y }, then

\exp[z]=e^{x+i y } = e^x e^{i y} ~.\ \ \ \ \

So {y}, which is the imaginary part of {z}, determines the argument, hence the color. The zoomed out version makes it clear that the argument is periodic with period {2 \pi} in the imaginary direction.

While on the topic of the exponential function, let us take a look at Talyor polynomial expansions. The figure below shows the Talor polynomial of degree 5.


Notice the 5 zeroes, which lie on an arc like the letter “C” slightly to the left of the origin. The exponential function does not have zeroes, of course. We know that if we take the infinite degree Talyor polynomial, i.e. the infinite Taylor series expansion, then we recover the exponential function. We can already see that for positive real part and small imaginary part of {z}, the Taylor polymial above is starting to behave qualitatively like the exponential function.

Essential singularities

Having seen the exponential function, we can now look at essential singularities. Observe that the Laurent expansion of {\exp(1/z)} around the origin in {z} has an infinite number of terms of negative power in {z}. The singularity at the origin is thus stronger than a pole of any finite order. The figure below shows {z\mapsto \exp(1/z)}, shown at three different scales.  The third figure is a zoom of the second, which is a zoom of the first.


The software is apparently having some trouble near the origin, in the last figure! The reason for this is the Great Picard’s theorem, which says, loosely speaking, that an analytic function near an essential singularity takes all possible complex values, with at most 1 exception. In the case of {z\mapsto \exp(1/z)}, the function cannot become zero, which is the exceptional value. As we approach the origin, the argument (i.e. color) changes, cycles around, etc., increasingly quickly.

Let us now look at some trigonometric functions:


domain-coloring-sin-z-small domain-coloring-sin-z-large


domain-coloring-tan-z-small domain-coloring-tan-z-large

We can clearly see the zeroes in {\sin(z)} and the poles and zeroes of {\tan(z)}. Moreover, it is clear that these are periodic functions.

Compare the above trigonometric functions with their inverses:





Notice that on the real line, for {|z|>1} there is a discontinuity in color for arcsine. Similarly, for arctan there is a color discontinuity on the imaginary axis. To understand this jump in color, recall that {|\sin(x)|\leq 1}, which means that the inverse function {\sin^{-1}(x)=\arcsin(x) } is not defined for {x} outside the interval { -1\leq x\leq 1}. Recall also that {\sin(x)=\sin(x+2\pi)}, so that the inverse function {\sin^{-1}} must be multivalued. What is being shown above is the principal branch.

It helps to switch over to the logarithmic form. Recall that

\displaystyle \sin(z)= {e^{iz} - e^{-iz} \over 2i } ~. \ \ \ \ \

If we write {z=\arcsin(w)}, then {\sin(z)=w}. Substituting, we get

\displaystyle 2iw ={e^{iz} - e^{-iz} }~. \ \ \ \ \

To simplify the algebra, let {e^{iz}=Z}, so that

\displaystyle 2iw = {Z - Z^{-1} } ~, \ \ \ \ \

which gives us a quadratic equation:

\displaystyle Z^2 -2iwz -1 =0~, \ \ \ \ \

whose roots are

\displaystyle Z= iw \pm \sqrt{1-w^2} ~. \ \ \ \ \

So we finally get

\displaystyle z=\arcsin(w)=-i \log Z= -i \log\left( iw \pm \sqrt{1-w^2} \right) ~. \ \ \ \ \

So the branch cut in the {\arcsin} function is due to the 2 possible values of the square root. There is a branch point at {w=\pm 1} and there is actually another branch point at infinity.

Branch points and cuts

Let us look at branch points more closely. As we know, the square root is multivalued, and the figure below shows the two branches, with the principal branch at the bottom.


Note how in each branch alone the colors do not cycle all the way through the rainbow colors. The missing colors of one branch are on the other branch. To see both branches, one would need to visualize the Riemann surface for the square root, a topic beyong what I wish to cover here.

The figure below shows the 3 branches of the cube root function, with the principal branch at the top:


There is more than 1 type of branch point. Algebraic branch points are those that arise from taking square roots, cubic roots, and {n}-th roots (for positive integer {n}). In general there will be {n} well defined branches.

What happens if one takes {n} to be a positive irrational number? Here is a hint:

z^{\alpha} = e^{\alpha \ln z} ~.

If we choose {\alpha} rational, say {\alpha=p/q}, then putting {z=r e^{i\theta + 2n\pi i}} we get {\ln z= \ln r + i\theta + 2n\pi i}, so that

\displaystyle z^\alpha = r^{p/q} e^{i \alpha \theta} e^{ 2\pi i (np /q)} ~.

But {n p/q} is rational, and so there can be at most {q} branches. But if {\alpha} is irrational, this argument does not work. Instead of {np/q} we get {n\alpha}, which can never equal an integer. So the branches never cycle through and the number of branches is infinite. Indeed, the logarithm above leads to a infinite branching, as we will see below.

There is a more complicated type of branch point in the function {z\mapsto \exp(1/z^{1/n})} for integer {n}. If we loop around the origin {n} times, the function returns to the original starting point (i.e. there is finite monodromy). However, there is an essential singularity at the origin. In other words, there is the unhappy coincidence of the algebraic branch point of {1/z^{1/n}} sitting exactly on top of an essential singularity. Such branch points are known as transcendental branch points.


The figure above shows {z\mapsto \exp(1/z^{1/2} )}. The essential singularity coincides with the branch point.

Finally, as we saw above, there are branch points where the number of branches is infinite. Consider the complex logarithm. If we again write {z=r e^{i \theta}} then {\log z= \log r + i \theta}. Since {\theta} and {\theta \pm 2\pi} give same value of {z}, the logarithm is multivalued. The branch cut is usually taken at {\theta=\pm \pi}. Here is the logarithm:


The zero at {z=1} is clearly visible, as is the branch cut along the negative real axis.

The complex logarithm, like other multivalued functions, could instead be visualized as Riemann surfaces. Here is an example of the Riemann surface of the complex logarithm.

Euler’s reflection formula

Let us next consider another topic related to Weierstrass’s beautiful factorization theorem, which states that every entire function can be expressed as an infinite product. Among the best known infinite products are the one used by Euler to solve the Basel problem, and the product formula for Euler’s gamma function. Indeed, the gamma function is, in a sense, one half of the sine function (or cosecant). Put differently, the sine function is the product of two gamma functions:

\Gamma(z)\Gamma(1-z) = {\displaystyle \pi \over \displaystyle \sin (\pi z) } .

Proofs can be found in textbooks. Here I wish to focus on the zeroes and poles. Take a look at these figures:


The figures above show {1/\Gamma(z)} and {1/ [\Gamma(z)\Gamma(1-z) ]}. Note the zeroes at the non-positive integers in the first image, correponding to the poles of the gamma function. If we multiply two gamma functions, so that there are zeroes at all integers, we basically get {\sin(2\pi z)} upto a constant! Indeed, the last figure above is identical, up to scale, to that of the sine function seen earlier.

When are holomorphic functions conformal?

Finally, let us take a closer look at conformality, i.e. the angle-ṕreserving property found in many holomorphic functions. The examples above of monomials of degree greater than 1 and poles shows clearly that conformality can break down at zeroes and poles.

Consider again the function {z \mapsto z^2} below:


Conformality indeed breaks down at the zero at the origin, as expected.

But if a function is holomorphic with no zeroes in a region, is it necessarily conformal in that region? The answer is NO, as seen from the following counter-example:

{z \mapsto z^2+1}


There is no zero at the origin, yet conformality breaks down!

To understand why, recall that to preserve angles, the map must locally be a scaled rotation (upto a translation). So the Jacobian determinant of the conformal map must be some positive constant. It is easy to show that the Cauchy-Riemman equations lead to a scaled rotation, provided the derivative is not zero. If the derivative is zero, however, the function need no longer be a scaled rotation, so angles need not be preserved.

In the example above, the function is holomorphic at the origin, but the derivative is zero at the origin, in other words, the origin is a critical point. Moreover, as {z} moves around the origin once, the function {z^2+1} moves around 1 twice, hence the breakdown in conformality.

Conversely, if a holomorphic function has a critical point at the origin, then its Taylor series does not contain a term of degree 1. But all higher order monomials {z^n} break the conformal property at the origin, so the function cannot be conformal at the critical point. By translating arbitrary functions such that a critical point is on the origin, we can understand the following well known result:

A holomorphic function is conformal if and only if there are no critical points in the region of interest.


[1]. Below is the Mathematica code I used for the plot of the identity map. The code has been adapted from the discussion on stackexchange here and also here.

f[z_] := z;

paint[z_] :=
Module[{x = Re[z], y = Im[z]},
color = Hue[Rescale[ArcTan[-x, -y], {-Pi, Pi}]];
shade = Mod[Log[2, Abs[x + I y]], 1];
Darker[color, shade/4]];

ParametricPlot[{x, y}, {x, -3, 3}, {y, -3, 3},
ColorFunctionScaling -> False,
ColorFunction -> Function[{x, y}, paint[f[x + y I]]], Frame -> True,
MaxRecursion -> 1, PlotPoints -> 300, PlotRangePadding -> 0,
Axes -> False , Mesh -> 29,
MeshFunctions -> {(Re@f[#1 + I #2] &), (Im@f[#1 + I #2] &)},
PlotRangePadding -> 0, MeshStyle -> Opacity[0.3], ImageSize -> 400]