Marco has worked hard to create this superb introduction to complex analysis which is one of the most beautiful and useful branches of mathematics.

I strongly recommend this study group to everyone!

About applications of the Riemann sphere and the stereographic projection, others in this forum have mentioned cartography, but that does not require complex numbers. The only "practical" application of the Riemann sphere I know of is to Lorentz transformations. The situation is this: imagine two observers moving at very large relative velocities. By a relativistic effect, the two observers will see the stars in different positions. How to relate these positions? In this problem it can actually be useful to think of the sky as a Riemann sphere for the purpose of calculation. Then, the position of a star will be given by a complex number and the problem becomes how to relate the position of the same star as seen by the two observers, or to relate the two complex numbers. More precisely, the Lorentz transformations of the "celestial sphere" can be described by conformal transformations of the Riemann sphere. So the theorems about the stereographic projection (circles mapped to circles etc.) become useful to derive properties about Lorentz transformations. For example, a perfectly circular constellation should remain circular as seen by the second observer. Anyway I am no expert, so I refer you to Chapter 1 of Vol. 1 of "Spinors and Spacetime" by Penrose and Rindler, or perhaps to Chapter 18 of "The road to reality" by Penrose (and I apologize for any wrong statements).

I tried to describe the only application I know of in a reply to Joseph Smith, above.

In Lesson 13, in the proof of Cauchy's Theorem, I think there is a sign error in the first (left) integrand. I think partial of u with respect to y (du/dy) should have a negative sign in accordance with Green's Theorem. Then, the Cauchy-Riemann equations yield that the first integrand is 0. Is that the case?

Perhaps the contributed function RiemannSphereComplexPlot[] -- a 3D rotatable Riemann sphere version of ComplexPlot -- is what you are seeking: https://resources.wolframcloud.com/FunctionRepository/resources/RiemannSphereComplexPlot/

Hi,

I'm not exactly sure what you want, but here are formulas for the various elements of the projection. You can put them together in graphics as you see fit:

(* elements *)
pole = {0, 0, 2};
numberPt = {Re[z], Im[z], 0};
projectionPt = { (* main formula: proj. onto R. sphere *)
  (4 Re[z])/(4 + Abs[z]^2), 
  (4 Im[z])/(4 + Abs[z]^2), 
  (2 Abs[z]^2)/(4 + Abs[z]^2)};
riemann = Sphere[{0, 0, 1}, 1];
plane = InfinitePlane[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}];

 z = 2 - I; (* need to give z a value *)
 Graphics3D[{
   Opacity[0.6],(* change style as desired *)
   Point[{pole, numberPt, projectionPt}],
   Line[{pole, numberPt}],
   riemann,
   plane},
  PlotRange -> {{-3, 3}, {-3, 3}, {0, 2}},
  FaceGrids -> { (* may omit *)
    {{0, 0, -1}, {Range[-3, 3], Range[-3, 3]}}}
  ]

Changing FaceGrids to the following will highlight the real and imaginary axes:

FaceGrids -> { (* may omit *)
  {{0, 0, -1},
   {Range[-3, 3], Range[-3, 3]} /.
     0 -> {0, AbsoluteThickness[1]}}}

Omitting FaceGrids gives a plain plane below the sphere. Omitting plane and using FaceGrids gives a nice look, too.

Please look at the 3rd question in Quiz 1. The question asks whether a given point is stereographically projected into the lower or the upper hemisphere of the Riemann sphere. In fact, the magnitude of the point is exactly 1, so that the given point lies on the equator. But that's not an available answer.

So, what's the right response?

I find the projection of the Reimann sphere is extremely distorted. I prefer that both sides of the sphere be projected on the inner disk separately so that the two projections have symmetry.

So, I asked this question during the lecture and Marco did not have an answer for it. I had never heard of the stereographic projection of complex numbers. I was wondering if anybody knows of a practical application for it. It seemed to me that it might be used in modulation/coding theory or maybe image recognition problems but I'm just shooting in the dark. Anybody know?

Attached is an updated list of the References provided, with a link to the publisher’s page first and then a link to the corresponding Amazon page.

Attachments:

Updated References.pdf

Lesson 2 The Complex Plane

There might be an error in the proof for the formulas of the stereographic projection: "... And because the triangles (0,z,N) and (z',z,Overscript[z, ^]) are similar, then |z'|/|z|=1/(1-Z)...."

correct: |z|/|z'|=1/(1-Z) <--- This formula was used for the rest of the proof. The formulas itself are valid.

Quiz 1 Question 3: Is the stereo graphic representation of z1=1/2+i*(3^1/2)/2 in the northern (upper?) or southern (lower?) hemisphere? The text says: "Numbers with absolute value less than 1 are mapped to the southern hemisphere, and 0 to the south pole. Numbers with absolute value greater than 1 are mapped to the northern hemisphere."

Abs[z1] = 1, but the answer "in the upper hemisphere" comes back wrong. Can you explain?

You are right, I'll have the exercise fixed. Thanks

Although I have encountered and used complex numbers over the decades, I never understood what motivated their introduction until your first lecture. Thanks.

I am wondering about the history of complex numbers, Girolamo and Rafel, in 16C. How did they find the need, and what profession both were in? Did mathematician careers exist in 16C?

Hi Michael, What references (texts) would you recommend for studying the topics "(differential) topology" and integrals of the form (2) that you referred to?

Trying to catch up here. Surely this is a simple question. When plotting Re[z] and Im[z], shouldn't these plots be confined to either the {z, Re[z]} or the {z,Im[z]} planes? I don't see why these plots appear to be surfaces in 3 dimensions.

Thanks!

The idea behind that plot is to look at the function f(z)=z^2 as a complex function of the complex variable z. The horizontal plane is the z plane, so if you write z = x + I * y, the x axis corresponds to Re[z] and the y axis to Im[z]. Remember that Re[f(z)]=Re[z^2] is real-valued. So, it will be a surface in the plot, constructed as follows: the height of the surface at coordinates (x,y) is Re[(x+I*y)^2]. Don't confuse the meaning of z (a complex number) with the vertical axis of the plot. For example, try calculating the height of the yellow curve at the point (1,2) which corresponds to z=(1+2* I) . It will be Re[(1+2* I)^2] = -3.

In Lesson 3, why does the Postfix instruction //Labeled[#,Text[ ...]& result in just one Text for the entire row rather than a separate text for each of the 3 ComplexPlot3D[...] entries in the Row[...] for the following code?

Row[{
ComplexPlot3D[z,{z,3},PlotRange->{{-3,3},{-3,3},{0,6}},BoxRatios->{1,1,1},PlotStyle->Directive[Opacity[0.8`]],AxesLabel->{Style[ToString[Re[z],TraditionalForm],12],Style[ToString[Im[z],TraditionalForm],12],Style[ToString[Abs[z],TraditionalForm],12]},ImageSize->160],
ComplexPlot3D[Conjugate[z],{z,3},PlotRange->{{-3,3},{-3,3},{0,6}},BoxRatios->{1,1,1},PlotStyle->Directive[Opacity[0.8`]],AxesLabel->{Style[ToString[Re[z],TraditionalForm],12],Style[ToString[Im[z],TraditionalForm],12],Style[ToString[Abs[Overscript[z, _]],TraditionalForm],12]},ImageSize->160],
ComplexPlot3D[1/z,{z,3},PlotRange->{{-3,3},{-3,3},{0,6}},BoxRatios->{1,1,1},PlotStyle->Directive[Opacity[0.8`]],AxesLabel->{Style[ToString[Re[z],TraditionalForm],12],Style[ToString[Im[z],TraditionalForm],12],Style[ToString[HoldForm[Abs[1/z]],TraditionalForm],12]},ImageSize->160,PlotLegends->Automatic]
}]//Labeled[#,Text[StringJoin["Fig. 4. Plots of the functions ",ToString[z,TraditionalForm],", ",ToString[Overscript[z, _],TraditionalForm],", ",ToString[1/z,TraditionalForm],"."]]]&

I think I see the answer to my question. The Postfix code is just a function; no Map (/@) is specified.

Thxs, Marco.

I am wondering how you got into complex number analysis. What trigger made you enter this field?

Hi Gerald,

I learned these things back in the 1980s, some facts as an undergraduate but mainly in graduate school from lectures. The professors sometimes recommended books as ancillary materials, but only a few used books as textbooks and followed them. Further, I've been at an institution teaching only 1st and 2nd year students for 30 years, so I haven't taught this or related material for a long time. Nonetheless, I can give you some recommendations, but I personally have not field-tested them, so to speak.

Integrals of the form (2) first showed up for me as an undergraduate in physics. Integrals with the square of 3rd and 4th degree polynomials are called elliptic integrals. The arc length of an ellipse can be expressed with such an integral, hence the name, and they first came to the attention of mathematicians in the mid 18th century. They were studied somewhat intensely in the early 19th century by Gauss, Jacobi, Legendre, and in particular by Abel, who extended the scope to include integrals involving square roots of higher degree polynomials. These are called hyperelliptic or "Abelian* integrals. Much knowledge of them was developed before the Riemann sphere or topology was even invented. These facts give you some keywords to google, but they also warn you that you will find a lot of information about (2) not relating to my remarks above.

I came to know more about elliptic integrals from a sideways direction. I was studying number theory and algebraic geometry in grad school; in particular, "elliptic curves" was a hot topic. They come from the square-roots in the elliptic integrals, that is, curves of the form $y^2 = \text{cubic or quartic}$. In the 1980s, they were thought to hold a key to proving Fermat's Last Theorem (which turned out to be the case). They also yielded a powerful factorization algorithm, which was important because of the then relatively new RSA encryption algorithm. It's truly amazing the connections the integral (2) has to seemingly unrelated branches of mathematics.

It turns out there is a book that "does it all", or almost:

Elliptic Curves: Function Theory, Geometry, Arithmetic by McKean and Moll

The book was recommended to me by a friend, and on that basis, I am recommending it here. It gives an explicit construction of the differential manifold structure on the Riemann sphere, which is fairly simple, considering. It also gives a definition of manifold. So it covers everything that is necessary. However, it is question in my mind whether, having given someone the needed tool, they will know why, when, and how to use it. But it's less likely to be a problem than the general case, because the goal is to understand this manifold, namely the Riemann sphere, not manifolds in general. That aside, it covers elliptic integrals, elliptic functions, elliptic curves (though not the connections to Fermat's Last Theorem and factorization), and a few other things.

For differential topology, it's a bit harder to suggest a book. Many assume the reader is familiar with topology, especially manifolds. General introductions to topology include manifolds and much else that isn't strictly necessary for the question at hand. The following was my undergraduate textbook. I remember liking it and using it as a reference in grad school from time to time. And it seems to have the right scope, at least through, say, Chapter 5, which covers differentiable manifolds:

Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry.

This Math.SE Q&A contains some good recommendations:

https://math.stackexchange.com/questions/46482/introductory-texts-on-manifolds

These two have a good reputation:

• L. Tu, An Introduction to Manifolds (includes differentiable a.k.a. smooth manifolds†)
• John M. Lee, Introduction to Smooth Manifolds

†Note: Usually there is a slight difference between these two; however, every smooth manifold is a differentiable manifold.

I hope that helps.

I studied physics at the university and complex analysis was a required course.

You are correct, I'll have it fixed. Thank you!

Hi;

Is there a straight-Forward method of producing a Stereographic Projection of a complex number? The examples given in our discussion materials seems extremely convoluted with several complex plots joined together with possibly some primitives added in.

Thanks,

Mitch Sandlin

This is not a specific answer to your question...but

The Schaum's Outline text gives some physical examples relating to fluid flow in the chapter on Conformal Transformations.

Michael, I've written some code that enhances the ComplexPlot3D display of the function

f[z_] := ((z + Conjugate[z])^2 (z - 
      Conjugate[z]))/((z - Conjugate[z])^2 - (z + Conjugate[z])^4)

The code is in the attached Notebook, file ComplexLimitCounterExample_16nov24.nb .

From the plots, it appears that the values of function f have only two arguments (angles with the real axis) and that they are determined by whether Im[z} is positive or negative. I have not analyzed f to confirm this. Am I correct? Do you see a simple analysis to determine this?

BTW, "Calculus Volume II" by Tom M. Apostol, Copyright 1962 by Blaisdell Publishing Company has on page 70 a simple real-valued function of two real variables that is not continuous at 0 even though approaching 0 along either axis has limit 0. However, when approaching 0 along the line x = y, its limit is 1/2. Here is the function:
f(x,y) = x*y / (x^2 + y^2) if (x,y) != (0,0) and f(0,0) = 0.

Attachments:

ComplexLimitCoun...nb

I am reading the book. The legend for chapter 3 Fig.9 in the book says “Fig. 9 Vector Plot of z2+1=z2”. This puzzled me, so I checked notebook 3, and it says “Fig. 9 Vector Plot of f(z)=z2”, which I think is correct.

What happens if one keeps moving the sphere down the y axis? At some point the north pole would overlap the origin and infinity becomes 0 and then nothingness, yes?

The same problem comes up in the following figure 10

Thank you, This is nice and will, I hope, give me some basis for experimentation. I would like to define my own shapes on the sphere and see the mapping not the plane.

Michael, I took a look at the function f(z)=(z−z∗)(z∗+z)^2 / ( (z−z∗)^2 − (z∗+z)^4 ). I assume that you intend z* to denote the complex conjugate of z. f(z) is a real valued function because ( z - z* ) = 2 * Im(z) and ( z* + z ) = 2 * Re(z). For that reason, I don't understand why i (the square root of -1) appears in the value of the function f along either the straight line ( f((a+ib)t) ) or the parabola ( f(ta+ibt^2) ).

On a side note, if a real valued function of a complex variable has a derivative at any point, that derivative must be zero. So, if it has a derivative everywhere in an open set (region), the function must be a constant. To see that, if we approach a complex point z0 along a vertical line, then the derivative is the limit as h approaches 0 of ( f(a + i h) - f(a) ) / ( i h ). The numerator is real and the denominator is complex but f is real. So the derivative must be zero. A slightly more detailed description of this is provided on page 23 of "Complex Analysis" second edition by Lars V. Ahlfors, copyright McGraw-Hill, 1966.

I suppose if we choose to use lines instead of rays for projection then after the sphere moved below the complex plane the projection will be an inverse image of the one from above?

Marco, My mistake. Thanks for the correction.

Hi;

In working with the properties of Harmonic Functions, how is the mean value calculated when you are given the formula of (x,y) along with the radius and center of a circle? I have gone through the course materials up to include unit 18 and have not found anything to help me set-up the problem. Thanks, Mitch Sandlin

You are right, I'll have it fixed. Thank you

If f(z)=2+2z+ Sum(z^n), where n = 2,3, ..., Infinity, the output of SeriesCoefficient[f[z],{z,0,n}] is:

1 n>1

2 n==0||n==1

0 True

What does "0 True" mean?

It refers to all other possibilities for n, so n<0. So there are no negative-power terms in the series.

In Example 5.3, shouldn't the text read: "In other words, the branch cut corresponds to either z purely imaginary or to z real between -1 and 1."

You are right, much appreciated. I'll get it fixed.

Are branch cuts in a unique location? In Lesson 5 exercise 3 we are asked to "describe the branch cuts" of the function Sqrt[(e^2+1)]. Given the definition from Wolfram Mathworld "A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments." The solution states:

I can see from the figure that there are discontinuities for any line parallel to the Im axis for x>0. Is the branch cut where these discontinuities start or can the branch cut be anywhere x>0?

You are right... thanks again... I'll get it fixed

Question on notation in Section 12. Should the equation read the contour integral over C1 equals the contour integral over C2 (it says it equals the contour integral over negative C2)? I believe that C2 is defined as being in the clockwise direction, so the negative sign will naturally arise when the integral is done in the counterclockwise direction.

I intended C2 to be defined clockwise in that specific figure (because I drew the arrow that way) So the integral over (C1+C2) is zero by Cauchy. So int_C1 = - int_C2 or int_C1 = int_(-C2) . So I don't think there are typos there. In this example it is (-C2) which is counterclockwise. An exception to our regular convention since normally one always takes paths to be counterclockwise. But it is marked by an arrow so I hope it's clear.

Hi Michael, Thanks for the simple and clear analysis that shows that the only 2 arguments of f(z) are Pi/2 and -Pi/2, where

f[z_] := ((z + Conjugate[z])^2 (z - Conjugate[z]))/((z - Conjugate[z])^2 - (z + Conjugate[z])^4)

I had not seen previously the Option TargetFunctions that you used to quickly resolve my question in the instruction

ComplexExpand[f[z], z, TargetFunctions -> {Re, Im}] // Simplify