Complex analysis: Harmonic functions

[@ericvosselmans5657]

This is pure gold

[@staj6236]

Excellent lecture

[@omarhayat6016]

thank you for these lectures on complex analysis! Merry Christmas!

[@thomasjefferson6225]

Not gonna lie, I'm enjoying this dudes lectures

[@robertfulton6397]

I'm having trouble understanding why the space of homogeneous harmonic polynomials of degree d > 0 is two-dimensional (5:13). Can anybody explain why this is?

[@deljohnson3264]

What is the dimension of the space of all monomials of degree d in x and y? The Laplacian sends this space to the space of monomials of degree d-2. This map is linear. Use some Linear Algebra reasoning to get the dimension of the kernel.

[@robertfulton6397]

thank you

[@hyperduality2838]

Two dimensions implies duality! Mappings from domains to co-domains are dual. Complex numbers are dual to real numbers (in physics). Homology is dual to co-homology. Integration is dual to differentiation. Photons or light, the electro-magnetic field are dual. Electro is dual to magnetic -- Maxwell's equations. The wave equation in physics which measures probability contains "i" and this imposes a conservation of duality on the physics and mathematics. Space is dual to time -- Einstein. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Questions are dual to answers. The answer to your question is duality (physics). "Always two there are" -- Yoda. The conservation of duality (energy) will be known as the 5th law of thermodynamics, energy is duality, duality is energy.

[@analander9222]

It's just the real and imaginary part of (x+iy)ⁿ

[@hasirama]

@Robert For checking the dimension of the harmonic polynomials, this is what I got. let (x+iy)^n=x_n+iy_n, and represent a polynomial of degree d as f(z)=X+iY where X and Y are linear combinations of {x_0,...,x_d} and {y_0,...,y_d}. Now, if X=r_0*x_0+r_1*x_1+r_2*x_2+... and Y=s_0y_0+s_1y_1+s_2y_2+..., after double differentiation and using the harmonic condition that Delta(X+iY)=0, if we consider x and y such that they are not roots of the respecitve polynomials, we see that r_2,... and s_2,... have to be 0 for the condition to hold. So in the basis, only r_0,r_1 and s_0,s_1 remain for X and Y respectively which makes the dimension 2. Hope I did not make any mistake

[@martinsoderstrom449]

Fantastic lecture and series.

[@lilbthebasegod8606]

25:18 I think the compactness of [0, 1] is necessary to guarantee that v is constant.. is it right?

[@PaPa-kr5yt]

Not really. You can show the set of reachable points in U from the origin by chains of finitely many disks is clopen. So the connectedness is enough.

[@lefximusic4962]

While watching this video, I have several problems going through the lecture First I don’t really know the exact definition of all these connected sets, which professor have been mentioning quite regularly And during the explanation of why we cannot extend u to a holomorphic functions in a region with a hole Can someone explain these to me? With full appreciation

[@lefximusic4962]

@Peter J. i'd taken real analysis courses so i know what connected set means in real analysis, but im not sure if it's with same definition in this lecture

[@dannycrytser7268]

Your second question is a bit too vague, but I'll try to give an idea of what Borcherds means when he refers to connected sets. One formal definition of a connected subset X of the complex plane: if you write X as the disjoint union of non-empty subsets A and B, then either A contains a limit point of B, or B contains a limit point of A, or both. Here a limit point of B is a point z such that any circle centered at z will contain a point of B, no matter how small you take the radius of the circle. The informal idea is that you cannot separate U into parts A and B without those parts being "infinitely close" to one another. Another definition of connectedness, which applies to open subsets of the complex plane: a subset X is path-connected if given any points w,z in X, you can define a continuous path p(t) for 0≤t≤1 with p(t) in X for all t, p(0)=w, and p(1)=z. The important bit is that every intermediate point p(t) along the path must belong to X. There are connected subsets of the plane which do not satisfy this condition (google "closed topologist's sine curve") but any connected open set, or "region," will be path-connected (Edit: and path-connected sets are *always* connected) This definition is a bit more useful because if you have the subset sketched on a piece of paper it is usually quite simple to verify that any two points are connected by a continuous path. Also most holomorphic functions studied in complex analysis are defined on regions. Often, mathematicians will sketch prototype curves between points in a subset without going to the trouble of writing down the formulas. Edit: A basic topological fact used in complex analysis is that an open subset of the plane is connected if and only if it is path-connected. It's important to distinguish between "connected" and "simply-connected." To say that a subset X is simply-connected means that (i) X is path-connected and (ii) any paths p,q in X connecting the point w to the point z can be "continuously deformed" into one another using a continuous family of paths in X which connect the point w to the point z. This video does a good job of touching on the meaning of this, but it's a pretty fundamental idea, and if you want to understand it better I'd suggest finding a book on complex analysis or algebraic topology and reading about path homotopy. The basic example to distinguish the concepts is: if we take X to be the set of nonzero complex numbers, this is path connected: to connect nonzero complex numbers w and z, draw any arc that curves around the origin. For example, to connected the number w = 1 to z = -1, you could draw the upper semi-circle of the unit circle, or you could draw the lower semi-circle of the unit circle. However, there is no way to deform/bend the upper semi-circle down onto the lower semi-circle without touching the origin, which we have excluded from the set X. (Proving this rigorously takes a modest amount of work, but hopefully it is plausible.) Thus X is an example of a path-connected subset that fails to be simply-connected. The degree to which a path-connected subset fails to be simply-connected is measured by an algebraic object called the fundamental group, which is often taken as the first topic in an algebraic topology course. In my experience, it's pretty rare to actually verify the conditions in the formal definitions of connectedness, especially when you are moving through examples at a brisk pace (like Borcherds is). There are various theorems that simplify the matter of proving that a subset is connected (for example, if two subsets are connected, and they share at least one point, their union is connected). But most of the time, when it comes to identifying connected subsets of the complex plane, "you know it when you see it": an unbroken line, line segment, arc, curve, circle, disk, rectangle, half-plane, or plane is connected, whereas a set that has been cut in half or otherwise written in several disjoint parts is disconnected. The plane with the single point 0 removed is connected (one way to see this is that this set is the image of the complex numbers under the continuous map z->e^z, although there are simpler options). The set U of complex numbers that aren't real (that is, the complex numbers with non-zero imaginary part) is disconnected: we can write U as A union B, where A is the set of complex numbers with positive imaginary part and B is the set of complex numbers with negative imaginary part, and neither of these sets contains a limit point from the other. A singleton set {z} is connected, because you can't write it as a union of disjoint non-empty sets. A non-singleton discrete point set such as the integers is a disconnected set, as is any infinite countable subset, like the rational numbers (or the "complex rationals" {a+ib: a,b rational}).

[@dannycrytser7268]

@DnB and Psy Production True for open subsets but not subsets in general (closed topologist's sine curve is a a subset of the plane).

[@migarsormrapophis2755]

ye