The Cauchy-Riemann Equations -- Complex Analysis 8
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- Опубліковано 24 гру 2024
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My favorite way to think about the Cauchy-Riemann Equations is geometric. It's a simple picture, but it requires some setup to explain the idea.
The big idea of the derivative is finding the best linear approximation to a function at a point. For functions ℝ->ℝ, this means if you zoom in close, as long as its differentiable, the curve starts looking like a line. (And a different line, depending on which point you zoom in on, of course.) For functions ℝᵐ->ℝⁿ, it means that when you zoom in on a point, the function starts looking like a linear map ℝᵐ->ℝⁿ (over the field ℝ). (Which, as usual in linear algebra, we can represent with an n×m matrix; in this context, that matrix is called the Jacobian.) For functions ℂ->ℂ, if we zoom in close to an input point, the function should start looking like a linear map ℂ->ℂ (over the field ℂ)
Now taking a detour through linear algebra: But what does a linear map ℂ->ℂ look like? It's much more restrictive than linear maps ℝ²->ℝ², even though we picture ℝ² and ℂ similarly. As a vector space over itself, ℂ is *_one-dimensional,_* not two; the basis is {1}, not {1,i}. So, as soon as you know where 1 goes, you automatically know where i goes: L(i) = i*L(1), by linearity. What that's saying is that L(i) is a 90° turn from L(1). If you think this through, it implies that complex linear maps ℂ->ℂ must always be a combination of scaling and rotating! This is more restrictive than linear maps ℝ²->ℝ², which include shears and reflections - those aren't linear on ℂ! (Maybe you see where this is going...)
Okay, bringing it back to calculus. If we have a complex function f : ℂ->ℂ, f(x+yi) = u(x,y) + i*v(x,y), we can make a corresponding real function g : ℝ²->ℝ² with g(x,y) = (u(x,y), v(x,y)), which "looks" the same geometrically.
The Jacobian of g, ie the closest linear map to it, is [uₓ, uᵧ; vₓ, vᵧ] (i'm writing it row-by-row). (Sidenote: the partials need to be continuous here, or else the Jacobian doesn't necessarily approximate g arbitrarily well as you zoom in.)
But remember, just because g is real-differentiable doesn't mean f is complex differentiable - not every linear map ℝ²->ℝ² corresponds to a linear map ℂ->ℂ. Namely, we need to add the restriction that it the map just rotates-and-scales. Which, if you think about the linear algebra... means that uₓ = vᵧ and uᵧ=-vₓ!! That's the Cauchy-Riemann equations! They just state that the Jacobian is, geometrically, a rotation-and-scaling!
So in sum: to be complex differentiable is to be locally linear. To be linear ℂ->ℂ is to be a combination of rotating and scaling. So to be complex-differentiable is to locally look like rotation-and-scaling. Meaning that the Jacobian of the corresponding ℝ²->ℝ² function is a rotation-and-scaling matrix (and the partials are continuous). Which is equivalent to uₓ = vᵧ and uᵧ=-vₓ. (I think it's possible to make that informal argument rigorous)
This is perfectly brilliant. Thank you for contributing it!
Thank you
Complex conjugate derivatives have two dual limits -- non holomorphic or non complex differentiable.
Holomorphic is dual to non holomorphic.
Points (limits, singularities) are dual to lines -- the principle of duality in geometry.
Homotopic is dual to non homotopic.
"Always two there are" -- Yoda.
Conformal invariance is dual to non conformal invariance.
Same is dual to difference, homo is dual to hetero.
Integration is dual to differentiation.
Two paths implies duality -- the Cauchy Riemann equations require two paths!
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature, gravitation (forces) are dual.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Commutation is dual to non commutation -- forces are dual.
Attraction is dual to repulsion, push is dual to pull -- forces are dual.
This is a great explanation, it adds some intuition, thank you! ❤ It was partially known to me, but it's good to read the whole thing here. Thank you for your efforts.
I learned a lot from this series.
Thanks Prof. Penn
My answers for these warm-up questions:
Q1
u_x=3x^2-3y^2=v_y
u_y=-6xy=-v_x
Q2
f(z)=e^(alpha*z), where alpha=i
Q3
when b=c=0
Thank you professor for your wonderful explanation.
there is infinitely many possibilities for Q3), you have to solve a linear system of equations.
At 10.21, you have inadvertently written i partial dv/dy. This should be i partial dv/dx. (This has been corrected in the next screen)
Counter-example to the last statement (around 33:00) if D = R (set of real numbers) and f is any analytic function with real values for real input. Finding why and which additional property the set needs to make the theorem correct is left as an exercise :-)
I suppose you mean openness, the set D must be open (a domain). That's always assumed, otherwise (complex) differentiability doesn't even make sense.
I like how Serge Lang (in his book _Complex Analysis_ ) shows the equivalence of CR and analytic. He thinks of the derivative at a point as being a linear map and expresses it as a matrix. It then becomes clear that CR holds if and only if f is analytic.
Analytic (wholes to parts) is dual to synthetic (parts to wholes) -- Immanuel Kant.
Deductive inference (mathematics) is dual to inductive inference (physics).
Differentiation (analytic, divergence) is dual to integration (synthetic, convergence).
Complex conjugate derivatives have two dual limits -- non holomorphic or non complex differentiable.
Holomorphic is dual to non holomorphic.
Points (limits, singularities) are dual to lines -- the principle of duality in geometry.
Homotopic is dual to non homotopic.
"Always two there are" -- Yoda.
Conformal invariance is dual to non conformal invariance.
Same is dual to difference, homo is dual to hetero.
Integration is dual to differentiation.
Two paths implies duality -- the Cauchy Riemann equations require two paths!
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature, gravitation (forces) are dual.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Commutation is dual to non commutation -- forces are dual.
Attraction is dual to repulsion, push is dual to pull -- forces are dual.
Love your videos
17:46 don't those fractions require modulus operators or are they bound to be in the reals? Keep in mind that z is in Complex so by extension delta z is also in Complex.
Well, bear in mind we showed that the modulus operator was continuous last video, so R1/Δz -> 0 and |R1/Δz| -> 0 are equivalent. You can certainly prefer to think of the latter if you want to keep it all in the reals.
Let's be honest, as a French math teacher (modestly in high school), my first thought was to criticize you. I was pretty sure you wouldn't have our French rigor.
But I never found anything to criticize and I must even say that I really appreciate your way of doing things simply.
I only managed to watch a tiny part of your huge work on youtube and now you offer us an excellent course of complex analysis that allows me to review things I didn't always understand at university.
I have recommended your videos to some of my students who want to go a little further.
I absolutely must send you a huge thank you and express my total respect :-)
French too here. I felt an itch when Prof Penn used the notation f' for derivatives before proving f was differentiable. But he explained verbally that he was abusing the notation here. "Faute avouée est à moitié pardonnée" :-) . So all fine. Great videos. Also, he organizes his board and uses the colours so well, that should be a model to all teachers.
Analytic (wholes to parts) is dual to synthetic (parts to wholes) -- Immanuel Kant.
Deductive inference (mathematics) is dual to inductive inference (physics).
Differentiation (analytic, divergence) is dual to integration (synthetic, convergence).
Complex conjugate derivatives have two dual limits -- non holomorphic or non complex differentiable.
Holomorphic is dual to non holomorphic.
Points (limits, singularities) are dual to lines -- the principle of duality in geometry.
Homotopic is dual to non homotopic.
"Always two there are" -- Yoda.
Conformal invariance is dual to non conformal invariance.
Same is dual to difference, homo is dual to hetero.
Integration is dual to differentiation.
Two paths implies duality -- the Cauchy Riemann equations require two paths!
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature, gravitation (forces) are dual.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Commutation is dual to non commutation -- forces are dual.
Attraction is dual to repulsion, push is dual to pull -- forces are dual.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line.
why would you assume American professors wouldn't teach rigorously? these are standard undergraduate classes that have more or less consistent curricula at any respectable university in the world
Will you be covering several variable complex analysis?
I don’t usually see analytic defined with f’(z) being continuous but it comes for free as was said anyway so it doesn’t matter, just saying usually isn’t part of the definition
Can someone explain why you get continuity for free? Was just stated and shown with an example in the video, but not really formally proven.
@@sebastiandierks7919 Analytic functions are continuous. Just like from calc 1, being differentiable implies continuous but idk if that was proven here but is true
@@Happy_Abe Being differentiable implies f is continuous, how does it imply f' is continuous?
@@Sriram-fl5hm Because as proven either in this video or a different one, if f is analytic in some disk then all its derivatives exist:
1st derivative, 2nd derivative, 3rd derivative,…etc.
Thus any derivative of f is continuous because it will also be itself differentiable.
This is of course differs from real functions where this property does not exist.
@@Happy_Abe Oh thanks
Do you plan to cover the Wirtinger derivatives at some point? I know they don't usually come up when there's only a single complex variable, but I personally find it pretty cool that you can differentiate with respect to z and its complex conjugate as though they were independent variables.
Yes please! From the Wirtinger derivatives, you can also give a super-short proof of Cauchy's integral theorem using differential forms like so:
f(z) holomorphic ∂*f = 0
let w = f(z) dz
dw = ∂f dz∧dz + ∂*f d*z∧dz = ∂*f d*z∧dz = 0 f holomorphic
=> ∫_∂S f(z) dz = ∫_∂S w = ∫_S dw = ∫_S 0 = 0 f holomorphic
I support this
The need for the path-connectedness premises aren't really explained.
To take the derivative, you are taking a limit to z along a path in the complex plane. For that you need path connectedness.
@@praharmitra why? Doesn't the fact that the set is open give you that each point in the set is surrounded by an open ball, and then you can do all your limits within the open ball. Thus the whole thing is generaliseable to an arbitrary open set.
@@edskev7696 But if your open set is the union of disjoint open sets A and B, then how could there be a path between a point in A to a point in B that stays in the open set?
@@Alex_Deam for what would we need a path from point in A to a point B?
@@synaestheziac To keep continuity.
If you didn't have path connectedness you could have a situation where the derivative is 0 but different values.
Think of analytic as trying to maintain the idea of differentiabilty from the Real numbers.
Really enjoy your videos, could you also give a different proof why conjugate of z function is not differentiable by sequence of convergence , thanks
As usual thank you for the video.
It's called subscript, but you do do it in LaTeX with an underscore
This seems a very roundabout method, via two lots of complex conjugation. Also the path variation along dz->0 can be handled by taking Re(dz) and Im(dz) as independent variables.
Reading deltas for d's and partial derivatives for d/dx, d/dy where appropriate: put
f(x+iy) = g(x, y) + ih(x, y) g and h real
df = dx dg/dx + dy dg/dy + idx dh/dx + idy dh/dy
For f to be analytic we require there exists complex f'(z) such that
df = dz f'(z)
Put f'(z) = G(x,y) + iH(x,y) G and H real
dx dg/dx + dy dg/dy + idx dh/dx + idy dh/dy = (dx + idy) (G(x,y) + iH(x,y))
Equating real and imaginary parts
dx dg/dx + dy dg/dy = dx G - dy H
dx dh/dx + dy dh/dy = dy G + dx H
We regard dx and dy as independent free variables. Hence in detail
dg/dx = G
dg/dy = -H
dh/dx = H
dh/dy = G
dg/dx = dh/dy [1]
dg/dy = -dh/dx [2]
[1] and [2] are the Cauchy-Riemann equations. We get
f'(x+iy) = d/dx (g+ih) = df/dx
= d/dy (h - ig) = -i df/dy
Analytic (wholes to parts) is dual to synthetic (parts to wholes) -- Immanuel Kant.
Deductive inference (mathematics) is dual to inductive inference (physics).
Differentiation (analytic, divergence) is dual to integration (synthetic, convergence).
Complex conjugate derivatives have two dual limits -- non holomorphic or non complex differentiable.
Holomorphic is dual to non holomorphic.
Points (limits, singularities) are dual to lines -- the principle of duality in geometry.
Homotopic is dual to non homotopic.
"Always two there are" -- Yoda.
Conformal invariance is dual to non conformal invariance.
Same is dual to difference, homo is dual to hetero.
Integration is dual to differentiation.
Two paths implies duality -- the Cauchy Riemann equations require two paths!
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature, gravitation (forces) are dual.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Commutation is dual to non commutation -- forces are dual.
Attraction is dual to repulsion, push is dual to pull -- forces are dual.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line.
At the proof we implicitly took real and imaginary parts of the analytic function. My question is how can we be sure that for analytic functions, those real part and imaginary parts are differentiable in the multivariable sense? Their partials won't exist if they are not differentiable right? I mean if f behaves nicely what's stopping u(x,y) and v(x,y) from not behaving nicely?
I think this was in fact proved in the first part starting around 07:38 up until around 13:42. The existence (and continuity) of the partial derivatives is a consequence of the assumption that f is complex-differentiable and that its derivative is continuous (i.e. f is analytic). Also use the fact that if a function converges at a point in the complex plane, its real and imaginary parts must both converge in the real numbers. Those are exactly the partial derivatives u_x and v_x (when taking a path in the direction of the real axis).
why does D need to be path connected?
Great ❤❤❤
Also lol nice ending.
Teaching exactly this at the moment I am writing this comment
Analytic (wholes to parts) is dual to synthetic (parts to wholes) -- Immanuel Kant.
Deductive inference (mathematics) is dual to inductive inference (physics).
Differentiation (analytic, divergence) is dual to integration (synthetic, convergence).
Complex conjugate derivatives have two dual limits -- non holomorphic or non complex differentiable.
Holomorphic is dual to non holomorphic.
Points (limits, singularities) are dual to lines -- the principle of duality in geometry.
Homotopic is dual to non homotopic.
"Always two there are" -- Yoda.
Conformal invariance is dual to non conformal invariance.
Same is dual to difference, homo is dual to hetero.
Integration is dual to differentiation.
Two paths implies duality -- the Cauchy Riemann equations require two paths!
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature, gravitation (forces) are dual.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Commutation is dual to non commutation -- forces are dual.
Attraction is dual to repulsion, push is dual to pull -- forces are dual.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line.