I Intuition behind Cauchy–Riemann equations

I know that if a complex function is analytic , it means that i can reach the neighborhood of every complex point using a certain "stretch and rotation".
In which way this fact conducts us to the "Cauchy Riemann equations" ? What's the intuition behind them ?
Thanks
 

FactChecker

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The derivative of an analytic function at a point must be a single complex number, regardless of the direction that a path approaches that point. So coming in from the x direction must be the same as coming in from the iy direction.

That leads to consider ∂x = ux + ivx and ∂iy = -i∂y.
f(z) = u(x,y) + iv(x,y),
∂f/∂x = ux + ivx and
∂f/∂iy = -iuy +i(-ivy) = -iuy + vy.
Since we need ∂f/∂x = ∂f/∂iy, setting the real and imaginary parts equal gives the Cauchy Riemann equations.
 
∂f/∂x = ux + ivx and
∂f/∂iy = -iuy +i(-ivy) = -iuy + vy.
Thanks so much.
The last doubt : geometrically, why this partial derivatives have this form ? In particular why in ∂f/∂iy there is a i in the denominator ? It's the first time i see a partial derivative with the imaginary number i
 

FactChecker

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I'm not sure how to describe it geometrically. The derivative of a vector is the derivative of the individual coordinates.
So ∂f/∂x = ∂u/∂x + i ∂v/∂x.
And because d/dy (iy) = i, we have ∂iy = i ∂y. So ∂f/∂iy = ∂u/∂iy + i ∂v/∂iy = ∂u/i∂y + i ( ∂v/i∂y)) = -i(∂u/∂y) + (∂v/∂y) = ∂v/∂y - i ∂u/∂y
 

fresh_42

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I've tried to give a bit of a different explanation, better different representation, since the math is the same in here:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/

Basically the Cauchy Riemann equations arise from the fact that ##\mathbb{C}## is a field, and as such more than simply ##\mathbb{R}^2##. However, we want to calculate and draw pictures with real and imaginary parts, which means this discrepancy between the field and the real vector space has to show up somewhere. ##i \cdot i## doesn't take place in the imaginary dimension, it leaves it and ends up in the real dimension - a property we don't have in ##\mathbb{R}^2##. Therefore there is an inherent mixture which results in the Cauchy Riemann equations.
 
Thanks so much.
The last doubt : geometrically, why this partial derivatives have this form ? In particular why in ∂f/∂iy there is a i in the denominator ? It's the first time i see a partial derivative with the imaginary number i
up
 

lavinia

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A function ##f(x,y)=(u(x,y),v(x,y))## is conformal if its Jacobian ##J(f)## is multiplication by a complex number. That is: ##J(f)## is a rotation multiplied by a scale factor. ##J(f)## is then of the form

##\begin{pmatrix}∂u/∂x&∂u/∂y\\
∂v/∂x&∂v/∂y\end{pmatrix} =r\begin{pmatrix}cos(θ)&-sin(θ)\\
sin(θ)&cos(θ)\end{pmatrix}## from which one has ##∂u/∂x=∂v/∂y## and ##∂u/∂y=-∂v/∂x##.

If ##f## is twice continuously differentiable then the partial derivatives with respect to ##x## and ##y## commute and ##u## and ##v## are both harmonic.

If one thinks of ##u(x,y)## as a velocity potential then its gradient can be thought of as the velocity of a fluid in the plane. Since the Laplacian is zero, the flow has zero divergence and is incompressible. Since the flow has a potential, its curl is zero so it is also irrotational. The curves ##u=##constant are the lines of constant potential and the curves ##v=##constant are the stream lines of the flow.
 
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WWGD

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I've tried to give a bit of a different explanation, better different representation, since the math is the same in here:
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/

Basically the Cauchy Riemann equations arise from the fact that ##\mathbb{C}## is a field, and as such more than simply ##\mathbb{R}^2##. However, we want to calculate and draw pictures with real and imaginary parts, which means this discrepancy between the field and the real vector space has to show up somewhere. ##i \cdot i## doesn't take place in the imaginary dimension, it leaves it and ends up in the real dimension - a property we don't have in ##\mathbb{R}^2##. Therefore there is an inherent mixture which results in the Cauchy Riemann equations.
But, don't we have something similar with irrational numbers in that, e.g. ##\sqrt2 \times \sqrt 2=2 ## is not Irrational? Why don't we then have an analog in Real-differentiability? EDIT: True that Irrationals do not occupy a different dimension than Rationals; actually EDT Irrationals are "most" of the Reals by many measures, so then one can argue that they "go into nowhere" or something similar when they disappear this way?
 
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fresh_42

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True that Irrationals do not occupy a different dimension than Rationals
Yes, we don't consider ##\mathbb{R}## or even only the algebraic numbers (Btw, do they have a letter? ##\mathbb{A}## perhaps?) as ##\mathbb{Q}-## vector space, at least not in analysis. I know you would like to!
I found @lavinia's explanation good, to emphasize that the derivative is a ##\mathbb{C}-##linear map. This is the point: it has to be complex linear and not just real linear.
 

FactChecker

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actually Rationals are "most" of the Reals by many measures
This statement surprises me, since the rationals have Lebesgue measure zero and are countably infinite -- far fewer than the irrationals.
 

WWGD

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This statement surprises me, since the rationals have Lebesgue measure zero and are countably infinite -- far fewer than the irrationals.
Yes, right, I meant the Irrationals. Let me edit.
 

lavinia

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Thanks so much.
The last doubt : geometrically, why this partial derivatives have this form ? In particular why in ∂f/∂iy there is a i in the denominator ? It's the first time i see a partial derivative with the imaginary number i
Dividing by i rotates the vector clockwise by 90 degrees.

For every directional derivative one gets a rotation. For instance derivative in the negative x-direction is rotated by 180 degrees(multiplied by -1). What does this tell you about the complex derivative?
 
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WWGD

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Indeed: The Complex Jacobian/Differential of an Analytic function , with entries M:= ( a b -b a ) acts like multiplication by a+ib , i.e., when you left-multiply (c+id) by M, you get the same as multiplying (a+ib)(c+id), i.e., you scale by the modulus and rotate by the argument .
 

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