Wolfram Mathworld's article on Cauchy-Riemann Equations

[Swamp Thing]

TL;DR Summary

Need help understanding this : "Along the real, or x-axis, partialf/partialy=0"

On this page https://mathworld.wolfram.com/Cauchy-RiemannEquations.html ...
... above equations 8 and 9 it says :

Along the real, or x-axis, $\partial f / \partial y = 0$
Along the imaginary, or y-axis, $\partial f / \partial x = 0$

These statements seem to be saying that f is independent of y at the x axis, and independent of x at the y axis. Is this necessarily correct for any f?

I can see intuitively how (7) reduces to (8) and (9) -- if we're moving along the x axis, we can ignore partial derivative terms that have $\partial y$ in the denominator. (Non-rigorously, we are setting $\Delta y=0$). But I don't quite understand the two statements quoted above.

 

[WWGD]

I understand it to mean that f=u+iv is purely imaginary along the y-axis and purely real along the x axis.

 

[BvU]

I understand it to mean that f=u+iv is purely imaginary along the y-axis and purely real along the x axis.

Counter example: $f = {\bf z}^2$

 

[WWGD]

Counter example: $f = {\bf z}^2$

I meant it as a tautology. Just like points (x,y) in the plane restrict to (x,0) on the Real axis and (0,y) on the y-axis. I should have said 'on'the axis instead of 'along' the axis .

 

[BvU]

$f = {\bf z}^2$ is purely real on the y-axis

 

[BvU]

Non-rigorously, we are setting $\Delta y=0$

Have not found a rigorous proof, but Joel Feldman at least shows the steps here

Check out his example 7

 

[WWGD]

$f = {\bf z}^2$ is purely real on the y-axis

It takes values i2xy which are imaginary. This is what I meant. It is a tautology from the way things are defined.

 

[BvU]

That is not $f$ but $iv$

 

[WWGD]

That is not $f$ but $iv$

True, I miswrote. Let me edit. But it is tautological that v is Real along the y-axis.

 

[WWGD]

Edit: I understand it to mean that f=u+iv and so iv is purely imaginary along the y-axis and purely real along the x axis.

 

[WWGD]

True, I miswrote. Let me edit. But it is tautological that v is Real along the y-axis.

Ok, I just edited to clarify.

 

[BvU]

I think the idea is that $v$ is real everywhere ...

 

[FactChecker]

The notation of partial derivatives is not really appropriate in those intermediate steps. The idea is that the change in $f$ due to a change in $x$ is zero as $z=x+iy$ goes along the imaginary axis. That is clear because there is no change in $x$. It is hard to say what notation would be best to use, but they want the conclusions to be in terms of partial derivatives, so they use that notation throughout. IMHO, introducing another non-standard notation would only be more confusing.

I guess they could have used $\frac {\partial f}{\partial x} dx$, which would be zero because $dx=0$.

 

[WWGD]

I think the idea is that $v$ is real everywhere ...

That would force f to be constant if f is analytic by e.g. a Cauchy-Riemann argument.

 

[mathwonk]

The original 2 statements linked seem false to me on their face. E.g. if f(z) = z = x+iy, then ∂f /∂x = 1 and ∂f/∂y = i everywhere, including along both axes. So it is hard for me to understand what the author means by his statements. This article uses the practice of manipulating notation in a suggestive way, without rigorously defining what is meant. Unless you find it entertaining, I suggest ignoring this confusing derivation and reading one more clear and correct, such as that of Henri Cartan, in Theory of functions of one and several complex variables, or Lang in his Complex Analysis, or for that matter, even Goursat's Course of mathematical analysis, tr. by Hedrick, volume II.

To me the clearest explanation of the Cauchy Riemann equations is one I have trouble finding a reference for at the moment, but which was also due to Cartan, in a book long out of print, until recently, called simply Diifferential Calculus, (now ... on Banach spaces, p.37). Namely regard f:C-->C as a map f:R^2-->R^2, assumed real differentiable, and take the derivative at a point p, which is of course a real linear mapping R^2-->R^2. If in fact that linear mapping is not only real linear, but also complex linear, which is to say it commutes with multiplication by i, or rotation counterclockwise through 90 degrees, then we say that f is complex differentiable, or holomorphic at p. That's it, the conditions expressing complex linearity of the derivative are the "Cauchy Riemann" equations.

To wit: A real linear map has a 2x2 matrix of real numbers, namely whose entries are the real partials ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y, and (you can check) this matrix defines a complex linear map (i.e. the matrix commutes with the matrix with rows [0 -1], [1 0], for a rotation), if and only if the rows have form [ a -b], [b a], i.e. iff ∂u/∂x = ∂v/∂y, and ∂u/∂y = -∂v/∂x.

I see I expressed it more briefly in 2006, in post #11 of this thread:
https://www.physicsforums.com/threads/cauchy-riemann-relations.140294/

Well maybe I am the last one to see this, but after re-reading the usual derivations of the C-R equations, including that of Riemann himself, the author seems to be trying to say that the limit of the difference quotient delta(f)/delta(z), should be independent of the direction in which delta(z) points. Riemann says that df/dz (he uses dw/dz) should be independent of dz. I am still unable to see how the author's words say this.

OK: I claim he is at least somewhat off the mark, and (some of) his statements are false. He asserts, as a consequence of his questionable and (I think) false statement that ∂f/∂y is "zero along the x axis", that therefore
∂f/∂z = (1/2)(∂u/∂x + i ∂v/∂x). But this is wrong. As everyone knows, for a holomorphic function,
∂f/∂z = ∂f/∂x, not 1/2 ∂f/∂x. E.g. just take f(z) = z again, so that ∂f/∂z = 1, but u=x so ∂u/∂x = 1, and v = y so ∂v/∂x = 0; hence (1/2)(∂u/∂x + i ∂v/∂x) = (1/2), and 1 ≠ 1/2.

In my opinion, this is going to make it hard to find a viewpoint from which his derivation works.

Although Cartan's explanation is the clearest theoretical one to me, the simplest derivation is the usual one, that we want df/dz to be a directional limit, independent of the direction dz points in. So when we approach in the x direction, i.e. when dz = dx, we get ∂f/∂x, and when we approach in the y direction, i.e. when dz = idy, we get ∂f/i∂y, so these must be equal, i.e. the C-R equations just say that

∂f/∂z = ∂f/∂x = (1/i)(∂f/∂ y) = -i∂f/∂y.

If you write out f = u+iv,
you get ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.I think this is what he is trying to say. Maybe someone can make sense of his approach, as suggested by FactChecker above, by working with differentials and setting dx = 0 along the y axis, etc..

maybe this? ∂f/∂z dz = ∂f/∂z (dx + idy) = ∂f/∂z dx + ∂f/∂z (idy). Now along the x axis, where dy = 0, and dz = dx, this says ∂f/∂z dx = ∂f/∂x dx, so we must have ∂f/∂z = ∂f/∂x. But along the y-axis where dx=0, and dz = idy, we have ∂f/∂z idy = ∂f/∂iy idy, so we must have ∂f/∂z = ∂f/i∂y. Thus we must have ∂f/∂z = df/∂x = ∂f/i∂y.

I still prefer Cartan's explanation: a real differentiable function has as derivative its best real linear approximation; a complex differentiable function has as derivative its best complex linear approximation; since every complex linear map is also real linear, thus every complex differentiable function is also real differentiable, with the same derivative, and a real differentiable function is complex differentiable if and only its unique best real linear approximation is also complex linear.

 

[Tendex]

The original 2 statements linked seem false to me on their face. E.g. if f(z) = z = x+iy, then ∂f /∂x = 1 and ∂f/∂y = i everywhere, including along both axes. So it is hard for me to understand what the author means by his statements. This article uses the practice of manipulating notation in a suggestive way, without rigorously defining what is meant.

It seems to me the Wolfram page is not using the (1, i) real basis but (z, zbar) one instead but not specifying it clearly that the Wirtinger linear operator was defined for the partial of f respecto z. That explains the 1/2 vs. 1 discrepancy when expressing x in terms of z and zbar over 2.
My doubt is if the assumption of real differentiability can always be made without loss of generality, I guess it can as long as mappings between smooth manifolds is assumed which is always the case for linear spaces like R^2.

 

[mathwonk]

Since in fact ∂f/∂z = ∂f/∂x for all holomorphic functions, I don't see how a choice of bases can explain away claiming that ∂f/∂z = (1/2)∂f/∂x, (except when f is constant).

And indeed you are right that using the basis z, zbar does introduce a (1/2). Hence his factor of 1/2 is correct when writing x as a function of z and zbar, namely x = (1/2)(z + zbar). Hence in some sense, ∂x/∂z = (1/2). And this leads him to the correct formula that ∂f/∂z = (1/2){∂f/∂x - i ∂f/∂y}.

I.e. by definition of the limits concerned, as I said, we have ∂f/∂z = ∂f/∂x, if we take the limit in the x direction, and ∂f/∂z = ∂f/∂iy = -i ∂f/∂y, if we take the limit in the y direction.

Therefore in the correct equation above we do indeed have ∂f/∂z = (1/2){∂f/∂x - i ∂f/∂y}
= (1/2){∂f/∂z + ∂f/∂z} = (1/2){(2) ∂f/∂z}.

It is when he brazenly sets ∂f/∂y = 0 in this correct equation, that he renders it incorrect, by a factor of 1/2.

Indeed the Cauchy Riemann equations say exactly that ∂f/∂x = -i∂f/∂y, (everywhere). Hence one of these two expressions cannot be zero anywhere that the other one is not, e.g. along the x axis.

I.e. the Cauchy Riemann equations imply that ∂f/∂x is zero exactly where ∂f/∂y is zero. So his conclusions, i.e. the C-R equations, contradict his own statements. It seems he just slipped up, or "misspoke", it happens to all of us.

 

[Tendex]

Since in fact ∂f/∂z = ∂f/∂x for all holomorphic functions, I don't see how a choice of bases can explain away claiming that ∂f/∂z = (1/2)∂f/∂x, (except when f is constant).

The second expression is different, it is not actually a partial derivative, even if it usually has the same notation, it is a Wirtinger derivative and it is defined in that way as a linear operator. As you wrote in the thread linked by yourself, just in this case applied to df/dz: " the point is that df/dzbar is not defined as a limit wrt some "variable" zbar, but as a coefficient of a basis vector in a linear space."

 

[Tendex]

In other words the definitions in the Wolfram page that include the 1/2 factor are previous to applying the Cauchy-Riemann conditions, when they are applied to both then the halves cancel each other. When setting the partials of f respect to y and x equal to zero is before applying the CR conditions, precisely to obtain them when equating 8 and 9 to demand complex-differentiability.

 

[BvU]

That would force f to be constant if f is analytic by e.g. a Cauchy-Riemann argument.

I think one of us is looking the domain of $f$ and another at the range.

 

[zinq]

In the past I've found a number of mistakes in Wolfram's MathWorld. This article on Cauchy-Riemann equations seems to have several more. (For instance, it claims that for an analytic function f(z), along the x-axis, ∂f/∂y = 0.)

 

[benorin]

Iirc the Mathworld page argued similarly to this text (Fundamentals of Complex Analysis by Snaff & Snider, pg. 73):

 

[WWGD]

I think one of us is looking the domain of $f$ and another at the range.

Could be, I may have misunderstood you, apologies if so. The argument in Wolfram is one I have seen before. It considers the change along the X ( resp Y) axis so that dx=0 ( dy=0) along the axis to prove C-R.

 

[Tendex]

The argument in Wolfram is one I have seen before. It considers the change along the X ( resp Y) axis so that dx=0 ( dy=0) along the axis to prove C-R.

It can be seen in a number of complex variables textbooks, and it is not so hard to understand if one considers what I wrote in #18 and #19.