Talk:Cauchy–Riemann equations

Holomorphic implies analytic

If f(z)=u(x,y)+ i v(x,y)satisfies cauchy reimann equation alongwith condition that u,v and all its 1st order patial derivatives are continuous, then the function is analytic ...........plz prove that --anon

See Holomorphic functions are analytic. Oleg Alexandrov (talk) 21:30, 9 November 2005 (UTC)

"Not necessarily sufficient" ??!!

If a condition is "not necessarily sufficient" then it is not sufficient, surely?

I don't think the use of the more complex term is helpful. I'll take it out, unless someone else wants to, or wants to make a case for it.

David Young

I'm working on the survey article "complex analysis". I'm puzzled by the "necessary but not sufficient" language in this article. I'd like to rewrite this, because I don't think it's true. But I don't want to start another controversy inadvertently.

• If the Cauchy-Riemann equations are satisfied in a neighborhood, f(z) has a first derivative there. It doesn't matter whether the first partial derivatives are continuous functions or not ... we just have to be able to integrate them to get u and v.
• By the Cauchy-Goursat theorem, if f(z) has a first derivative in a neighborhood, it's analytic there. No requirement of continuity has to be imposed on the partials to do this proof, either.
• So the Cauchy-Riemann equations are in fact sufficient to prove analyticity, indirectly.

Am I missing something here? DavidCBryant 15:48, 7 December 2006 (UTC)

I've been thinking about this some more, and I think the current language (necessary but not sufficient) may have been written by somebody who only studied one textbook. Some authors develop the theory of functions by assuming that the partial derivatives in C-R are continuous, and eventually get around to Cauchy-Goursat. Other authors start off with complex integration, and prove Cauchy-Goursat first. Just a thought. DavidCBryant 17:05, 7 December 2006 (UTC)

I'm pretty sure the article is correct as it stands, as some sort of continuity assumption is required as well as the Cauchy-Riemann equations. For example: let f(x+iy) = 0 if one or both of x, y is zero, and f(x+iy) = 1 if neither of x, y is zero. Then, for this f, all the partial derivatives exist at the origin, and all are zero, so that the C-R equations hold. All this is for 'differentiability at a point'. of course, and you are probably right if we are considering C-R equations in a region. Madmath789 17:21, 7 December 2006 (UTC)

Oops! Please forgive me for getting some facts mixed up. It's just been too long since I thought about some of this basic stuff. Proceeding directly from the C-R equations and trying to prove that holomorphic ==> analytic is the wrong way to go. Anyway, I probably shouldn't have even asked this question here in the first place, if I'd been thinking more carefully the other day. DavidCBryant 20:04, 8 December 2006 (UTC)

I don't know enough to contribute to this article, but I did notice that "necessary but not sufficient" and "if and only if" are both used to describe the same thing; one way or another something is wrong there. 192.5.109.49 (talk) 01:11, 15 May 2008 (UTC)

The C-R conditions are necessary, but not sufficient. However, if a function is smooth enough (C^1 I think), the C-R conditions become sufficient. That is to say, the C-R conditions are necessary, and with a small extra assumption they become sufficient. Oleg Alexandrov (talk) 04:58, 15 May 2008 (UTC)

So most people agree that the C-R are not sufficient conditions for differentiability - indeed the textbook I'm reading has a chapter called necessary condition for differentiability and another one called sufficient cfd, where continuity and existence are required. This analysis book agrees too: http://books.google.com/books?id=61BO2UY9QN4C&pg=PA46&source=gbs_toc_r&cad=0_0&sig=ACfU3U0v8aA5HRrpzKZXSmy-ZPk_WJVqvA Here is an example of a function who's first order partial derivatives do not exist at z=0, the "manual" x and y limits tell us that the C-R holds but if z approaches 0 along the line x=y we get a different answer than in the horizontal and vertical limits and hence the derivative does not exist at 0: f(z) = conj(z)^2/z if z is not 0, f(0) = 0. —Preceding unsigned comment added by 24.80.104.219 (talk) 08:13, 15 October 2008 (UTC)

Me again: By the way, planetmath, mathworld and answers.com all mention the existence and continuity along with the C-R as the sufficient condition. Some may not mention existence directly, but continuity at a point implies existence at a point. —Preceding unsigned comment added by 24.80.104.219 (talk) 08:28, 15 October 2008 (UTC)

The statement is this: If f=u + iv, where u,v:R2->R, then provided u and v are differentiable at a point (x,y) (associated with z=x+iy) then f is analytic at z if and only if the Caucy-Riemann equations are satisfied. This was kind of stated in the article, but not really so I'm going to modify the article to make this more clear. --67.193.128.233 (talk) 21:30, 13 April 2009 (UTC)

We must fix this: This entry on the Cauchy-Riemann equations is particularly disturbing. My contention: The Cauchy-Riemann equations are (only) necessary conditions for a function to be differentiable at a point. They are not sufficient. Let's first agree on the fact that the opening sentence (as of 22 October 2010) is wrong: "In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides a necessary and sufficient condition for a differentiable function to be holomorphic in an open set." The incorrect phrasing is "and sufficient." This is demonstrably incorrect. Perform the following exercise: verify that the function 1/z satisfies the Cauchy-Riemann equations. You will find that the real and imaginary parts of this function (which are x/(x^2+y^2) and -y/(x^2+y^2), respectively) satisfy the Cauchy-Riemann conditions: u_x = v_y and u_y = -v_x. Please verify this now. If we integrate 1/z over the unit circle, we find that the value of this integral is 2*pi*i. This violates the Cauchy-Goursat theorem, which states that the integral of a function that is differentiable on and within a closed contour is zero. A violation of the Cauchy-Goursat theorem means that the derivative of the function does not exist somewhere in the domain enclosed by the contour. In other words, the function is not differentiable within or on the contour. Simply put: 1/z (and its derivative) is divergent at z=0. Sufficiency is a stronger condition and requires both the satisfaction of the Cauchy-Riemann equations and the continuity of the derivatives. Let's fix this before more people reading about complex variables are confused. MarkWayne (talk) 04:37, 22 October 2010 (UTC)

Dear Mark Wayne: Your counterexample is wrong: It does not prove that we need continuity in the partial derivatives. This is because the C-R equations hold for z different from zero. C-R are not saying that it is differentiable at z=0 because C-R are not defined at z=0!, so your contradiction never arises. I am sure it is necessary and sufficient. The confusion is brought about by following a single textbook, as mentioned above. Please try to prove that indeed, by assuming only the real differentiability of u and v, we can get complex differentiability of f at that point. I am posting the proof below. 190.118.29.30 (talk)Arestes

Proof that Cauchy-Riemann equations with u and v just (real) differentiable are sufficient for differentiability at a point

(We can easily see that C-R are necessary) NOTE: Before reading the proof, which is just the same as the one in the article I have to point out the reason why there is so much confusion: Real differentiability of a real function of two real variables at a point is a condition STRONGER THAN the existence of its partial derivatives at that point but WEAKER THAN the continuity of its partial derivatives at that point. The existence of the partial derivatives of u (or v) does not guarantee the real-differentiability of u (correspondingly, v). But if we require u and v to be differentiable then that, together with C-R, is enough to say that f=u+iv is (complex)differentiable. This holds even if the partial derivatives of u and v are not continuous. 190.118.29.30 (talk)Arestes

REAL DIFFERENTIABILITY OF u AND v + C-R EQUATIONS IS SUFFICIENT FOR COMPLEX DIFFERENTIABILITY OF f at a point. So, what many textbooks do is requiring that u and v be continuously differentiable, which is an OVERKILL. We only need to say that u and v are real differentiable at a point. That, together with C-R, is a necessary and sufficient condition for complex differentiability at a point. Below is the proof 190.118.29.30 (talk)Arestes

Suppose the complex function f is of the form f=u(x,y)+iv(x,y) with u and v being functions from R^2 to R(defined in an open subset of C\ equivalently, R^2 for u and v\ so we can take derivatives by considering variations of the arguments small enough to stay in the domain). Then, if they are differentiable, then, by DEFINITION, we have u(x+a, y+b) = u(x,y) + [d_x u(x,y)]*a + [d_y u(x,y)]*b +Ru(a,b) and v(x+a, y+b) = v(x,y) + [d_x v(x,y)]*a + [d_y v(x,y)]*b +Rv(a,b) where d_x and d_y denote partial derivatives and Ru(a,b) and Rv(a, b) denote continuous functions of a and b such that Lim_{(a,b)->(0,0)} [R_u(a,b)/||(a,b)|| ] = Lim_{(a,b)->(0,0)} [R_v(a,b)/||(a,b)|| ] = 0.

This is just the definition guys. It only says that u and v are differentiable, that is, that they can be approximated linearly. This doesn't need any continuity assumptions for d_x u , d_y u ,d_x v or d_y v.

The proof relies on just plugging in these values in the difference: (z=x+iy=(x,y)) f(z+ a+ib) - f(z)= [ u(x+a,y+b)-u(x,y) ] + i [ v(x+a,y+b)-v(x,y) ] =[d_x u(x,y)]*a + [d_y u(x,y)]*b +Ru(a,b) +i [ [d_x v(x,y)]*a + [d_y v(x,y)]*b +Rv(a,b) ] manipulating this expression and using C-R equations to substitute adequately, please verify we get: f(z+ a+ib) - f(z) = [d_x u(x,y) - i*d_y u(x,y) ] (a+ib) + [ R_u(a,b) + i Rv(a,b) ] with Lim_{(a,b)->(0,0)} [ [R_u(a,b)+ iRv(a,b)]/||(a,b)|| ] = 0+i0=0 which means that f is C-differentiable at z=x+iy (because we were able to factor out the "complex difference" vector a+ib). Moreover, f'(z) = d_x u(x,y) - i*d_y u(x,y) = d_y v(x,y) + i*d_x v(x,y). QED.

NOW, to get a holomorphic function in an open subset of C, let's call it U , we need to see that f is differentiable in the whole of U. We just need to check C-R for all z in U AND check that u and v are real differentiable. We don't need any mention of continuity for the partial derivatives. 190.118.29.30 (talk)Arestes

History

According to ru:Условия Коши — Римана, these equations were first published by D'Alambert in 1752. In 1777 Euler connected these equations to the analyticity of complex functions. Cauchy used these equations to construct the theory of functions in 1814. Riemann's dissertation on the theory of functions was published in 1851. Unfortunately, the Russian article does not have any references. Can anyone look into this?(Igny 15:09, 7 July 2006 (UTC))

I have edited the article, adding this information. Also I deleted the following nonsense

The relation has this interpretation: ${\mathit {x}}$ and ${\mathit {y}}$ must be constant with respect to ${\bar {z}}$. This expresses the concept that an analytic function is "truly" a function of a single complex variable, rather than of a real vector.

It simply contradicts with $x=.5(z+{\bar {z}}),y=.5(z-{\bar {z}})$.

I find your use of a Russian language reference very unusual. Your references should be available to all; in this case those that speak English. Given that the per centage of English speakers that can also speak Russian is very small, using a Russian reference is not acceptable. Declan Davis (talk) 12:08, 17 September 2008 (UTC)

A good reference for this is: Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Johns Hopkins University Press, p. 419. I wish to put the reference on the main page but I don't know how to.

Why the Cauchy-Riemann Equations are Satisfied

An complex analytical function is continuous at a point Zo if and only if it's limit exist and is the same independent of what direction the limit takes. Here Cauchy is taking advantage of the fact that $\lim _{\Delta x\rightarrow 0,\Delta y=o}$ is equivalent to $\lim _{\Delta x=0,\Delta y\rightarrow 0}$.

—The preceding unsigned comment was added by 72.71.218.71 (talk • contribs) .

is that supposed to make any sense? --MarSch 14:32, 8 December 2006 (UTC)

I think we both know what he's trying to say. Besides, the article now includs a section on the derivation of the CREs. Namely that, for real r and θ,

$\lim _{r\to 0}{\frac {d}{dr}}f(z_{0}+re^{i\theta })$

is well defined, i.e. exists and is independent of θ. Since first order differentiation is linear this can be checked by computing the limits for θ = 0 and θ = π/2. Declan Davis (talk) 11:28, 17 September 2008 (UTC)

More general form

There is a higher dimensional analogue of the CauchyRiemann equations involving a complex structure $J_{\nu }^{\mu }$ See e.g. http://en.wikipedia.org/wiki/Pseudoholomorphic_curve It would be nice if this could be added here.

I think the polar form section would be better represented with the "r" terms. It suggests divide by zero problems in a way that could be confusing. The constants should simply be moved o the other side. You can see this form in Sarason's "Notes on Complex Function Theory." Comments? Are there any reasons that the current form is better? 24.7.83.80 (talk) 09:25, 13 February 2008 (UTC) Alex Kaiser ( UC Berkeley )

Rewrite attempt

The current version of Alternative formulation is not incorrect. I would just like to point out a certain slight abuse of notation, f in $f(x,y)$ is not the same as in $f(z,{\bar {z}})$. I tried to make it clear that when you calculate df/dz, you mean $d/dzf(x(z,{\bar {z}}),y(z,{\bar {z}}))$, not $d/dzf(z,{\bar {z}})$. (Igny 04:11, 7 February 2007 (UTC))

There is a slight abuse of notation indeed. But the calculation is done correctly, using the chain rule, as far as I see. In your edit you spelled out fully the Jacobian matrix in the chain rule for 2 variables, but I don't think that it adds rigor, and it looked harder to understand. Oleg Alexandrov (talk) 04:51, 7 February 2007 (UTC)

I agree the calculation is correct in principle, and in general the current version is ok by me. I consider my concern about the notation noted (no puns intended). By the way, the Jacobian is calculated in the current version as well, just not named such. (Igny 05:16, 7 February 2007 (UTC))

You are right that in the current version one calculates the jacobian also (you have to, it is a variable change we are talking about). But the way it is now done is easier to follow than in your proof I think.

So the question is then whether it is worth making the proof more formal (and more complicated) for the sake of fixing a slight abuse of notation which I think is used quite a lot in such cases. I don't have a good answer. Oleg Alexandrov (talk) 16:23, 7 February 2007 (UTC)

Does anyone have a print reference for this alternative formulation, because to me it look very dodgey. In particular, it appears to assume that ${\frac {\partial {\bar {z}}}{\partial z}}=0$ which is just plain wrong no matter how you look at it. I think this section needs to go until someone can provide a print reference. IN any case, I'm putting a citation needed flag on that section. ObsessiveMathsFreak (talk) 18:51, 7 December 2007 (UTC)

This formulation is the standard one in some physics applications (CFT and strings). See lectures by Paul Ginsparg or Polchinski Vol I chapter II. —Preceding unsigned comment added by 83.57.132.73 (talk) 20:17, 24 February 2008 (UTC)

Analyticity

Is there a difference between analytic and holomorphic in this article? It sounds confusing to me.

There is no difference, a holomorphic function of a complex variable is the same as an analytic function, although the latter term is more general, being used for functions of real variables too. Oleg Alexandrov (talk) 15:33, 8 August 2007 (UTC)

Derivation

I was trying to look for a derivation of the CR eqs. Is it possible to have it on the article? --Hamsterlopithecus (talk) 18:57, 7 October 2008 (UTC)

There already is. Try the section on complex differentiability.  Δεκλαν Δαφισ   (talk)  21:25, 7 October 2008 (UTC)

This is only necessity, not sufficiency. --67.193.128.233 (talk) 21:32, 13 April 2009 (UTC)

Goursat section is not clear

Text says

The hypotheses of Goursat's theorem can be weakened significantly. If f = u+iv is continuous in an open set Ω and the partial derivatives of f with respect to x and y exist in Ω, then f is holomorphic (and thus analytic). This result is the Looman–Menchoff theorem.

but does not recall Cauchy-Riemann equations. Isn't it misleading? --Bdmy (talk) 19:40, 1 January 2009 (UTC)

Also, why not state Goursat by saying simply: C-differentiability at every point is enough, rather than what the article says: [differentiable as a function on R² + CR equations]? Of course it is the same, but why choose the longer equivalent form? --Bdmy (talk) 19:45, 1 January 2009 (UTC)

This is more or less related but isn't any (distribution) solution of the homogeneous Cauchy-Riemann equations are infinitely differentiable, for ${\partial \over \partial {\overline {z}}}u=0\Rightarrow \Delta u=0\Rightarrow u\in C^{\infty }$ (For example, by the elliptic regularity theorem)? -- Taku (talk) 02:22, 2 January 2009 (UTC)

The last result quoted in the article follows from elliptic regularity. Would it be worth mentioning this? I'm not exactly sure how the other Goursat-like results compare with this one, since they deal only with classical notions of differentiability and continuity. The corresponding classical regularity results obviously assume more differentiability than needed here. siℓℓy rabbit (talk) 03:34, 2 January 2009 (UTC)

Physical interpretation

The assertion "irrotational + solenoidal" implies "to be a potential flow" is not always true. It depends on the topology of the domain. For example 1/z does not have a potential because of the singularity in z=0. To correct the text it suffices to add that the domain U is simply connected. 80.174.196.107 (talk) 06:28, 6 May 2014 (UTC)

f-bar

The f-bar notation in the "physical interpretation" section is very poorly chosen, as it may be mistaken for complex conjugate. Tkuvho (talk) 11:13, 12 April 2010 (UTC)

It looks like the vector field needs to be the real and imaginary components of the complex conjugate of a holomorphic function, otherwise the Cauchy-Riemann equations do not give exactly the right conditions. But the section does a rather poor job of explaining this. Sławomir Biały (talk) 12:33, 23 April 2010 (UTC)

Differentiable as a prerequisite?

The first sentence of the article:

In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations that provides a necessary and sufficient condition for a differentiable function to be holomorphic in an open set.

But if the function is differentiable, it is already holomorphic, by definition (and vice versa). I suppose the word "differentiable" should be omitted (or changed to smth like real and imaginary parts are differentiable as real-valued functions). Semifinalist (talk) 22:05, 30 August 2010 (UTC)

I have changed that part. There was a confusion about holomorphy and continuity of partials... I think it's clearer now — Preceding unsigned comment added by 190.118.29.30 (talk) 08:32, 31 October 2011 (UTC)

Euler 1797 ??

Euler died in 1783. Tkuvho (talk) 09:07, 8 December 2010 (UTC)

Not continuous at zero?

...it looks like it is. It's certainly continuous on the real line. On which isn't it? 174.254.163.152 (talk) 02:20, 26 February 2011 (UTC)

Along the line parameterised by z=t(1+i), as t→0, z^4→-4t^4, so f(z)=e^(1/4t^4), so f→∞ as t→0. The point is that this function is continuous along the real and imaginary axes, for which z^4 is a positive real number. The CR equations check f(z) along these axes only, so from the point of view of the CR equations it is continuous (though not generally). Hope that made sense (and is correct...) — Preceding unsigned comment added by 82.6.96.22 (talk) 05:20, 8 June 2011 (UTC)

Missing condition?

The article claims in more than one place that the differentiability of the corresponding R2 function and the C-R equations are necessary and sufficient to imply the complex function is holomorphic. However, is this claim not omitting that the partial derivatives must be continuous too in a neighbourhood of the point in question? This is not implied by differentiability in R2. --94.169.117.101 (talk) 20:38, 16 April 2011 (UTC)

Proposal of a New Section

Should we by any chance add a section for the polar forms of C-R?

They are:

v_t = ru_r and u_t = -rv_r

and yield the following Laplace Equations:

r^2u_rr + ru_r + u_tt = r^2v_rr + rv_r - v_tt,

the proof for which is trivial.

I feel it would make the article feel more complete for people who are looking for it.71.234.166.64 (talk) 01:26, 26 August 2013 (UTC)

conjugate harmonic?

"Viewed as conjugate harmonic functions, the Cauchy–Riemann equations are a simple example of a Bäcklund transform."

This sentence does not make any sense. You view the equations as functions, and then they become transforms.  ?????--345Kai (talk) 05:55, 24 September 2014 (UTC)--345Kai (talk) 05:55, 24 September 2014 (UTC)

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No need to refer to the d-bar operator

I think the generalization of the equations to higher dimensions is far more straightforward: the article on several complex variables makes no reference to the d-bar operator at all, and instead provides the much more elementary description using Wirtinger derivatives. I think we should use that here. The d-bar operator may provide the most general formulation, but is unnecessary for a simple extension to several variables.--Jasper Deng (talk) 10:04, 2 December 2017 (UTC)

Opaque Lede

This article needs a more accessible lede. The first sentence is not understandable to the average reader, and it just goes deeper into the weeds from there. It needs a more general description for someone who did not major in math in college, and some reference to practical applications. I see that it is used to "check the differentiability of a complex function", to move a point function from cartesian to polar coordinates using D'Alembert–Euler condition and in triangular Mesh generation in computational fluid dynamics, for example to study flow around theoretical airfoil shapes without the time and expense of wind tunnel testing [1] [2]. But this is probably better included by someone more knowledgeable than me. Dhaluza (talk) 17:36, 13 March 2021 (UTC)

Definition in Clifford Algebra, what is $\sigma_1$ and $\sigma_2$?

Where do they come from? They must be some kind of operator like object, given that a product of them produces the identity matrix. 23.93.95.115 (talk) 00:11, 9 January 2024 (UTC)

The notation was bad. The product isn't the identity matrix, but is a matrix whose square is minus the identity. I have changed it to $\sigma _{1}\sigma _{2}=J$. A representation of the Clifford algebra is $\sigma _{1}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}$ and $\sigma _{2}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}$. Tito Omburo (talk) 11:49, 9 January 2024 (UTC)

Mistake in the first image

Hi, I'm not really a wikipedian and i have very little experience with editing wikipedia but I've noticed a mistake in the first image of this article (Cauchy-Riemann.svg). the two expressions on the bottom right shouldn't have a Z inside of the function since they describe the situation where you only multiply by z after applying the mapping. these expressions should be zdf(X) and df(X). I don't know how to fix this myself but I didn't see anyone else pointing the mistake. 109.253.198.52 (talk) 13:16, 22 April 2024 (UTC)

Yes, this is an error. Tito Omburo (talk) 09:57, 24 April 2024 (UTC)