take some complex valued function f(z) = f(x + iy) = u(x,y) +iv(x,y) with u,v,x,y real valued variables/functions there are relationships between the partial derivites of u and v that must hold in order for the function to be analytic. I'm assuming by analytic that means that the function is differentiable. secondly, the anti-symmetry minus sign is reminiscent of a conformal map... but i'm a little lost as to the physical significance can someone explain this. I honestly don't have a good question other than i d like someone to discuss a simple example of this or something of why these conditions must hold and how that makes sense. say we had the function f(z) = 1 over the whole complex plane. would that satisfy the cauchy riemann conditions or would there be some discontinuity? can you give me an example of a function that satisfies this over the complex plane?
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closed as off topic by Raskolnikov, Kostya, Ron Maimon, David Zaslavsky♦ Jun 3 '12 at 21:40
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This is quite probably best suited to math.SE, but here goes: The Cauchy-Riemann equations are the conditions the real and imaginary part of a function $f=u+iv:\mathbb{C}\rightarrow\mathbb{C}$ for it to be differentiable in the complex sense. More specifically, if you take $u$ and $v$ to be differentiable at some point $z_0$ then the limits $$\lim_{|h|\rightarrow0}\frac{f(z_0+|h|e^{i\theta})-f(z_0)}{|h|e^{i\theta}}$$ will exist for any given $\theta\in\mathbb{R}$. However, you'd like these limits to be independent of $\theta$, in order for the (complex) limit $$\lim_{h\rightarrow0}\frac{f(z_0+h)-f(z_0)}{h}$$ to exist. This enforces an extra condition on the derivatives of $u$ and $v$, the Cauchy-Riemann equations. Because they are a differential-equations kind of condition (as opposed to local, limit-existence kind of conditions), their solutions are far more rigid than real-differentiable functions. For example, if a function is complex-$C^1$ in some open set, then it is complex-$C^\infty$ and equal to its Taylor series there. Also, path integration is path-independent so long as you stay within domains of analyticity. Regarding conformal mapping, you're quite right: any analytical function is a conformal mapping except where its derivative vanishes. In terms of real-world examples, pretty much anything that looks differentiable will be analytic, with the notable exception of things that involve complex conjugates or (equivalently) moduli. Thus all polynomials, radicals, trigonometric functions, logarithms and exponentials (i.e. $e^{\alpha z}$) will be analytic just about everywhere, though care must be taken with poles, branch cuts and other singularities. Given the practically endless number of applications of complex numbers and complex functions in physics, it's hard to come up with a "physical significance" for the C-R equations, as that will depend on the specific problem you're dealing with. Some examples that come to (my) mind are in the theory of 2D airfoils and in quantum field theory |
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