The Cauchy-Riemann equations are probably the most straightforward way to determine complex differentiability of $ f(z) = |z|^{10}Re(z^2) $, but there's supposedly a good way to solve this without using them. However, I'm failing to see it.
Differentiability of $ |z|^{10} Re(z^2) $ without Cauchy-Riemann equations
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complex-analysis
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0Hint: purely real-valued non-constant functions cannot be complex-differentiable... – 2017-01-23
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0@paulgarrett that implicitly uses CR equations. No? – 2017-01-23
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1@OpenBall, that purely real-valued (non-constant) functions are not complex differentiable can certainly be done via the Cauchy-Riemann equations, which at least is more economical than actually applying the C-R equations to a complicated explicit formulaic object. But, also, for example, composing with (for example) the sine function, one would have a bounded entire (non-constant) function, contradicting Liouville's theorem. Also, for example, a purely real-valued (non-constant) function contradicts the open-mapping property of holomorphic functions. Lots of possibilities... – 2017-01-23
1 Answers
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Let $z=x+yi$
And complex numbers as a real plane.
$f:\mathbb{R}^2 \mapsto \mathbb{R}$
$f(x,y)=(x^2+y^2)^5(x^2-y^2)=|z|^{10}Re(z)$
With derivatives:
$\frac{d}{dx}f(x,y)=20x^2(x^2+y^2)^4$, $\frac{d}{dy}f(x,y)=-20y^2(x^2+y^2)^4$