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Question:

Find all points at which the complex valued function $f$ define by $$f(z)=(2+i)z^3-iz^2+4z-(1+7i)$$ has a derivative.


I know that $z^3$,$z^2$, and $z$ are differentiable everywhere in the domain of $f$, but how can I write my answer formally? Please can somebody help?

Note:I want to solve the problem without using Cauchy-Riemann equations.

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So $z\in \mathbb{C}$, I guess? –  draks ... Mar 1 '12 at 7:40
    
@draks: yes $z$ is in $\mathbb C$ –  Hassan Muhammad Mar 1 '12 at 9:28
    
Hint: put z=x+iy and then get f(x+iy)=U(x,y)+iV(x,y). You can check the question and follow ways in answers math.stackexchange.com/questions/102623/… –  Mathlover Mar 1 '12 at 9:54
    
@Mathlover: A good hint, but I don't want to use Cauchy-Riemann equations. –  Hassan Muhammad Mar 1 '12 at 10:20
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You can just write the definition of a derivative (ie. take $z\in\mathbb{C}$ and find $\lim_{h\rightarrow 0}{f(z+h)-f(z) \over h}$), considering it's a polynomial it shouldn't be too hard to find the answer. –  Najib Idrissi Mar 1 '12 at 12:23

1 Answer 1

up vote 1 down vote accepted

I'm not sure where the question is coming from (what you know/can know/etc.).

But some things that you might use: you might just give the derivative, if you know how to take it. Perhaps you might verify it with the Cauchy-Riemann equations. Alternatively, differentiation is linear (which you might prove, if you haven't), and finite linear combination of differentiable functions is also differentiable. Or you know the series expansion, it's finite, and converges in infinite radius - thus it's holomorphic.

But any of these would lead to a complete solution. Does that make sense?

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yes it make sense, but supposing I do not know how to use Cauchy-Riemann equations in fact I don't want to use C-R equation. If I understand you, since finite linear combination of differentiable functions is also differentiable then the whole function $f$ is differentiable. –  Hassan Muhammad Mar 1 '12 at 9:36

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