Consider the ball $$B = \{ \left| z_1 \right|^2 + \left| z_2 \right|^2 <1 \} \subset \mathbb{C}^2.$$ I want to construct a power series with domain of convergence $B$. My thoughts so far have been to consider a function of the form $$\frac{z_1 z_2}{1 - z_1^2 - z_2^2} = \sum_{\mu = 0}^{\infty} (z_1z_2(z_1^2-z_2^2))^{\mu}.$$ But I don't think this is correct.
Construct power series with domain of convergence
2
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complex-analysis
power-series
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0Why not add two series one in $z_1$ and the other in $z_2$ with the relevant ROC? – 2017-02-05
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0@copper.hat Forgive me, I wrote the incorrect ROC. Please see edit. – 2017-02-05
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0Your example converges whenever $z_1=\pm z_2$ regardless of the value of $|z_1|^2+|z_2|^2.$ – 2017-02-05