Radius of convergence of an analytic function

**hatsoff** · Jun 6th 2011, 05:42 AM

This is from a practice exam:

Let $f(z)=\sum_{k=0}^\infty a_kz^k$ have a radius of convergence 2, and let $g(z)=\sum_{k=0}^\infty b_k(z-3)^k$ have a radius of convergence 2. Let $I=\{z:f(z),g(z)\text{ both converge}\}$ .

(a) If $f(z)=g(z)$ for $z\in I$ , prove there is an analytic function $w$ for which $w(z)=f(z)$ when $|z|<2$ and $w(z)=g(z)$ when $|z-3|<2$ .

(b) For the function $w$ of part (a), determine the radius of convergence $R$ for the power series $w(z)=\sum_{k=0}^\infty c_k\left(z-\frac{3}{2}\right)^k$ .

(c) What relationship exists between the coefficient sequence $\{a_k\}$ and the coefficient sequence $\{b_k\}$ ?

Part (a) is fairly straightforward. Let $A=\{z:|z|<2\}$ and $B=\{z:|z-3|<2\}$ . Then $A\cap B\subseteq I$ , which means we can define $w$ by $w=f'\cup g'$ , where $f'$ and $g'$ are the restrictions of $f$ and $g$ to $A$ and $B$ , respectively. It follows immediately that $w$ is analytic on $A\cup B$ .

However, I'm lost for parts (b) and (c). Any help would be much appreciated!

**girdav** · Jun 6th 2011, 06:28 AM

The series is convergent for the $z=-2+0.00000001$ and $z=5-0.0000000001$ for example, hence we can expect that the radius of convergence is greater than $\frac 72$ . Now you have to show it's indeed $\frac 72$ .

**hatsoff** · Jun 6th 2011, 08:13 AM

Hmm. I'm sorry to say I don't understand your reasoning here.

By Taylor's theorem $R\geq \sup\{r : D(3/2;r)\subseteq A\cup B\}=\frac{\sqrt{7}}{2}$ . (Here I use the notation $D (z_0;r)=\{z:|z-z_0|<r\}$ ). However, I don't see how we can get $R\geq\frac{7}{2}$ , nor $R\leq\frac{7}{2}$ .

**girdav** · Jun 7th 2011, 04:44 AM

The series is convergent on set a set which a diameter $\geq 7$ , because the series converges on points of the form $-2+\delta$ end $5-\delta$ for $1>\delta>0$

**hatsoff** · Jun 7th 2011, 02:47 PM

Oh, of course... I had forgotten every power series converges precisely on a disc and a subset of its boundary. Sorry about that.

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