Radius of convergence of laurent series

1. What is the radius of convergence of Laurent series?
2. What is region of convergence in Laurent series?
3. How do you find the radius of convergence?
4. Do Laurent series converge uniformly?

What is the radius of convergence of Laurent series?

= lim |z| n + 1 = 0. Since L < 1 this series converges for every z. Thus, by Theorem 7.1, the radius of conver- gence for this series is ∞.

What is region of convergence in Laurent series?

Therefore, the Laurent series is. f ( z ) = 1 2 . 1 z − i + 1 4 i ∑ n = 0 ∞ ( − z − i 2 i ) n. As we know, the principal part is given by the first term. And, the region of convergence is 0 < |z − i| < 2.

How do you find the radius of convergence?

The radius of convergence is half of the length of the interval of convergence. If the radius of convergence is R then the interval of convergence will include the open interval: (a − R, a + R). To find the radius of convergence, R, you use the Ratio Test.

Do Laurent series converge uniformly?

Theorem 0.1. For the Laurent series above, if 1/R1 < R2, then the Laurent series 0.1 converges for all z ∈ C such that 1/R1 < |z - a| < R2. Moreover, the convergence is uniform and absolute in the region r1 ≤ |z - a| ≤ r2 for any r1,r2 satisfying 1/R1 < r1 < r2 < R2.