Chapter 1 Definitions

Question 1

Region

Answer 1

A connected open set

Question 2

Bounded

Answer 2

The set G is bounded if G is a subset D[0,r] for some r.

Question 3

Open Set

Answer 3

A Set is open if all it’s points are interior points

Question 4

Closed Set

Answer 4

A set is closed if it contains all its boundary points.

Question 5

Interior points

Answer 5

Suppose G is a subset of C

a € G is an interior point if some open disk with center a is a subset of G

Question 6

Isolated points

Answer 6

G is a subset of C

A point d € G is an isolated point of G if some open disk centered at d contains no points of G other than d.

Question 7

Holomorphic

Answer 7

The function f is holomorphic on the open set E subset G if it is differentiable at every point in E.

If a function is holomorphic on C it is said to be an entire function.

For it to be holomorphic is must be and open and nonempty set.

Question 8

Open Disk

Answer 8

D[a,r]= { z€ C : |z-a| < r } is an open disk centered at a with radius r located inside circle C[a,r]= { z€ C : |z-a| < r }

Question 9

Entire

Answer 9

A function that is differentiable in the whole complex plane C

Question 10

Conformal

Answer 10

A holomorphic function with nonzero derivative preserves angles.

Question 11

Continuous

Answer 11

Suppose f: G -> C and z_o € G

f is continuous at z_o if for very positive real number epsilon there is a positive real number delta such that

|f(z)-f(z_o)| < epsilon for all z € G

Satisfying |z-z_o| < delta

Question 12

Boundary Point

Answer 12

Suppose G is a subset of C, b € G is a boundary point of G if every open disk centered at b contains a point in G and also a point that is not in G

Question 13

Accumulation Point

Answer 13

Suppose G is a subset of C, c € G is an accumulation point of G if every open disk centered at c contains a point of G different from c