Chapter 1

Question 1

Open ball

Answer 1

The open ball with centre zo and radius e, B(zo , e), is the set {z e (C : | z - zo| < e}

Question 2

Punctured open ball

Answer 2

The punctured open ball with centre zo and radius e, B(zo , e), is the set { z e (C : 0 < |z - zo| < e.

Sometimes called a disc

Question 3

Interior point

Answer 3

Suppose that S C_ (C. For any point zo in (C, provided that e is a sufficiently small positive real number, B(zo, e) is a subset of S, that is, B( zo, e) n S = B(zo, e).

In this case, zo is an interior point.

Question 4

Exterior point

Answer 4

Provided that e is sufficiently small positive real number, B(zo , e) does not meet S, that is, B(zo , e) n S = empty set.

In this case, zo is an exterior point of S.

Question 5

Boundary point

Answer 5

No matter how small the positive real number e is, neither of the above holds, that is empty C B(zo, e) n S C B(zo , e)

In this case, zo is the boundary point of S.

Question 6

Open set

Answer 6

The set S is open if all the points are interior points

Question 7

Closed set

Answer 7

The set S is closed if it contains all of its boundary points or equivalently the complement of the set is open

Question 8

Bounded set

Answer 8

The set is bounded is SC_ B(0, R) for some number R.

Question 9

Compact

Answer 9

The set S is compact if it is closed and bounded

Question 10

Polygonal path

Answer 10

A polygonal path is a finite sequence of finite line segments, where the end point of one line segment is the initial point of the next one.

Question 11

A closed polygonal path

Answer 11

A closed polygonal path is a polygonal path where the final point of the last segment is the initial point of the first segment.

Question 12

Connected set

Answer 12

Suppose that S C_ (C and that S is open.

The set S is connected if any two points of S can be joined by a polygonal path lying inside S

Question 13

Simply connected

Answer 13

The S is simply connected if any closed polygonal path can be shrunk to a point, staying inside the set.

(It has no holes)

Question 14

Domain (set)

Answer 14

The set S is a domain if it is connected and open

Question 15

Domain (function)

Answer 15

The domain is the set of numbers you put in

Question 16

Natural domain

Answer 16

The largest domain possible

Question 17

Co-domain

Answer 17

The co-domain is the set of number you can get out and perhaps others

Easier to calculate than the range

Question 18

Range

Answer 18

The range is the set of the numbers you can get out and no others

Sometimes hard to calculate

Question 19

Complex function

Answer 19

A complex function is one whose domain, or range, or both is a subset of the complex plane (C that is not a subset of the real line R

Question 20

Function of a complex variable

Answer 20

When the domain is complex, not real

Question 21

Complex-value function

Answer 21

When the range is complex, not real

Question 22

Complex polynomial

Answer 22

A complex polynomial is a function p: (C -> (C of the form
P(z) = amz^m + … + a1z + a0
where am, …, a1, a0 e (C. If am is not equal to zero, we say that polynomial is of degree m.

Question 23

Fundamental theorem of algebra

Answer 23

Every nonconstant complex polynomial p of degree m factorises: there exists complex number a1, …, am and c such that
p(z) = c product (from j=1, …, m) of (z-aj)

Question 24

Range and domain of any complex polynomial is (C

Answer 24

The natural domain of any complex polynomial is (C.

If the polynomial is nonconstant then the range is also (C