Chapter 1 Flashcards
Open ball
The open ball with centre zo and radius e, B(zo , e), is the set {z e (C : | z - zo| < e}
Punctured open ball
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
Interior point
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.
Exterior point
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.
Boundary point
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.
Open set
The set S is open if all the points are interior points
Closed set
The set S is closed if it contains all of its boundary points or equivalently the complement of the set is open
Bounded set
The set is bounded is SC_ B(0, R) for some number R.
Compact
The set S is compact if it is closed and bounded
Polygonal path
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.
A closed polygonal path
A closed polygonal path is a polygonal path where the final point of the last segment is the initial point of the first segment.
Connected set
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
Simply connected
The S is simply connected if any closed polygonal path can be shrunk to a point, staying inside the set.
(It has no holes)
Domain (set)
The set S is a domain if it is connected and open
Domain (function)
The domain is the set of numbers you put in
Natural domain
The largest domain possible
Co-domain
The co-domain is the set of number you can get out and perhaps others
Easier to calculate than the range
Range
The range is the set of the numbers you can get out and no others
Sometimes hard to calculate
Complex function
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
Function of a complex variable
When the domain is complex, not real
Complex-value function
When the range is complex, not real
Complex polynomial
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.
Fundamental theorem of algebra
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)
Range and domain of any complex polynomial is (C
The natural domain of any complex polynomial is (C.
If the polynomial is nonconstant then the range is also (C
