Definitions

Question 1

Starlike

Answer 1

A domain is called starlike if there exists a point in D such that for any other point in D there is a straight line connecting a and b entirely inside D

Question 2

Biholomorphic

Answer 2

Let D and D’ be domains. We say that f:D -> D’ is biholomorphic if f is holomorphic and a bijection, the inverse of which is also holomorphic.

Question 3

Domain

Answer 3

An open path connected set.

Question 4

Metric Space

Answer 4

A set with a function satisfying positivity, symmetry and the triangle inequality.

Question 5

Meromorphic

Answer 5

A function is meromorphic on a domain D if f is holomorphic on D-S where S has no isolated points and f has a pole at each element of S.

Question 6

Principal Part

Answer 6

The negative section of a laurent series.

Question 7

Simple Closed Curve

Answer 7

A closed curve is called simple if f(t1) = f(t2) for any t1

Question 8

Simply connected

Answer 8

An open set is called simply connected if every closed contour is homologous to 0 i.e. winding number 0.

Question 9

Holomorphic on an Open set

Answer 9

A complex function is holomorphic on an open set if it is complex differentiable at every point in the open set.

Question 10

Holomorphic at a point.

Answer 10

A complex function is holomorphic at a point in an open set if it is holomorphic on some open ball around that point.

Question 11

Conformal

Answer 11

We say a real differentiable map on a domain is conformal at a point if it preserves the angle and orientation between any two tangent vectors at that point.

Question 12

Uniform Convergence

Answer 12

A sequence of functions converges uniformly if for every epsilon greater than zero there exists an N in the natural numbers such that for every n>N d(fn(x),f(x))

Question 13

Complex Differentiable

Answer 13

A function is complex differentiable at a point a in the open set if the limit from z to a of f(z)-f(a)/z-a exists.

Question 14

Non isolated point

Answer 14

Given a subset S of C we say that a point w in S is a non isolated point if for every epsilon greater than 0 there exists a z in S such that the 0

Question 15

Harmonic Function

Answer 15

A real valued function is harmonic on a domain if it has continuous second order partial differentials and Uxx+Uyy = 0

Question 16

Local Uniform Convergence

Answer 16

A sequence of functions converges locally uniformly on X to f if for every x in X there exists an open set U in X such that x is in U and fn converges to f uniformly.

Question 17

Length of a contour

Answer 17

The integral between the end points of the modulus of the differential of the contour.

Question 18

Compactness

Answer 18

A non empty subset K of a metric space X is called compact if for any sequence in K there exists a convergent subsequence with limit in K.

Question 19

Open Set

Answer 19

A subset is open if for every x in the subset there exists an epsilon such that the ball of radius epsilon around x is contained in the subset.

Question 20

Closed Set

Answer 20

A subset is closed if its complement is open.

Question 21

Convergence in a metric space

Answer 21

A sequence is convergent in a metric space if for every epsilon greater than zero there exists an N in the natural numbers such that d(xn,x)N

Question 22

Continuity

Answer 22

For every epsilon greater than 0 there exists a delta greater than 0 such that for every x in X1, d(x,x0)

Question 23

Pointwise convergent

Answer 23

For every x in X and for every epsilon>0 there exists an N in the natural numbers such that for every n >N d(fn(x),f(x))