Complex Analysis

accumulation point

A point z0 is an accumulation point of S in C if every deleted of z0 contains at least one point of S.

Bounded

A subset S of C is bounded if there exists r>0 s.t. S is a subset of {z | |z| < r}

differentiable at z0

If f'(z0) exists

Connected

If given any two points z1, z2 in S, there is a path in S from z1 to z2

harmonic conjugate

If u(x,y) is harmonic on a domain D, then a function v(x,y) defined on D so that f(z)=f(x+iy)=u(x,y)+iv(x,y) is analytic on domain D is called a harmonic conjugate of u on D

Real & Imaginery

If z=x+iy, then the real of part z is Re(z)=x and the imaginary part of z is Im(z)=y.

analytic

f(z) is analytic at z0 if f is differentiable at every point in some nbhd of z0

entire

f(z) is entire if f is differentiable and hence analytic at every z0 in C

definite integral of w(t)

int(w(t)dt)=int(u(t)dt) + i int(v(t)dt)

log z

log(z)=ln |z| + iarg(z) multivalued

Product of two complex numbers

z1z2=(x1+iy1)(x2+iy2)=(x1x2-y1y2)+i(x1y2+y1x2) Geometrically:

Exponential form

z=re^itheta

complex exponents

z^c=exp(c*logz)

deleted neighborhood

{ z | 0 < |z - z0| < E}

neighborhood of z0

{ z | |z-z0| < E}

Neighborhood of ∞

{ z | |z| > 1/E

periodicity

y1=y2+2nπ for n in Z

Simple Closed Contour (SCC)

A contour C is a SCC if there is a parametrization z=[a,b] -> C of Contour s.t. z(a)=z(b), but z is otherwise 1-1

Domain and region

Domain: A domain is a connected open set Region: A region is a domain together with none, some, or all of its boundary points

triangle inequality

For z1, z2 in C, |z1+z2| ≤ |z1| + |z2|

length

L(c)= int(a,b) of |z'(t)|

ML Theorem

Let C be a contour of length L and let f(z) be piecewise continuous on C. Suppose there exists M>0 s.t. |f(z)| -< for all z on C. Then | int(C) of f(z)dz| -< ML.

contour integral

Let Contour C be a smooth arc and f bee a complex function defined on C. Then int(C) of f(z)dz =int(C) of f(z)dz = int (a,b) of f(z(t))z'(t)dt where z(t): [a,b] -> C is a smooth parametrization of C.

harmonic

Let H(x,y) have continuous second partials in a domain D. Then H is harmonic in D if Hxx+Hyy=0.

Interior, exterior, and boundary points

Let S be a subset of C. Interior: z is an interior point of S if there is some epsilon nbhd of z that is completely contained in C. Exterior: z is an exterior point of S if there is some epsilon nbhd of z that is disjoint from S. Boundary point: z is a boundary point of S if every epsilon nbhd of z contains at least one point in S and at least one point not in S

Open and closed sets

Let S be a subset of C. Open: If S contains none of its boundary point. Closed: If S contains all of its boundary points

continuity

Let f be defined in a nbhd of z. Then f is continuous at z0 if lim z->z0 (f(z))=f(z0).

Lim z->z0 (fz)=w0

Let f be defined in a neighborhood of z0. Then lim z->z0 (fz)=w0 if for all E>0 there exists S>0 s.t. |f(z)-w0) < E for all z satisfying 0 < |z-z0| < S.

contour

a finite number of smooth arcs joined end to end

closure

S together with all its boundary points

exp(z)

exp(z)=exp(x+iy)=e^xcosy+e^xisiny

modulus

The modulus of a complex number is its distance to the origin in the complex plane/ |z|=sqrt(x^2+y^2) when z=x+iy

Principal argument Arg z

The unique angle in arg z so that -π≤Arg z≤π

arc

a set of pints C= {z(t) = x(t) +iy(t) | t in [a,b]} where x,y are continuous on [a,b]

argument z

arg z = {theta | z = |z| (cos(theta) + isin(theta)

hyperbolic functions

coshz=exp(z)+exp(-z)/2=coshxcosy+isinhxsiny sinhz=exp(z)-exp(-z)/2=sinhxcosy+icoshxsiny

trigonometric functions

cosz= exp(iz)+exp(-iz)/2=cosxcosh7-isinxsinhy sinz+exp(iz)-exp(-iz)/2i=sinxcoshy+icosxsinhy

derivative of f(z)

f'(z0)=lim z->z0 (f(z)-f(z0))/(z-z0)

principal root

r0^1/ne^i(Argz/n)

smooth arc

the image of a differentiable z(t): [a,b] -> C s.t. z'(t)=/0 on (a,b)