Yohance Defns Flashcards
Defn of a subset U being open
Defn of neighbourhood
Defn of Function being continious
Defn of subset being compact
Details of mobius transformation
Defn of function being analytic (power series)
Defn of CR eqns in both real form and imaginary form
Defn of partial derivative wrt to z and z*
Defn of a function u(x,y) being harmonic
defn of v being harmonic conjugate to u
Defn of a curve being closed
Suppose f is continious and gamma is a smoth curve, define the intergral of f.
also define the parametiztion f = P +iQ
Define L(z)
statement about polynomials factorisaiton and zeros, with order of zeros
Defn of isolated singularity
Defn of Laurent Expansion. When is z_0 a removable singularity? defn of pole of order m?
defn of essential singularity?
If z_0 is an isolated singularity, defn the principle part
Define the residue of f at z_0
Define the principle part of z_0 if;
i) singularity is removable
ii) singularity is a pole
iii) in the case of an essential singularity
Defn of entire. Defn of meromophic
Statement about continuity of f(z) where f(z) is described as a power series
Maclaurin Taylor expansion of f(z)
Requirment of conformality of a function f(z)
if f’(z) = 0 for all z in complex plane then …
Suppose u is harmonic in complex plane, give statement about existence of v
State Triangle inequality for intergrals
State ML Lemma
Give statement about path independence of intergral
State Fundamental Thm of Calculus for complex plane
give statement about existance of F’(z) = f(z)
State Cauchys Thm
State Cauchys Intergral Formula
state result about intergral of 1/(z-a) along curve |z|=R for |a| > R and |a| < R
State Cauchys formula for its derivatives
Describe f as its taylor expansion and cauchy inequalities
State Louivilles Thm
State Fundemental Thm of Algebra
Give statement about zeros of non-zero polynomials
State Laurents Theorem
State Residue Theorem for Meromorphic functions
Suppose f(z) = h(z)/g(z) , give details about resiude of such function
Suppose f(z) = g(z) / (z-z_0)^m, give statement about residue of pole
State Jordans Lemma
State Rouches Thm
