Sec: 25-27 Analytic functions, Harmonic functions

Question 1

Definition of function f analytic on an open set S

Answer 1

If f has a derivative at each point in S

Question 2

Definition of function f analytic at a point $z_0$

Answer 2

If f is analytic is some neighbourhood of $z_0$

neighbourhood includes the point itself

Question 3

Entire function

Answer 3

Function that is analytic at each point in C

Question 4

Singular point

$z_0$ is a singular point of f

Answer 4

if f is not analytic at $z_0$, but f is analytic at some point in every neighbourhood of $z_0$

Question 5

Isolated singularity

Answer 5

f is not analytic at point of isolated singularty $z_0$, but f is analytic in some deleted neighbourhood of $z_0$

check if this deleted part is just uselss

$\frac{1}{z}$ has an isolated singularity at z=0

Question 6

Laplace’s equation

Answer 6

$H_{xx}(x,y) + H_{yy}(x,y) = 0$

Question 7

Harmonic function

A real valued function H of 2 real variables x,y is said to be harmonic in a domain D of the xy plane if…

Answer 7

1. All its partial derivatives of the first AND second orders are continous
   and
2. It satisfied Laplace’s equation
   $$H_{xx}(x,y) + H_{yy}(x,y) = 0$$

Question 8

Theorem pg 78

Relation between analyticity and harmonic functions

Answer 8

If a function $f(z) = u(x,y) +iv(x,y)$ is analytic in a domain D, then its component functions u and v are HARMONIC in D

more so, they will actually have partials that are continuoes for all orders by other theorem CHECK

Question 9

Corallary p.169

Relation between analyticity of a function and partials

Answer 9

If $f(z) = u(x,y) +iv(x,y)$ is analytic at $x+iy$ then BOTH u and v have continuous partial derivatives of all orders at $(x,y)$

CHECK
all orders syas here but Harmonic functions theorem says 2 orders