Ch 2: Analytic functions, Sec 15-16: Limits

Question 1

Formal definition of a limit for a complex valued function

FIX

Answer 1

f has a limit $w_0$ at $z_0$ if $\forall \epsilon > 0 \ \exists \delta > 0 $ st if
$z \in D_f $ and
$0< |z-z_0|$ then

Question 2

Uniqueness of complex limits

Answer 2

If the limit of a complex valued function f exsists, then it is unique

Question 3

Limits of complex numbers means what for ‘components’

if $f(z) = u(x,y) + v(x,y)$ where $z = x+iy, \quad z_0 = x_0 +i y_0 \quad w_0 = u _0 + iv_0$ then…

Answer 3

the limit as $(x,y) \rightarrow (x_0,y_0)$ of $u(x,y) =u_0$

and

the limit as $(x,y) \rightarrow (x_0,y_0)$ of $v(x,y) =v_0$

iff
the limit as $z \rightarrow z_0$ of $f(z) =w_0$

Question 4

Limit rules hold as in the real case for

Answer 4

addition, ,ultiplication, division (given that deno limit not 0)