Complex Analysis- Lecture 13 - Continuity

Theorems of limits:-
Theorem 1:- (Uniqueness of limit)
If a function f has a limit at Z_0 then the limit of function f at Z_0 is unique. That is we can t find l_1,l_2 such that lim?(Z?Z_0 )?f(Z)=l_1 & lim?(Z?Z_0 )?f(Z)=l_2.
Theorem 2:- Let f(Z)=u(x,y)+iv(x,y), then lim?(Z?Z_0 )?f(Z)=u_0+?iv?_0=?_0 if and only if (iff) lim??(x?x_0@y?y_0 )??u(x,y)=u_0 ?, lim??(x?x_0@y?y_0 )?v(x,y)=v_0 where Z_0=x_0+?iy?_0 .
Theorem(3):- Let f(Z) and g(Z) are two functions, such that lim?(Z?Z_0 )??f(Z)=L ?& lim?(Z?Z_0 )?g(Z)=M, then
lim?(Z?Z_0 ) [f(Z)±g(Z) ]=lim?(Z?Z_0 ) f(Z)±lim?(Z?Z_0 ) g(Z)=L±M
lim?(Z?Z_0 ) [f(Z) ?g(Z) ]=(lim?(Z?Z_0 ) f(Z) )?(lim?(Z?Z_0 ) g(Z) )=LM
lim?(Z?Z_0 ) [f(Z)/g(Z) ]=(lim?(Z?Z_0 ) f(Z))/(lim?(Z?Z_0 ) g(Z) )=L/M
Theorem(4):-
The limit of a constant function is constant.
Let f(Z)=c, then lim?(Z?Z_0 ) c=c
Let f(Z)=Z ,then lim?(Z?Z_0 )?f(Z)=lim?(Z?Z_0 ) Z=Z_0
Let f(Z)=Z^2 , then lim?(Z?Z_0 )?f(Z)=lim?(Z?Z_0 ) Z^2=Z_0^2
If P(Z)=?_(k=0)^n??a_k Z^k=a_0+a_1 Z+a_2 Z^2+?+a_n Z^n ? is polynomial function with coefficient a_1,a_2,….,a_n are complex numbers, then lim?(Z?Z_0 ) P(Z)=P(Z_0 ).
If lim?(Z?Z_0 )??f(Z)=L?, then lim?(Z?Z_0 ) |f(Z) |=|lim?(Z?Z_0 ) f(Z) |=|L|
EX:-Find
lim?(Z?(1+i) )??(5Z+1)/(5Z-i)=(lim?(Z?(1+i) ) (5Z+1))/(lim?(Z?(1+i) ) (5Z-i) )?=(lim?(Z?(1+i) ) (5Z)+lim?(Z?(1+i) ) 1)/(lim?(Z?(1+i) ) (5Z)-lim?(Z?(1+i) ) i)
= (5(1+i)+1)/(5(1+i)-i)=(5+5i+1)/(5+5i-i)=(6+5i)/(5+4i)?(5-4i)/(5-4i)
=([30-(-20) ]+[(-24)+25]i)/(25+16)=((30+20)+i)/41=50/41+i 1/41
lim?(Z?2i)??(Z^2+4Z-2)=? lim?(Z?2i) Z^2+lim?(Z?2i) 4Z-lim?(Z?2i) 2
=(2i)^2+4(2i)-2=-6+8i
lim?(Z?i)??(iZ^3-1)/(Z+i)? H.W.
2.4 Continuity:-
Def:- Let f(Z) be defined and single-valued function in a neighborhood of Z_0. The function f(Z) is said to be continuous at Z_0 if lim?(Z?Z_0 )?f(Z) is exist .
Note that this implies three conditions which must be met in order that f(Z) be continuous at Z_0 :-
lim?(Z?Z_0 ) f(Z)=L must exist.
f(Z_0 ) must exist , i.e. f(Z) is defined at Z_0 .
L=f(Z_0 ).
Equivalently, if f(Z) is continuous at Z_0, we can write this in the suggestive formlim?(Z?Z_0 ) f(Z)=f(lim?(Z?Z_0 ) Z).
Alternative to above definition of continuity, we can defined f(Z) as continuous at Z_0 if for any ?>0 we can find ?>0 such that |f(Z)-f(Z_0)|