المحاضرة السادسة عشر

52
????? ول ا ? ت??? ر??
Uniqueness of Limits 1
Z0 f
2
f (z) = u (x,y) + iv (x,y)
Lim f(z) = u0 + iv0 = w0
Lim u (x , y) = u0
Lim v (x,y) = v0
z0 = x0 + iy0
3
z g (z ), f (z)
Lim g(z) = M , Lim f(z) = L
1- Lim [ f (z) ? g (z) ] = L ? M
2- Lim f (z) . g (z) = L . M
3- Lim = ; M ?0
Lim
z z0
x x0
y y0
x x0
y y0
z z0
z z0
z z0
z z0
f(z)
g(z)
L
M
5Z + 1
z 1+i 5Z - i
z z0
53
Lim = [ ]
= =
= +
?? و??? ن? ر???
1
Lim
Q3/ Show that
Q4/ Prove that
Q5/ Find
Q6/ Compute
Q7/ by using the def. of limit find
Q8/ Find
Q9/ Compute
Q10/ Find
5Z + 1
5Z - i
5( 1 + i) + 1
5(1 + I ) + i
6 + 5i
5 + 4i
50
41
i
41
Z3 + 8
Z 2ei Z4 + 4z2 + 16
3
z 1+i
Lim(5Z) + Lim 1
Lim(5Z) + Lim i
z 1+i z 1+i
z 1+i z 1+i
2 2
lim1
iz i
z
?
?
z
z
z 0 lim?
1
lim 2 ?
?
? z
z i
z i
lim2 ? ?2
? iz z i
z z
z
z?1 ? lim
1
lim 3 3 1
3 2
1 ?
? ? ?
? ? z
z z z
z
z i
z i
z ?
?
? ?
3
1 lim
?
?
iz iz z z lim3 ? 3
?52
????? ول ا ? ت??? ر??
Uniqueness of Limits 1
Z0 f
2
f (z) = u (x,y) + iv (x,y)
Lim f(z) = u0 + iv0 = w0
Lim u (x , y) = u0
Lim v (x,y) = v0
z0 = x0 + iy0
3
z g (z ), f (z)
Lim g(z) = M , Lim f(z) = L
1- Lim [ f (z) ? g (z) ] = L ? M
2- Lim f (z) . g (z) = L . M
3- Lim = ; M ?0
Lim
z z0
x x0
y y0
x x0
y y0
z z0
z z0
z z0
z z0
f(z)
g(z)
L
M
5Z + 1
z 1+i 5Z - i
z z0
53
Lim = [ ]
= =
= +
?? و??? ن? ر???
1
Lim
Q3/ Show that
Q4/ Prove that
Q5/ Find
Q6/ Compute
Q7/ by using the def. of limit find
Q8/ Find
Q9/ Compute
Q10/ Find
5Z + 1
5Z - i
5( 1 + i) + 1
5(1 + I ) + i
6 + 5i
5 + 4i
50
41
i
41
Z3 + 8
Z 2ei Z4 + 4z2 + 16
3
z 1+i
Lim(5Z) + Lim 1
Lim(5Z) + Lim i
z 1+i z 1+i
z 1+i z 1+i
2 2
lim1
iz i
z
?
?
z
z
z 0 lim?
1
lim 2 ?
?
? z
z i
z i
lim2 ? ?2
? iz z i
z z
z
z?1 ? lim
1
lim 3 3 1
3 2
1 ?
? ? ?
? ? z
z z z
z
z i
z i
z ?
?
? ?
3
1 lim
?
?
iz iz z z lim3 ? 3
?