المحاضرة العشرون

61
u ,v 1
2 1 2
f (z) = z2 + 5iz + 3 – i
z = x + iy
f(z) = (x2 – y2 – 5y +3) + i (2xy + 5x -1)
u = x2 – y2 – 5y + 3
v = 2xy + 5x – 1
u x = 2x = v y
v x = 2y +5 = -u y
u , v
f (z)
= 2x + i (2y + 5)
= 2z + 5i
f (z)
w = f (z) = 2
= x2 + y2
z = 0 z = 0
u = x2 + y2 ; v = 0
vx = 0 ; vy = 0 ux= 2x ; uy = 2y
f(z) = ?z?2 z = 0 + i0
z
f ?(z)
62
1
ez = u + i v i
v = ex siny u = ex cosy
d x
e y d u
d x
d v ? x cos ?
d y
e y d u
d x
d v ? x sin ??
y, x v, u ii
ez ez
d z
d ( ) ?
z ez ez
ez
2
u = Log r , v= ? Log z = u + i v = Log v + i ?
? d ?
d u
d r r
d v
d r
d v
d r r
d u ? 1 ? 1 , ? 0 ? ? 161
u ,v 1
2 1 2
f (z) = z2 + 5iz + 3 – i
z = x + iy
f(z) = (x2 – y2 – 5y +3) + i (2xy + 5x -1)
u = x2 – y2 – 5y + 3
v = 2xy + 5x – 1
u x = 2x = v y
v x = 2y +5 = -u y
u , v
f (z)
= 2x + i (2y + 5)
= 2z + 5i
f (z)
w = f (z) = 2
= x2 + y2
z = 0 z = 0
u = x2 + y2 ; v = 0
vx = 0 ; vy = 0 ux= 2x ; uy = 2y
f(z) = ?z?2 z = 0 + i0
z
f ?(z)
62
1
ez = u + i v i
v = ex siny u = ex cosy
d x
e y d u
d x
d v ? x cos ?
d y
e y d u
d x
d v ? x sin ??
y, x v, u ii
ez ez
d z
d ( ) ?
z ez ez
ez
2
u = Log r , v= ? Log z = u + i v = Log v + i ?
? d ?
d u
d r r
d v
d r
d v
d r r
d u ? 1 ? 1 , ? 0 ? ? 161
u ,v 1
2 1 2
f (z) = z2 + 5iz + 3 – i
z = x + iy
f(z) = (x2 – y2 – 5y +3) + i (2xy + 5x -1)
u = x2 – y2 – 5y + 3
v = 2xy + 5x – 1
u x = 2x = v y
v x = 2y +5 = -u y
u , v
f (z)
= 2x + i (2y + 5)
= 2z + 5i
f (z)
w = f (z) = 2
= x2 + y2
z = 0 z = 0
u = x2 + y2 ; v = 0
vx = 0 ; vy = 0 ux= 2x ; uy = 2y
f(z) = ?z?2 z = 0 + i0
z
f ?(z)
62
1
ez = u + i v i
v = ex siny u = ex cosy
d x
e y d u
d x
d v ? x cos ?
d y
e y d u
d x
d v ? x sin ??
y, x v, u ii
ez ez
d z
d ( ) ?
z ez ez
ez
2
u = Log r , v= ? Log z = u + i v = Log v + i ?
? d ?
d u
d r r
d v
d r
d v
d r r
d u ? 1 ? 1 , ? 0 ? ? 1