المحاضرة الثامنة عشر

57
( Cauchy – Riemann Theorem ) ن??? ر ?? و? ??? ر??
z0 1
z0 w = f (z) = u (x , y) + iv (x ,y)
= - , = ……………. ( * )
z0 2
z0
x = r cos ? , y = r sin? , Z = r ei?
f (z) = u ( r , ? ) + i v ( r , ? )
r Ur = V? , r Vr = - U?
Ex: find if it exist when f(z) = r ei?
Sol .
f(z) = r ei?
= r ( cos ? + i sin ?) = r cos ? + i r sin ?
?u
?y
?v
?x
?u
?x
?v
?y
( ) ( , ) ( , )
( ) ( , ) ( , )
0 0 0 0 0
0 0 0 0 0
f z V x y iU x y
or
f z U x y iV x y
y y
x x
? ? ?
? ? ?
? ? ? ? ? ?
? e ? v iu
r
f z e u iv or f z i
r r
?( )? ? i ? ?( ) ? 1 ?
f ?(z)
y
v
x
v
y
u
x
u
?
?
?
?
?
?
?
?
, , ,
58
= u + i v
u = r cos ? ? u r = cos ? , u? = - r sin ?
v = r sin ? ? v r = sin ? , v? = r cos ?
Now r Ur = V? = r cos ? & r Vr = - U? = - r sin ?
Since Ur , U? , Vr , V? are cont. we get
f(z) is differentiable and equal
? ? ? ?
( ) 1
. 1
( ) cos sin
? ? ?
? ?
? ? ? ? ?
?
? ?
f z
e e
f z e u iv e i
i i
i
r r
i
? ?
? ? ? ?57
( Cauchy – Riemann Theorem ) ن??? ر ?? و? ??? ر??
z0 1
z0 w = f (z) = u (x , y) + iv (x ,y)
= - , = ……………. ( * )
z0 2
z0
x = r cos ? , y = r sin? , Z = r ei?
f (z) = u ( r , ? ) + i v ( r , ? )
r Ur = V? , r Vr = - U?
Ex: find if it exist when f(z) = r ei?
Sol .
f(z) = r ei?
= r ( cos ? + i sin ?) = r cos ? + i r sin ?
?u
?y
?v
?x
?u
?x
?v
?y
( ) ( , ) ( , )
( ) ( , ) ( , )
0 0 0 0 0
0 0 0 0 0
f z V x y iU x y
or
f z U x y iV x y
y y
x x
? ? ?
? ? ?
? ? ? ? ? ?
? e ? v iu
r
f z e u iv or f z i
r r
?( )? ? i ? ?( ) ? 1 ?
f ?(z)
y
v
x
v
y
u
x
u
?
?
?
?
?
?
?
?
, , ,
58
= u + i v
u = r cos ? ? u r = cos ? , u? = - r sin ?
v = r sin ? ? v r = sin ? , v? = r cos ?
Now r Ur = V? = r cos ? & r Vr = - U? = - r sin ?
Since Ur , U? , Vr , V? are cont. we get
f(z) is differentiable and equal
? ? ? ?
( ) 1
. 1
( ) cos sin
? ? ?
? ?
? ? ? ? ?
?
? ?
f z
e e
f z e u iv e i
i i
i
r r
i
? ?
? ? ? ?