Page 102 - Calculus with Complex Numbers
P. 102
A p p e n d i x C auchy's theorem
j ./'(z) dz = 0.
/
Proof
Case 1: y = Unit square W riting y = n + yz + n + y4 where n , yz, n , y4 are
the four sides taken in anti-clockwise order starting from 0 andwriting f (x +ïy) =
r
u (.'r, y) + i ?? (.', y), we have
1
/
j ./*(2) dz =
Therefore
J flz) dz + J ./'U) dz
J /1 J /7
'
/' 'u (.x, ylou -s,- i /' '''(.x, yldxdy.o o ?,
.o ?,
- - j . ./'
