Re: [複變] 複變積分3題
Problem # 1
If C is simple closed curve enclosing a region of area A, prove that
_
A = (1/2i)∮z dz
C
Use this to evaluate the integral when C is the circle │z-2│= 3.
f= P+iQ
∮(P+iQ)(dx+idy)
c
= ∮Pdx-Qdy + i*∮Qdx+Pdy
c c
dQ dP dP dQ
= -∫∫(─+─)dxdy + i∫∫((─-─)dxdy
dx dy dx dy
dP dQ dP dQ
= i∫∫[ (─-─) + i*(─+─) ]dxdy
dx dy dy dx >請問這個步驟是怎麼化簡?
>
df >
= 2i∫∫─ dxdy >
d(zbar)
之後把f代zbar，再移項=>
1
─∮(zbar) dz= ∫∫dA
2i c
(得證)
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