z=x+iy,则dz=dx+idy
由曲线是摆线,则x=a(t-sint),y=a(1-cost)
dx=a(1-cost)dt,dy=asintdt
这样dz=dx+idy=a(1-cost)dt+iasintdt=a(1-cost+isint)dt
原积分=∫ [0-->2π] a(1-cost+isint)dt
=a(t+sint-icost) [0-->2π]
=2πa
x=a(θ-sinθ) dx=a(1-cosθ)dθ
y=a(1-cosθ) dy=asinθdθ
dz=dx+idy=a[(1-cosθ)+iasinθ ]dθ
∫[0,2π] dz=∫[0,2π] [a(1-cosθ)+iasinθ]dθ=2πa
∫[0,2π] [-cosθ+isinθ]dθ=∫[0,2π] d[ -sinθ-icosθ]=0