proof of closed curve theorem
Let
f(x+iy)=u(x,y)+iv(x,y). |
Hence we have
∫Cf(z)𝑑z=∫Cω+i∫Cη |
where ω and η are the differential forms
ω=u(x,y)dx-v(x,y)dy,η=v(x,y)dx+u(x,y)dy. |
Notice that by Cauchy-Riemann equations ω and η are closed differential forms. Hence by the lemma on closed differential forms on a simply connected domain we get
∫C1ω=∫C2ω,∫C1η=∫C2η. |
and hence
∫C1f(z)𝑑z=∫C2f(z)𝑑z |
Title | proof of closed curve theorem |
---|---|
Canonical name | ProofOfClosedCurveTheorem |
Date of creation | 2013-03-22 13:33:34 |
Last modified on | 2013-03-22 13:33:34 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 6 |
Author | paolini (1187) |
Entry type | Proof |
Classification | msc 30E20 |