Complex analysis review 1
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Argument
Stereographic Projection
Jordan’s Theorem
Every Jordan curve divides the plane into an “interior” region bounded by the curve and an “exterior” region containing all of the nearby and far away exterior points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere.
Heine-Borel Theorem
Suppose that
Bolzano-Weierstrass Theorem
There is at least an accumulating point in an infinit set.
Cauchy-Riemann Identity
Suppose that
For any path
Let
Let
Alltogether,
Theorem 1
Complex function
There is a fact that if
Which means that
Two Important Operators
And after a simple calculation
If
Conformality
Suppose that
Therefore,
Now suppose that there are two smooth curves pass through
Under the mapping
On the other hand, since
For any smooth curve through
Integration of Complex Functions
Suppose that
∫baf(t)dt=∫bau(t)dt+i∫bav(t)dt .γ is a rectifiable curve,f(z)=u(z)+iv(z),dz=dx+idy , then∫γf(z)dz=∫γudx−vdy+i∫γvdx+udy.
Define
Theorem 2
For any smooth
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