unit disk upper half plane conformal equivalence theorem

Theorem 1.

There is a conformal map from $\\Delta$ , the unit disk, to $U H P$ , the upper half plane.

Proof.

Define $f\\colon\\hat{{\\mathbb{C}}}\\to\\hat{{\\mathbb{C}}}$ (where $\\hat{{\\mathbb{C}}}$ denotes the Riemann Sphere) to be $f(z)=\\displaystyle{\\frac{z-i}{z+i}}$ . Notice that $f^{-1}(w)=\\displaystyle{i\\frac{1+w}{1-w}}$ and that $f$ (and therefore $f^{-1}$ ) is a Mobius transformation.

Notice that $f(0)=-1$ , $f(1)=\\displaystyle{\\frac{1-i}{1+i}}=-i$ and $f(-1)=\\displaystyle{\\frac{-1-i}{-1+i}}=i$ . By the Mobius Circle Transformation Theorem, $f$ takes the real axis to the unit circle. Since $f(i)=0$ , $f$ maps $U H P$ to $\\Delta$ and $f^{-1}:\\Delta\\to UHP$ . ∎

单词	UnitDiskUpperHalfPlaneConformalEquivalenceTheorem
释义	unit disk upper half plane conformal equivalence theorem Theorem 1. There is a conformal map from $\\Delta$ , the unit disk, to $U H P$ , the upper half plane. Proof. Define $f\\colon\\hat{{\\mathbb{C}}}\\to\\hat{{\\mathbb{C}}}$ (where $\\hat{{\\mathbb{C}}}$ denotes the Riemann Sphere) to be $f(z)=\\displaystyle{\\frac{z-i}{z+i}}$ . Notice that $f^{-1}(w)=\\displaystyle{i\\frac{1+w}{1-w}}$ and that $f$ (and therefore $f^{-1}$ ) is a Mobius transformation. Notice that $f(0)=-1$ , $f(1)=\\displaystyle{\\frac{1-i}{1+i}}=-i$ and $f(-1)=\\displaystyle{\\frac{-1-i}{-1+i}}=i$ . By the Mobius Circle Transformation Theorem, $f$ takes the real axis to the unit circle. Since $f(i)=0$ , $f$ maps $U H P$ to $\\Delta$ and $f^{-1}:\\Delta\\to UHP$ . ∎
随便看	locally cyclic group LocallyEuclidean locally Euclidean LocallyFiniteCollection locally finite collection LocallyFiniteGraph locally finite graph LocallyFiniteGroup locally finite group LocallyFinitePoset locally finite poset LocallyFiniteQuiver locally finite quiver LocallyFree locally free LocallyHomeomorphic locally homeomorphic LocallyIntegrableFunction locally integrable function LocallyMaximal locally maximal LocallyNilpotentGroup locally nilpotent group LocallyRingedSpace locally ringed space