Complex Analysis
Conformal Mapping
Draft for Information Only ContentComplex Function source/reference: Complex FunctionConformal MappingIntuitively, a conformal mapping is a "mapping that preserves angles between curves". In other words. when defining curves, the angles between curves must also be defined. PathsBy definition. A path in the complex plane from a point A to a point B is a continuous function 𝛾:[𝑎,𝑏]→ℂ such that 𝛾(𝑎)=𝐴 and 𝛾(𝑏)=𝐵. Examples
CurvesBy definition. A path 𝛾:[𝑎,𝑏]→ℂ is smooth if the functions 𝑥(𝑡) and 𝑦(𝑡) in the representation 𝛾(𝑡)=𝑥(𝑡)+𝑖𝑦(𝑡) are smooth, that is, have as many derivatives as desired. In the above examples, (1), (2), and (3) are smooth, whereas (4) is piecewise smoth, i.e. put together ("concateated") from finitely many smooth paths. The term curve is typically used for a smooth or piecewise smooth path. If 𝛾=𝑥+𝑖𝑦:[𝑎,𝑏]→ℂ is a smooth curve and 𝑡_{0}∈(𝑎,𝑏), then 𝛾′(𝑡_{0})=𝑥′(𝑡_{0})+𝑖𝑦′(𝑡_{0})
is a tangent vector to 𝛾 at 𝑧_{0}=𝛾(𝑡_{0}) The Angle between CurvesBy definition. Let 𝛾_{1} and 𝛾_{2} be two smooth curves, intersecting at a point 𝑧_{0}. The angle between the two curves at 𝑧_{0} is defined as the angle between the two tangent vectors at 𝑧_{0}. ExampleLet 𝛾_{1}:[0,𝜋]→ℂ, 𝛾_{1}(𝑡)=ℯ^{𝑖𝑡} and 𝛾_{2}: Then 𝛾_{1} 𝛾′_{1}(𝑡)=𝑖ℯ^{𝑖𝑡}, 𝛾′_{1}
The angle between these curves at 𝑖 is thus ConformalityBy definition. A function is conformal if it preserves angles between curves. More precisely, a smooth complexvalued function 𝑔 is conformal at 𝑧_{0} if whenever 𝛾_{1} and 𝛾_{2} are two curves that intersect at 𝑧_{0} with nonzero tangents, the 𝑔∘𝛾_{1} and 𝑔∘𝛾_{2} have nonzero tangents at 𝑔(𝑧_{0}) that intersect at the same angle. A conformal mapping of a domain 𝐷 onto 𝑉 is a continuously differentiable mapping that is conformal at each point in 𝐷 and maps 𝐷 onetoone onto 𝑉. Analytic FunctionsBy theorem. If 𝑓:𝑈→ℂ is analytic and if 𝑧_{0}∈𝑈 such that 𝑓′(𝑧_{0})≠0, then 𝑓 is conformal at 𝑧_{0}. Reason: If 𝛾:[𝑎,𝑏]→𝑈 is a curve in 𝑈 with 𝛾(𝑡_{0})=𝑧_{0} for some 𝑡_{0}∈(𝑎,𝑏), then
(𝑓∘𝛾)′(𝑡_{0})=𝑓′(𝛾(𝑡_{0}))⋅𝛾′(𝑡_{0})=𝑓′(𝑧_{0}) ∈ℂ\{0} ⋅𝛾′(𝑡_{0})
Thus (𝑓∘𝛾)′(𝑡_{0}) is obtained from 𝛾′(𝑡_{0}) via multiplication by 𝑓′(𝑧_{0}) (=rotation & stretching). If 𝛾_{1}, 𝛾_{2} are two curves in 𝑈 through 𝑧_{0} with tangent vectors 𝛾′_{1}(𝑡_{1}), 𝛾′_{2}(𝑡_{2}), then (𝑓∘𝛾_{1})′(𝑡_{1}) and (𝑓∘𝛾_{2})′(𝑡_{2}) are both obtained from 𝛾′_{1}(𝑡_{1}), 𝛾′_{2}(𝑡_{2}), respectively, via multiplication by 𝑓′(𝑧_{0}). The angle between them is thus preserved. Example
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