Teorema da área (conformal mapping)

Teorema da área (conformal mapping) In the mathematical theory of conformal mappings, the area theorem gives an inequality satisfied by the power series coefficients of certain conformal mappings. The theorem is called by that name, not because of its implications, but rather because the proof uses the notion of area.

Conteúdo 1 Declaração 2 Prova 3 Uses 4 References Statement Suppose that {estilo de exibição f} is analytic and injective in the punctured open unit disk {estilo de exibição mathbb {D} setminus {0}} and has the power series representation {estilo de exibição f(z)={fratura {1}{z}}+soma _{n=0}^{infty }uma_{n}z^{n},qquad zin mathbb {D} setminus {0},} then the coefficients {estilo de exibição a_{n}} satisfy {soma de estilo de exibição _{n=0}^{infty }n|uma_{n}|^{2}leq 1.} Proof The idea of the proof is to look at the area uncovered by the image of {estilo de exibição f} . Define for {displaystyle rin (0,1)} {displaystyle gamma _{r}(teta ):=f(r,e^{-ittheta }),qquad theta in [0,2pi ].} Então {displaystyle gamma _{r}} is a simple closed curve in the plane. Deixar {displaystyle D_{r}} denote the unique bounded connected component of {estilo de exibição mathbb {C} setminus gamma [0,2pi ]} . The existence and uniqueness of {displaystyle D_{r}} follows from Jordan's curve theorem.

Se {estilo de exibição D} is a domain in the plane whose boundary is a smooth simple closed curve {gama de estilo de exibição } , então {matemática de estilo de exibição {area} (D)=int_{gama }x,dy=-int _{gama }y,dx,,} provided that {gama de estilo de exibição } is positively oriented around {estilo de exibição D} . This follows easily, por exemplo, from Green's theorem. As we will soon see, {displaystyle gamma _{r}} is positively oriented around {displaystyle D_{r}} (and that is the reason for the minus sign in the definition of {displaystyle gamma _{r}} ). After applying the chain rule and the formula for {displaystyle gamma _{r}} , the above expressions for the area give {matemática de estilo de exibição {area} (D_{r})=int_{0}^{2pi }Re {De repente (}f(re^{-ittheta }){maior )},Eu estou {De repente (}-eu,r,e^{-ittheta },f'(re^{-ittheta }){maior )},dtheta =-int _{0}^{2pi }Eu estou {De repente (}f(re^{-ittheta }){maior )},Re {De repente (}-eu,r,e^{-ittheta },f'(re^{-ittheta }){maior )}teta .} Portanto, the area of {displaystyle D_{r}} also equals to the average of the two expressions on the right hand side. After simplification, this yields {matemática de estilo de exibição {area} (D_{r})=-{fratura {1}{2}},Re int _{0}^{2pi }f(r,e^{-ittheta }),{overline {r,e^{-ittheta },f'(r,e^{-ittheta })}},teta ,} Onde {estilo de exibição {overline {z}}} denotes complex conjugation. We set {estilo de exibição a_{-1}=1} and use the power series expansion for {estilo de exibição f} , para obter {matemática de estilo de exibição {area} (D_{r})=-{fratura {1}{2}},Re int _{0}^{2pi }soma _{n=-1}^{infty }soma _{m=-1}^{infty }m,^{n+m},uma_{n},{overline {uma_{m}}},e^{eu,(m-n),teta },teta ,.} (Desde {estilo de exibição int _{0}^{2pi }soma _{n=-1}^{infty }soma _{m=-1}^{infty }m,^{n+m},|uma_{n}|,|uma_{m}|,teta 0} , we may write the expression for the winding number of {displaystyle gamma _{s}} por aí {estilo de exibição z_{0}} , and verify that it is equal to {estilo de exibição 1} . Desde, {displaystyle gamma _{t}} does not pass through {estilo de exibição z_{0}} quando {displaystyle tneq r'} (Como {estilo de exibição f} é injetivo), the invariance of the winding number under homotopy in the complement of {estilo de exibição z_{0}} implies that the winding number of {displaystyle gamma _{r}} por aí {estilo de exibição z_{0}} is also {estilo de exibição 1} . This implies that {estilo de exibição z_{0}in D_{r}} and that {displaystyle gamma _{r}} is positively oriented around {displaystyle D_{r}} , como requerido.

Uses The inequalities satisfied by power series coefficients of conformal mappings were of considerable interest to mathematicians prior to the solution of the Bieberbach conjecture. The area theorem is a central tool in this context. Além disso, the area theorem is often used in order to prove the Koebe 1/4 teorema, which is very useful in the study of the geometry of conformal mappings.

References Rudin, Walter (1987), Análise real e complexa (3rd ed.), Nova york: McGraw-Hill Book Co., ISBN 978-0-07-054234-1, MR 0924157, OCLC 13093736 Categorias: Theorems in complex analysis