Théorème d'aire (conformal mapping)

Théorème d'aire (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.

Contenu 1 Déclaration 2 Preuve 3 Uses 4 References Statement Suppose that {style d'affichage f} is analytic and injective in the punctured open unit disk {style d'affichage mathbb {ré} setmoins {0}} and has the power series representation {style d'affichage f(z)={frac {1}{z}}+somme _{n=0}^{infime }un_{n}z ^{n},qquad zin mathbb {ré} setmoins {0},} then the coefficients {style d'affichage a_{n}} satisfy {somme de style d'affichage _{n=0}^{infime }n|un_{n}|^{2}leq 1.} Proof The idea of the proof is to look at the area uncovered by the image of {style d'affichage f} . Define for {displaystyle rin (0,1)} {displaystyle gamma _{r}(thêta ):=f(r,e ^{-c'est-à-dire }),qquad theta in [0,2pi ].} Alors {displaystyle gamma _{r}} is a simple closed curve in the plane. Laisser {displaystyle D_{r}} denote the unique bounded connected component of {style d'affichage mathbb {C} setminus gamma [0,2pi ]} . The existence and uniqueness of {displaystyle D_{r}} follows from Jordan's curve theorem.

Si {displaystyle D} is a domain in the plane whose boundary is a smooth simple closed curve {gamma de style d'affichage } , alors {style d'affichage mathrm {area} (ré)=int _{gamma }X,dy=-int _{gamma }y,dx,,} provided that {gamma de style d'affichage } is positively oriented around {displaystyle D} . This follows easily, par exemple, 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 {style d'affichage mathrm {area} (RÉ_{r})=int _{0}^{2pi }Re {soudain (}F(re^{-c'est-à-dire }){plus grand )},Je suis {soudain (}-je,r,e ^{-c'est-à-dire },F'(re^{-c'est-à-dire }){plus grand )},dtheta =-int _{0}^{2pi }Je suis {soudain (}F(re^{-c'est-à-dire }){plus grand )},Re {soudain (}-je,r,e ^{-c'est-à-dire },F'(re^{-c'est-à-dire }){plus grand )}thêta .} Par conséquent, the area of {displaystyle D_{r}} also equals to the average of the two expressions on the right hand side. After simplification, this yields {style d'affichage mathrm {area} (RÉ_{r})=-{frac {1}{2}},Re int _{0}^{2pi }F(r,e ^{-c'est-à-dire }),{surligner {r,e ^{-c'est-à-dire },F'(r,e ^{-c'est-à-dire })}},thêta ,} où {style d'affichage {surligner {z}}} denotes complex conjugation. We set {style d'affichage a_{-1}=1} and use the power series expansion for {style d'affichage f} , pour obtenir {style d'affichage mathrm {area} (RÉ_{r})=-{frac {1}{2}},Re int _{0}^{2pi }somme _{n=-1}^{infime }somme _{m=-1}^{infime }m,r ^{n+m},un_{n},{surligner {un_{m}}},e ^{je,(m-n),thêta },thêta ,.} (Depuis {style d'affichage entier _{0}^{2pi }somme _{n=-1}^{infime }somme _{m=-1}^{infime }m,r ^{n+m},|un_{n}|,|un_{m}|,thêta 0} , we may write the expression for the winding number of {displaystyle gamma _{s}} autour de {style d'affichage z_{0}} , and verify that it is equal to {style d'affichage 1} . Depuis, {displaystyle gamma _{t}} does not pass through {style d'affichage z_{0}} lorsque {displaystyle tneq r'} (comme {style d'affichage f} est injectif), the invariance of the winding number under homotopy in the complement of {style d'affichage z_{0}} implies that the winding number of {displaystyle gamma _{r}} autour de {style d'affichage z_{0}} is also {style d'affichage 1} . This implies that {style d'affichage z_{0}in D_{r}} et cela {displaystyle gamma _{r}} is positively oriented around {displaystyle D_{r}} , comme demandé.

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. En outre, the area theorem is often used in order to prove the Koebe 1/4 théorème, which is very useful in the study of the geometry of conformal mappings.

References Rudin, Walter (1987), Analyse réelle et complexe (3e éd.), New York: McGraw-Hill Book Co., ISBN 978-0-07-054234-1, M 0924157, OCLC 13093736 Catégories: Théorèmes en analyse complexe