Princípio de Phragmén–Lindelöf

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Em matemática, o princípio de Phragmén–Lindelöf é uma extensão de 1908 por Lars Edvard Phragmén (1863-1937) e Ernst Leonard Lindelöf do princípio do módulo máximo da análise complexa, para ilimitados domínios.

Cenário

Na teoria da função complexa, sabe-se que se uma função f é holomorfa em um domínio limitado D, e é contínua em uma fronteira de D, então o máximo de |f|, devem ser atingidos no limite de D. Se, no entanto, a região D não for limitada, então isso não pode ser verdadeiro, como podemos ver ao analisar a função $g(z)=\exp(\exp(z))$ na fita $-\pi /2<{\mbox{Im}}\{z\}<\pi /2.$ A dificuldade aqui é que a função g tende para o infinito 'muito' rapidamente como z tende para o infinito ao longo do eixo real positivo.

O Princípio de Phragmén–Lindelöf mostra que, em determinadas circunstâncias, e limitando a rapidez com a qual f é permitido que tendem para o infinito, é possível provar que f é, na verdade, limitada no domínio ilimitado.

Na literatura de análise complexa, existem muitos exemplos do princípio de Phragmén–Lindelöf aplicado para regiões acopladas de diferentes tipos, e também uma versão deste princípio pode ser aplicada em uma forma similar as funções subharmônica e superharmônica.

Princípio de Phragmén–Lindelöf para um setor no plano complexo

Seja F(z) uma função holomorfa em um setor,

S=\left\{z\,{\big |}\,\alpha <\arg z<\beta \right\}

com ângulo π/λ = β–α, e a contínua no seu limite.

Se $|F(z)|\leq 1$ para o z sobre a fronteira de S, e $|F(z)|\leq e^{C|z|^{\rho }}$ para todo z em S, onde 0 ≤ ρ < λ e C > 0, então (1) também vale para todos os z em S.

Comentários

A condição (2) pode torna-se mais flexível a $\liminf _{r\to \infty }\sup _{\alpha <\theta <\beta }{\frac {\log |F(re^{i\theta })|}{r^{\rho }}}=0\quad {\text{for some}}\quad 0\leq \rho <\lambda ~,$ , com a mesma conclusão.

Princípio de Phragmén–Lindelöf para tiras

Na prática, o ponto 0 é muitas vezes transformado no ponto ∞ da Esfera de Riemann. Isso dá uma versão do princípio de que aplica-se a tiras, por exemplo, delimitada por duas linhas de constantes parte real do complexo plano. Este caso especial é às vezes conhecido como Teorema de Lindelöf.

Outros casos especiais

Teorema de Carlson é uma aplicação do princípio das funções limitadas no eixo imaginário.

Aplicações

O princípio é usado para provar o Princípio da incerteza de Heisenberg, que estabelece que uma função e a sua transformada de Fourier, não podem deteriorar mais rápido que exponencialmente.

Referências

Phragmén, Lars Edvard and Lindelöf, Ernst (1908). «Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier». Acta Math. 31 (1): 381–406. ISSN 0001-5962. doi:10.1007/BF02415450
Riesz, Marcel (1920). «Sur le principe de Phragmén-Lindelöf». Proceedings of the Cambridge Philosophical Society. 20 (Corr. «Sur le principe de Phragmén-Lindelöf». 21. 1921 )
Titchmarsh, Edward Charles (1976). The Theory of Functions Second ed. [S.l.]: Oxford University Press. ISBN 0-19-853349-7 (See chapter 5)
E.D. Solomentsev (2001), «Phragmén–Lindelöf theorem», in: Hazewinkel, Michiel, Enciclopédia de Matemática, ISBN 978-1-55608-010-4 (em inglês), Springer
Elias M. Stein and Shakarchi, Rami (2003). Complex analysis. Col: Princeton Lectures in Analysis, II. Princeton, NJ: Princeton University Press. ISBN 0-691-11385-8

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