Monodromy theorem

Monodromy theorem Illustration of analytic continuation along a curve (only a finite number of the disks {estilo de exibição U_{t}} are shown). Analytic continuation along a curve of the natural logarithm (the imaginary part of the logarithm is shown only).

Em análise complexa, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply analytic function) along curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued.

Before stating this theorem it is necessary to define analytic continuation along a curve and study its properties.

Conteúdo 1 Analytic continuation along a curve 2 Properties of analytic continuation along a curve 3 Monodromy theorem 4 Veja também 5 Referências 6 External links Analytic continuation along a curve The definition of analytic continuation along a curve is a bit technical, but the basic idea is that one starts with an analytic function defined around a point, and one extends that function along a curve via analytic functions defined on small overlapping disks covering that curve.

Formalmente, consider a curve (a continuous function) {gama de estilo de exibição :[0,1]para mathbb {C} .} Deixar {estilo de exibição f} be an analytic function defined on an open disk {estilo de exibição U} centered at {gama de estilo de exibição (0).} An analytic continuation of the pair {estilo de exibição (f,você)} along {gama de estilo de exibição } is a collection of pairs {estilo de exibição (f_{t},VOCÊ_{t})} por {displaystyle 0leq tleq 1} de tal modo que {estilo de exibição f_{0}=f} e {estilo de exibição U_{0}=U.} For each {lata de estilo de exibição [0,1],VOCÊ_{t}} is an open disk centered at {gama de estilo de exibição (t)} e {estilo de exibição f_{t}:VOCÊ_{t}para mathbb {C} } is an analytic function. For each {lata de estilo de exibição [0,1]} existe {displaystyle varepsilon >0} tal que para todos {displaystyle t'in [0,1]} com {estilo de exibição |t-t'|0} and the complex logarithm defined in a neighborhood of this point, and one lets {gama de estilo de exibição } be the circle of radius {estilo de exibição a} centered at the origin (traveled counterclockwise from {estilo de exibição (uma,0)} ), then by doing an analytic continuation along this curve one will end up with a value of the logarithm at {estilo de exibição (uma,0)} qual é {displaystyle 2pi i} plus the original value (see the second illustration on the right).

Monodromy theorem Homotopy with fixed endopoints is necessary for the monodromy theorem to hold.

As noted earlier, two analytic continuations along the same curve yield the same result at the curve's endpoint. No entanto, given two different curves branching out from the same point around which an analytic function is defined, with the curves reconnecting at the end, it is not true in general that the analytic continuations of that function along the two curves will yield the same value at their common endpoint.

De fato, one can consider, as in the previous section, the complex logarithm defined in a neighborhood of a point {estilo de exibição (uma,0)} and the circle centered at the origin and radius {displaystyle a.} Então, it is possible to travel from {estilo de exibição (uma,0)} para {estilo de exibição (-uma,0)} in two ways, counterclockwise, on the upper half-plane arc of this circle, and clockwise, on the lower half-plane arc. The values of the logarithm at {estilo de exibição (-uma,0)} obtained by analytic continuation along these two arcs will differ by {displaystyle 2pi i.} Se, Contudo, one can continuously deform one of the curves into another while keeping the starting points and ending points fixed, and analytic continuation is possible on each of the intermediate curves, then the analytic continuations along the two curves will yield the same results at their common endpoint. This is called the monodromy theorem and its statement is made precise below.

Deixar {estilo de exibição U} be an open disk in the complex plane centered at a point {estilo de exibição P} e {estilo de exibição f:Uto mathbb {C} } be a complex-analytic function. Deixar {estilo de exibição Q} be another point in the complex plane. If there exists a family of curves {displaystyle gamma _{s}:[0,1]para mathbb {C} } com {pecado de estilo de exibição [0,1]} de tal modo que {displaystyle gamma _{s}(0)=P} e {displaystyle gamma _{s}(1)=Q} para todos {pecado de estilo de exibição [0,1],} a função {estilo de exibição (s,t)dentro [0,1]vezes [0,1]to gamma _{s}(t)em matemática {C} } é contínuo, and for each {pecado de estilo de exibição [0,1]} it is possible to do an analytic continuation of {estilo de exibição f} along {displaystyle gamma _{s},} then the analytic continuations of {estilo de exibição f} along {displaystyle gamma _{0}} e {displaystyle gamma _{1}} will yield the same values at {displaystyle Q.} The monodromy theorem makes it possible to extend an analytic function to a larger set via curves connecting a point in the original domain of the function to points in the larger set. The theorem below which states that is also called the monodromy theorem.

Deixar {estilo de exibição U} be an open disk in the complex plane centered at a point {estilo de exibição P} e {estilo de exibição f:Uto mathbb {C} } be a complex-analytic function. Se {estilo de exibição W.} is an open simply-connected set containing {estilo de exibição U,} and it is possible to perform an analytic continuation of {estilo de exibição f} on any curve contained in {estilo de exibição W.} which starts at {estilo de exibição P,} então {estilo de exibição f} admits a direct analytic continuation to {estilo de exibição W.,} meaning that there exists a complex-analytic function {estilo de exibição g:Wto mathbb {C} } whose restriction to {estilo de exibição U} é {displaystyle f.} See also Analytic continuation Monodromy References Krantz, Steven G. (1999). Handbook of complex variables. Birkhauser. ISBN 0-8176-4011-8. Jones, Gareth A.; Singerman, Davi (1987). Complex functions: an algebraic and geometric viewpoint. Cambridge University Press. ISBN 0-521-31366-X. Triebel, Hans (1986). Analysis and mathematical physics, English ed. D. Reidel Pub. Companhia. ISBN 90-277-2077-0. External links Monodromy theorem at MathWorld Monodromy theorem at PlanetMath. Monodromy theorem at the Encyclopaedia of Mathematics Categories: Theorems in complex analysis