Math 2.0

Functions of a complex variable Generalized function theories and metric analysis Problem

Riemann–Hilbert problem

Builds on Holomorphic functionHölder continuityWinding numberSokhotski–Plemelj theorem

Problem

Let Γ⊂ℂ\Gamma \subset \C be a smooth simple closed curve, oriented counterclockwise. Let D+D^+ be its interior domain and D−D^- its exterior domain. Let G ⁣:Γ→ℂ∖{0}G \colon \Gamma \to \C \setminus \set{0} and g ⁣:Γ→ℂg \colon \Gamma \to \C be Hölder continuous functions. Find a function Φ\Phi that is holomorphic on D+D^+ and on D−D^- and vanishes at ∞\infty. It must have continuous boundary values Φ+\Phi^+ from D+D^+ and Φ−\Phi^- from D−D^- on Γ\Gamma with Φ+(t)=G(t) Φ−(t)+g(t)for t∈Γ.\Phi^+(t) = G(t)\, \Phi^-(t) + g(t) \quad \text{for } t \in \Gamma. In the matrix form of the problem, Φ\Phi takes values in ℂn\C^n. Then GG is a Hölder continuous map from Γ\Gamma to the invertible n×nn \times n complex matrices.

Remarks

The scalar problem is solved explicitly. Let κ\kappa be the winding number of the curve G(Γ)G(\Gamma) about 00, called the index of the problem. Assume 0∈D+0 \in D^+; taking a branch of log⁡\log of t−κG(t)t^{-\kappa} G(t), which has winding number 00, turns the problem into a jump problem Ψ+−Ψ−=h\Psi^+ - \Psi^- = h. The Cauchy integral 12πi∫Γh(τ)τ−z dτ\frac{1}{2\pi i} \int_\Gamma \frac{h(\tau)}{\tau - z}\, d\tau solves it by the Sokhotski–Plemelj theorem. If κ≥0\kappa \ge 0, the problem is solvable for every gg, and the solutions form an affine space of dimension κ\kappa. If κ<0\kappa < 0, it is solvable if and only if gg satisfies −κ-\kappa linear conditions, and then the solution is unique.

Example: for G=1G = 1 the index is 00 and the unique solution is the Cauchy integral of gg.

The matrix problem cannot be solved by logarithms, since matrices do not commute. It is closely tied to Hilbert's twenty-first problem on linear differential equations with prescribed monodromy, and it underlies the Riemann–Hilbert method for integrable systems and random matrices.

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Complex numbers, object, no evidence levelComplex numbersobjectElement, definition, no evidence levelElementdefinitionFunction, concept, no evidence levelFunctionconceptHolomorphic function, concept, no evidence levelHolomorphic functionconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectContour integral, definition, no evidence levelContour integraldefinitionSet, definition, no evidence levelSetdefinitionLaplace transform, definition, no evidence levelLaplace transformdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionCauchy principal value, definition, no evidence levelCauchy principal valuedefinitionWinding number, definition, no evidence levelWinding numberdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionHeaviside step function, object, no evidence levelHeaviside step functi…objectExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionDirac delta function, object, no evidence levelDirac delta functionobjectSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionLipschitz map, definition, no evidence levelLipschitz mapdefinitionHölder continuity, definition, no evidence levelHölder continuitydefinitionSokhotski–Plemelj theorem, statement, no evidence levelSokhotski–Plemelj the…statementRiemann–Hilbert problem, problem, no evidence levelRiemann–Hilbert probl…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complex numbers object
  2. Element definition
  3. Function concept
  4. Holomorphic function concept
  5. Proposition concept
  6. Real numbers object
  7. Contour integral definition
  8. Set definition
  9. Laplace transform definition
  10. Implication definition
  11. Predicate definition
  12. Cauchy principal value definition
  13. Winding number definition
  14. Ordered pair definition
  15. Heaviside step function object
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Dirac delta function object
  20. Subset definition
  21. Uniqueness quantifier definition
  22. Binary relation definition
  23. Function definition
  24. Metric space definition
  25. Lipschitz map definition
  26. Hölder continuity definition
  27. Sokhotski–Plemelj theorem statement
  28. Riemann–Hilbert problem problem

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert_problem
About this page
TagRY8YQ9JZF9
URIm2://RY8YQ9JZF9
Kindproblem
WritingW0 stub
MSC30E25, 34M50
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AboutHolomorphic function
UsesHolomorphic function, Hölder continuity, Winding number, Sokhotski–Plemelj theorem
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Changed2 November 2026

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