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<title>Riemann–Hilbert problem</title>

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Riemann–Hilbert problem

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</style><div role="note" class="hatnote navigation-not-searchable">For the original problem of Hilbert concerning the existence of linear differential equations having a given monodromy group, see <a href="Hilbert's_twenty-first_problem" title="Hilbert's twenty-first problem">Hilbert's twenty-first problem</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, Riemann–Hilbert problems, named after <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> and <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>, are a class of problems that arise in the study of <a href="Differential_equation" title="Differential equation">differential equations</a> in the <a href="Complex_plane" title="Complex plane">complex plane</a>. Several <a href="Existence_theorem" title="Existence theorem">existence theorems</a> for Riemann–Hilbert problems have been produced by <a href="Mark_Krein" title="Mark Krein">Mark Krein</a>, <a href="Israel_Gohberg" title="Israel Gohberg">Israel Gohberg</a> and others.<a href="#cite_note-FOOTNOTEClanceyGohberg1981-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="The_Riemann_problem">The Riemann problem</h2></div>

Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> is a <a href="Smoothness" title="Smoothness">smooth</a>, <a href="Jordan_curve_theorem" title="Jordan curve theorem">simple, closed contour</a> in the <a href="Complex_plane" title="Complex plane">complex plane</a>.<a href="#cite_note-FOOTNOTEAblowitzFokas200371–72-2">[2]</a> Divide the plane into two parts denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{+}}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{+}}</annotation>

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</math><img src="./fca62b421791e4271b64c4dfaf7ae60ab679b173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Sigma _{+}}" loading="lazy"> (the inside) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{-}}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{-}}</annotation>

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</math><img src="./d44d9e03e23316f04655ddbdbcc3bb21be3b5bac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Sigma _{-}}" loading="lazy"> (the outside), determined by the <a href="Winding_number" title="Winding number">index</a> of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation, was that of finding a function

<msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{+}(t)=u(t)+iv(t),}</annotation>

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</math><img src="./719c8ca30bc16a2e611b308d38a895e1cd1bc153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.557ex; height:2.843ex;" alt="{\displaystyle M_{+}(t)=u(t)+iv(t),}" loading="lazy"></dd></dl>

analytic inside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma _{+}}">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma _{+}}</annotation>

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</math><img src="./fca62b421791e4271b64c4dfaf7ae60ab679b173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Sigma _{+}}" loading="lazy">, such that the boundary values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}}">

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<annotation encoding="application/x-tex">{\displaystyle M_{+}}</annotation>

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</math><img src="./77b1e5a6167655748c4d99e234d3f9a4e10bd688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.765ex; height:2.509ex;" alt="{\displaystyle M_{+}}" loading="lazy"> along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> satisfy the equation

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<annotation encoding="application/x-tex">{\displaystyle a(t)u(t)-b(t)v(t)=c(t),}</annotation>

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</math><img src="./02840feb50e9d5caa044d879480c89f82bc801a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.522ex; height:2.843ex;" alt="{\displaystyle a(t)u(t)-b(t)v(t)=c(t),}" loading="lazy"></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle t\in \Sigma }</annotation>

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</math><img src="./58710a4cf2b570e0492322687d5c9d67231741e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \Sigma }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(t)}">

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<annotation encoding="application/x-tex">{\displaystyle a(t)}</annotation>

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</math><img src="./bb7931a26b0d360eaf90aa45247d2de5c984d5d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.879ex; height:2.843ex;" alt="{\displaystyle a(t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(t)}">

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<annotation encoding="application/x-tex">{\displaystyle b(t)}</annotation>

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</math><img src="./6b522c57d8910b4438b26933339cf1e8769cf813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle b(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(t)}">

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<annotation encoding="application/x-tex">{\displaystyle c(t)}</annotation>

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</math><img src="./6e77d04462d26c4fdbfe8f988b182babd15059df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.656ex; height:2.843ex;" alt="{\displaystyle c(t)}" loading="lazy"> are given real-valued functions.<a href="#cite_note-FOOTNOTEBitsadze2001-3">[3]</a><a href="#cite_note-FOOTNOTEPandey1996-4">[4]</a> For example, in the special case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1,b=0}">

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<annotation encoding="application/x-tex">{\displaystyle a=1,b=0}</annotation>

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</math><img src="./a6c27bc5a8e91110e2f6cb470eae65ad1fa3ff99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.783ex; height:2.509ex;" alt="{\displaystyle a=1,b=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> is a circle, the problem reduces to deriving the <a href="Poisson_kernel" title="Poisson kernel">Poisson formula</a>.<a href="#cite_note-FOOTNOTEAblowitzFokas2003514-5">[5]</a>

By the <a href="Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a>, it suffices to consider the case when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> is the <a href="Circle_group" title="Circle group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}</annotation>

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</math><img src="./ec12bc40f28e1c65ce250bfcb870ccbe865568e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.16ex; height:2.843ex;" alt="{\textstyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}" loading="lazy">.<a href="#cite_note-FOOTNOTEPandey1996§2.2-6">[6]</a> In this case, one may seek <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}(z)}">

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<annotation encoding="application/x-tex">{\displaystyle M_{+}(z)}</annotation>

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</math><img src="./534ec0d875cd99c09a4fdc30eaf7cb7657f3a3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.662ex; height:2.843ex;" alt="{\displaystyle M_{+}(z)}" loading="lazy"> along with its <a href="Schwarz_reflection_principle" title="Schwarz reflection principle">Schwarz reflection</a>

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<annotation encoding="application/x-tex">{\displaystyle M_{-}(z)={\overline {M_{+}\left({\bar {z}}^{-1}\right)}}.}</annotation>

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</math><img src="./e394f6e54f25009158d98400e55ab0b69e4f7e1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.433ex; height:4.009ex;" alt="{\displaystyle M_{-}(z)={\overline {M_{+}\left({\bar {z}}^{-1}\right)}}.}" loading="lazy"></dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {T} }">

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<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {T} }</annotation>

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</math><img src="./054f5f18e6891d4c4ca6a5e5e127cf70536eecda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.479ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {T} }" loading="lazy">, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=1/{\bar {z}}}">

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<annotation encoding="application/x-tex">{\displaystyle z=1/{\bar {z}}}</annotation>

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</math><img src="./0610cc4ef4aa396273167a150e38350349d30436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.808ex; height:2.843ex;" alt="{\displaystyle z=1/{\bar {z}}}" loading="lazy"> and so

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<annotation encoding="application/x-tex">{\displaystyle M_{-}(z)={\overline {M_{+}(z)}}.}</annotation>

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</math><img src="./08f2e6305b30446a2ee56bcbe52fd9680e925120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.185ex; height:3.676ex;" alt="{\displaystyle M_{-}(z)={\overline {M_{+}(z)}}.}" loading="lazy"></dd></dl>

Hence the problem reduces to finding a pair of analytic functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}(z)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{+}(z)}</annotation>

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</math><img src="./534ec0d875cd99c09a4fdc30eaf7cb7657f3a3e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.662ex; height:2.843ex;" alt="{\displaystyle M_{+}(z)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-}(z)}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{-}(z)}</annotation>

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</math><img src="./196c9104f76c8d368acf1d9b3b4621c2fd98464c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.662ex; height:2.843ex;" alt="{\displaystyle M_{-}(z)}" loading="lazy"> on the inside and outside, respectively, of the <a href="Unit_disk" title="Unit disk">unit disk</a>, so that on the unit circle

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {a(z)+ib(z)}{2}}M_{+}(z)+{\frac {a(z)-ib(z)}{2}}M_{-}(z)=c(z),}</annotation>

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</math><img src="./2a8b84258010153d749361660480d1ab1a3b9fe0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.817ex; height:5.676ex;" alt="{\displaystyle {\frac {a(z)+ib(z)}{2}}M_{+}(z)+{\frac {a(z)-ib(z)}{2}}M_{-}(z)=c(z),}" loading="lazy"></dd></dl>

and, moreover, so that the condition at infinity holds:

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<annotation encoding="application/x-tex">{\displaystyle \lim _{z\to \infty }M_{-}(z)={\overline {{M}_{+}(0)}}.}</annotation>

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</math><img src="./c8e306698041cce1f5da6dcbc9301aef0374fa19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.702ex; height:4.676ex;" alt="{\displaystyle \lim _{z\to \infty }M_{-}(z)={\overline {{M}_{+}(0)}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="The_Hilbert_problem">The Hilbert problem</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Wiener-Hopf_method" class="mw-redirect" title="Wiener-Hopf method">Wiener-Hopf method</a></div>

Hilbert's generalization of the problem attempted to find a pair of analytic functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}(t)}">

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<annotation encoding="application/x-tex">{\displaystyle M_{+}(t)}</annotation>

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</math><img src="./57c920d0019f71d5e1c2cb209c23a245c42e34d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.414ex; height:2.843ex;" alt="{\displaystyle M_{+}(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-}(t)}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle M_{-}(t)}</annotation>

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</math><img src="./49c8c38fe53227c0a808045091df7dffe6d31860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.414ex; height:2.843ex;" alt="{\displaystyle M_{-}(t)}" loading="lazy"> on the inside and outside, respectively, of the curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">, such that for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \Sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t\in \Sigma }</annotation>

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</math><img src="./58710a4cf2b570e0492322687d5c9d67231741e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \Sigma }" loading="lazy"> one has

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<annotation encoding="application/x-tex">{\displaystyle \alpha (t)M_{+}(t)+\beta (t)M_{-}(t)=\gamma (t),}</annotation>

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</math><img src="./d55f0e4b6e667e2302146be589cd6dcf0b5dda10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.442ex; height:2.843ex;" alt="{\displaystyle \alpha (t)M_{+}(t)+\beta (t)M_{-}(t)=\gamma (t),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha (t)}</annotation>

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</math><img src="./a6cfa6e844c065b7c29c0a39cc4dcf40b237ea80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \alpha (t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \beta (t)}</annotation>

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</math><img src="./e1e442b7da324b393df6251802ab0b85248f4399.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.981ex; height:2.843ex;" alt="{\displaystyle \beta (t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma (t)}</annotation>

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</math><img src="./54fa4a5d64e164410e4a18106677bebefe1a1f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.911ex; height:2.843ex;" alt="{\displaystyle \gamma (t)}" loading="lazy"> are given complex-valued functions (no longer just complex conjugates).<a href="#cite_note-FOOTNOTENoble1958§4.2-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Riemann–Hilbert_problems">Riemann–Hilbert problems</h2></div>

In the Riemann problem as well as Hilbert's generalization, the contour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> was simple. A full Riemann–Hilbert problem allows that the contour may be composed of a union of several oriented smooth curves, with no intersections. The "+" and "−" sides of the "contour" may then be determined according to the index of a point with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">. The Riemann–Hilbert problem is to find a pair of <a href="Holomorphic_function" title="Holomorphic function">analytic functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{+}(t)}</annotation>

</semantics>

</math><img src="./57c920d0019f71d5e1c2cb209c23a245c42e34d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.414ex; height:2.843ex;" alt="{\displaystyle M_{+}(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{-}(t)}</annotation>

</semantics>

</math><img src="./49c8c38fe53227c0a808045091df7dffe6d31860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.414ex; height:2.843ex;" alt="{\displaystyle M_{-}(t)}" loading="lazy"> on the "+" and "−" side of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">, respectively, such that for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in \Sigma }</annotation>

</semantics>

</math><img src="./58710a4cf2b570e0492322687d5c9d67231741e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \Sigma }" loading="lazy"> one has

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha (t)M_{+}(t)+\beta (t)M_{-}(t)=\gamma (t),}</annotation>

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</math><img src="./d55f0e4b6e667e2302146be589cd6dcf0b5dda10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.442ex; height:2.843ex;" alt="{\displaystyle \alpha (t)M_{+}(t)+\beta (t)M_{-}(t)=\gamma (t),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha (t)}</annotation>

</semantics>

</math><img src="./a6cfa6e844c065b7c29c0a39cc4dcf40b237ea80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \alpha (t)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta (t)}</annotation>

</semantics>

</math><img src="./e1e442b7da324b393df6251802ab0b85248f4399.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.981ex; height:2.843ex;" alt="{\displaystyle \beta (t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (t)}</annotation>

</semantics>

</math><img src="./54fa4a5d64e164410e4a18106677bebefe1a1f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.911ex; height:2.843ex;" alt="{\displaystyle \gamma (t)}" loading="lazy"> are given complex-valued functions.

<div class="mw-heading mw-heading2"><h2 id="Matrix_Riemann–Hilbert_problems">Matrix Riemann–Hilbert problems</h2></div>

Given an oriented contour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> (technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane), a Riemann–Hilbert factorization problem is the following.

Given a matrix function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(t)}</annotation>

</semantics>

</math><img src="./e3d6c09ba5569413364689bf4837c7b71ef0892f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.476ex; height:2.843ex;" alt="{\displaystyle G(t)}" loading="lazy"> defined on the contour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">, find a <a href="Holomorphic" class="mw-redirect" title="Holomorphic">holomorphic</a> matrix function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)}</annotation>

</semantics>

</math><img src="./4b7703ad73053a9d1e8d6f43d873ac49825b1955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle M(z)}" loading="lazy"> defined on the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">, such that the following two conditions are satisfied<a href="#cite_note-FOOTNOTEAblowitzFokas2003§7.5-8">[8]</a>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{+}}</annotation>

</semantics>

</math><img src="./77b1e5a6167655748c4d99e234d3f9a4e10bd688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.765ex; height:2.509ex;" alt="{\displaystyle M_{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{-}}</annotation>

</semantics>

</math><img src="./335ebc73d1694048e81c2f28f4f31f4ab8711492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.765ex; height:2.509ex;" alt="{\displaystyle M_{-}}" loading="lazy"> denote the non-tangential limits of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> as we approach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}(t)=G(t)M_{-}(t)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{+}(t)=G(t)M_{-}(t)}</annotation>

</semantics>

</math><img src="./9125e277ef67510443e64349ba1cca3a1a1f5ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.402ex; height:2.843ex;" alt="{\displaystyle M_{+}(t)=G(t)M_{-}(t)}" loading="lazy">, at all points of non-intersection in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)}</annotation>

</semantics>

</math><img src="./4b7703ad73053a9d1e8d6f43d873ac49825b1955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle M(z)}" loading="lazy"> tends to the <a href="Identity_matrix" title="Identity matrix">identity matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{N}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{N}}</annotation>

</semantics>

</math><img src="./0b6c13021b679ca8a82c47e2e488a943d70e0ead.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.715ex; height:2.509ex;" alt="{\displaystyle I_{N}}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\to \infty }</annotation>

</semantics>

</math><img src="./380ee74715fed4cd83f76bacac1add6aad2d58c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:1.843ex;" alt="{\displaystyle z\to \infty }" loading="lazy"> along any direction outside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">.</li></ol>

In the simplest case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(t)}</annotation>

</semantics>

</math><img src="./e3d6c09ba5569413364689bf4837c7b71ef0892f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.476ex; height:2.843ex;" alt="{\displaystyle G(t)}" loading="lazy"> is smooth and integrable. In more complicated cases it could have singularities. The limits <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{+}}</annotation>

</semantics>

</math><img src="./77b1e5a6167655748c4d99e234d3f9a4e10bd688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.765ex; height:2.509ex;" alt="{\displaystyle M_{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{-}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{-}}</annotation>

</semantics>

</math><img src="./335ebc73d1694048e81c2f28f4f31f4ab8711492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.765ex; height:2.509ex;" alt="{\displaystyle M_{-}}" loading="lazy"> could be classical and continuous or they could be taken in the <a href="Square_integrable" class="mw-redirect" title="Square integrable"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

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</math><img src="./ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy">-sense</a>.

At end-points or intersection points of the contour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">, the jump condition is not defined; constraints on the growth of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> near those points have to be posed to ensure uniqueness (see the scalar problem below).

<div class="mw-heading mw-heading3"><h3 id="Example:_Scalar_Riemann–Hilbert_factorization_problem">Example: Scalar Riemann–Hilbert factorization problem</h3></div>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=2}</annotation>

</semantics>

</math><img src="./2beb4a0ec3b876fe73c5c10ef94beb0ca1bc8094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.088ex; height:2.176ex;" alt="{\displaystyle G=2}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma =[-1,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma =[-1,1]}</annotation>

</semantics>

</math><img src="./59de4cc01a16b61c9b02cac2417ef0b68e408e06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.237ex; height:2.843ex;" alt="{\displaystyle \Sigma =[-1,1]}" loading="lazy">. Assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is bounded, what is the solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">?

To solve this, let's take the <a href="Logarithm" title="Logarithm">logarithm</a> of equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{+}=GM_{-}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle M_{+}=GM_{-}}</annotation>

</semantics>

</math><img src="./0ebd467486b3dc08a879465b81bcc1a5ea5284d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.455ex; height:2.509ex;" alt="{\displaystyle M_{+}=GM_{-}}" loading="lazy">.

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \log M_{+}(z)=\log M_{-}(z)+\log 2.}</annotation>

</semantics>

</math><img src="./aa89c2ead5f92a824002e97bc6d3487a253a6c3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.149ex; height:2.843ex;" alt="{\displaystyle \log M_{+}(z)=\log M_{-}(z)+\log 2.}" loading="lazy"></dd></dl>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)}</annotation>

</semantics>

</math><img src="./4b7703ad73053a9d1e8d6f43d873ac49825b1955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle M(z)}" loading="lazy"> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log M\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log M\to 0}</annotation>

</semantics>

</math><img src="./de92c3ba47613d860f47cfc82c9eeb8759d19ad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.578ex; height:2.509ex;" alt="{\displaystyle \log M\to 0}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\to \infty }</annotation>

</semantics>

</math><img src="./380ee74715fed4cd83f76bacac1add6aad2d58c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:1.843ex;" alt="{\displaystyle z\to \infty }" loading="lazy">.

A standard fact about the <a href="Hilbert_transform" title="Hilbert transform">Cauchy transform</a> is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{+}-C_{-}=I}">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle C_{+}-C_{-}=I}</annotation>

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</math><img src="./21647c26857ea385146ccc6679637c28c7b98841.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.456ex; height:2.509ex;" alt="{\displaystyle C_{+}-C_{-}=I}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{+}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle C_{+}}</annotation>

</semantics>

</math><img src="./936f4d0c6533b0cbd25d003388a84fdeba485800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.173ex; height:2.509ex;" alt="{\displaystyle C_{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{-}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{-}}</annotation>

</semantics>

</math><img src="./f675d9eb4f31d1680b7ca545f6a58432a149775a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.173ex; height:2.509ex;" alt="{\displaystyle C_{-}}" loading="lazy"> are the limits of the Cauchy transform from above and below <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">; therefore, we get

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi i}}\int _{\Sigma _{+}}{\frac {\log 2}{\zeta -z}}\,d\zeta -{\frac {1}{2\pi i}}\int _{\Sigma _{-}}{\frac {\log {2}}{\zeta -z}}\,d\zeta =\log 2}</annotation>

</semantics>

</math><img src="./451ed732e59628022f3bb403bf308f3a66799036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.218ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{2\pi i}}\int _{\Sigma _{+}}{\frac {\log 2}{\zeta -z}}\,d\zeta -{\frac {1}{2\pi i}}\int _{\Sigma _{-}}{\frac {\log {2}}{\zeta -z}}\,d\zeta =\log 2}" loading="lazy"></dd></dl>

when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in \Sigma }</annotation>

</semantics>

</math><img src="./cb38f686318ce1d654087bbfc7660e1893d36449.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \Sigma }" loading="lazy">. Because the solution of a Riemann–Hilbert factorization problem is unique (an easy application of <a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem (complex analysis)</a>), the <a href="Sokhotski%E2%80%93Plemelj_theorem" title="Sokhotski–Plemelj theorem">Sokhotski–Plemelj theorem</a> gives the solution. We get

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log M={\frac {1}{2\pi i}}\int _{\Sigma }{\frac {\log {2}}{\zeta -z}}d\zeta ={\frac {\log 2}{2\pi i}}\int _{-1-z}^{1-z}{\frac {1}{\zeta }}d\zeta ={\frac {\log 2}{2\pi i}}\log {\frac {z-1}{z+1}},}</annotation>

</semantics>

</math><img src="./dd8d0adc902ad462ad9412387cb1bc60f3a9f0cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.678ex; height:6.343ex;" alt="{\displaystyle \log M={\frac {1}{2\pi i}}\int _{\Sigma }{\frac {\log {2}}{\zeta -z}}d\zeta ={\frac {\log 2}{2\pi i}}\int _{-1-z}^{1-z}{\frac {1}{\zeta }}d\zeta ={\frac {\log 2}{2\pi i}}\log {\frac {z-1}{z+1}},}" loading="lazy"></dd></dl>

and therefore

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}},}</annotation>

</semantics>

</math><img src="./ec2f8aa689e7e00be77aabdc1f08827757ab3b82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.262ex; height:7.509ex;" alt="{\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}},}" loading="lazy"></dd></dl>

which has a <a href="Branch_point#Branch_cuts" title="Branch point">branch cut</a> at contour <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">.

Check:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}M_{+}(0)&=(e^{i\pi })^{\frac {\log 2}{2\pi i}}=e^{\frac {\log 2}{2}}\\M_{-}(0)&=(e^{-i\pi })^{\frac {\log 2}{2\pi i}}=e^{-{\frac {\log 2}{2}}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./d2daf6937b845ae04c957bc6de7388b218dbc7a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:29.618ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}M_{+}(0)&=(e^{i\pi })^{\frac {\log 2}{2\pi i}}=e^{\frac {\log 2}{2}}\\M_{-}(0)&=(e^{-i\pi })^{\frac {\log 2}{2\pi i}}=e^{-{\frac {\log 2}{2}}}\end{aligned}}}" loading="lazy"></dd></dl>

therefore,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{+}(0)=M_{-}(0)e^{\log {2}}=M_{-}(0)2.}</annotation>

</semantics>

</math><img src="./4753136c329ffedd2549b81db2b3563171a4ad94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.842ex; height:3.176ex;" alt="{\displaystyle M_{+}(0)=M_{-}(0)e^{\log {2}}=M_{-}(0)2.}" loading="lazy"></dd></dl>

CAVEAT 1: If the problem is not scalar one cannot easily take logarithms. In general explicit solutions are very rare.

CAVEAT 2: The boundedness (or at least a constraint on the blow-up) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> near the special points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>

</semantics>

</math><img src="./704fb0427140d054dd267925495e78164fee9aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}" loading="lazy"> is crucial. Otherwise any function of the form

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}}+{\frac {a}{z-1}}+{\frac {b}{z+1}}}</annotation>

</semantics>

</math><img src="./0f9504809233ccdb0c5504d584fa3970c9d3da51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.15ex; height:7.509ex;" alt="{\displaystyle M(z)=\left({\frac {z-1}{z+1}}\right)^{\frac {\log {2}}{2\pi i}}+{\frac {a}{z-1}}+{\frac {b}{z+1}}}" loading="lazy"></dd></dl>

is also a solution. In general, conditions on growth are necessary at special points (the end-points of the jump contour or intersection point) to ensure that the problem is well-posed.

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="DBAR_problem">DBAR problem</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="DBAR_problem" title="DBAR problem">DBAR problem</a></div>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is some <a href="Simply_connected_space" title="Simply connected space">simply connected</a> <a href="Domain_(mathematical_analysis)" title="Domain (mathematical analysis)">domain</a> of the <a href="Complex_plane" title="Complex plane">complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> plane</a>. Then the scalar equation

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial M(z,{\bar {z}})}{\partial {\bar {z}}}}=f(z,{\bar {z}}),\quad z\in D,}</annotation>

</semantics>

</math><img src="./b6e57f45770ecd05b3aa75a1b93dc281d15045bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.285ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial M(z,{\bar {z}})}{\partial {\bar {z}}}}=f(z,{\bar {z}}),\quad z\in D,}" loading="lazy"></dd></dl>

is a generalization of a Riemann-Hilbert problem, called the DBAR problem (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\partial }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}}</annotation>

</semantics>

</math><img src="./6bd5dfe63f32552004b0d29d5ef7e8b046dda0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.487ex; height:3.009ex;" alt="{\displaystyle {\overline {\partial }}}" loading="lazy"> problem). It is the complex form of the nonhomogeneous <a href="Cauchy%E2%80%93Riemann_equations#Complex_differential_forms" title="Cauchy–Riemann equations">Cauchy-Riemann equations</a>. To show this, let

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=u+iv,\quad f={\frac {g+ih}{2}},\quad z=x+iy,}</annotation>

</semantics>

</math><img src="./e7df0ced555845836a2f14ed506bc11fc58c02a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.626ex; height:5.343ex;" alt="{\displaystyle M=u+iv,\quad f={\frac {g+ih}{2}},\quad z=x+iy,}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x,y)}</annotation>

</semantics>

</math><img src="./b57461332078df873c81bca2a395fd13f78bef50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.658ex; height:2.843ex;" alt="{\displaystyle u(x,y)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x,y)}</annotation>

</semantics>

</math><img src="./ca1cc47d643640c35af2867bd47f907af79574d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.456ex; height:2.843ex;" alt="{\displaystyle v(x,y)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x,y)}</annotation>

</semantics>

</math><img src="./bf358a54b0375e22ae5f3ab2c3e1a22c0c87e11c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.444ex; height:2.843ex;" alt="{\displaystyle g(x,y)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x,y)}</annotation>

</semantics>

</math><img src="./7235c6799c7d4112231c9941bd428fe6a4111fe4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.667ex; height:2.843ex;" alt="{\displaystyle h(x,y)}" loading="lazy"> all real-valued functions of real variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Then, using

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}</annotation>

</semantics>

</math><img src="./f924d4235b6c7582bc69761407e407d0a5ab50e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.826ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}" loading="lazy"></dd></dl>

the DBAR problem yields

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}=g(x,y),\quad {\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}=h(x,y).}</annotation>

</semantics>

</math><img src="./251527c7251a39f6a61a851eca40e1b251119176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.754ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial x}}-{\frac {\partial v}{\partial y}}=g(x,y),\quad {\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}}=h(x,y).}" loading="lazy"></dd></dl>

As such, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is holomorphic for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in D}</annotation>

</semantics>

</math><img src="./66c9016b90a20d376dbafef2db16f8dfc177a489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle z\in D}" loading="lazy">, then the Cauchy-Riemann equations must be satisfied.<a href="#cite_note-FOOTNOTEAblowitzFokas2003598-9">[9]</a>

In case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\to 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\to 1}</annotation>

</semantics>

</math><img src="./8c31da0dda0fb637464e43d3533beaa8b4d5b6d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.219ex; height:2.176ex;" alt="{\displaystyle M\to 1}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\to \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\to \infty }</annotation>

</semantics>

</math><img src="./380ee74715fed4cd83f76bacac1add6aad2d58c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.026ex; height:1.843ex;" alt="{\displaystyle z\to \infty }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D:=\mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D:=\mathbb {C} }</annotation>

</semantics>

</math><img src="./67bc342935e5fb8583752961a1133a1e2c34b401.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.348ex; height:2.176ex;" alt="{\displaystyle D:=\mathbb {C} }" loading="lazy">, the solution of the DBAR problem is<a href="#cite_note-FOOTNOTEAblowitzFokas2003600-10">[10]</a>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<mfrac>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z,{\bar {z}})=1+{\frac {1}{2\pi i}}\iint _{\mathbb {R} ^{2}}{\frac {f(\zeta ,{\bar {\zeta }})}{\zeta -z}}\,d\zeta \wedge d{\bar {\zeta }},}</annotation>

</semantics>

</math><img src="./adf088e50b27098ee772771c003708c06bf99d98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.456ex; height:6.343ex;" alt="{\displaystyle M(z,{\bar {z}})=1+{\frac {1}{2\pi i}}\iint _{\mathbb {R} ^{2}}{\frac {f(\zeta ,{\bar {\zeta }})}{\zeta -z}}\,d\zeta \wedge d{\bar {\zeta }},}" loading="lazy"></dd></dl>

integrated over the entire complex plane; denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy">, and where the <a href="Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a> is defined as

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\zeta \wedge d{\bar {\zeta }}=(d\xi +id\eta )\wedge (d\xi -id\eta )=-2id\xi d\eta .}</annotation>

</semantics>

</math><img src="./20d535e3c1e7eac97b747955014fe59ba5d16d13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.513ex; height:3.009ex;" alt="{\displaystyle d\zeta \wedge d{\bar {\zeta }}=(d\xi +id\eta )\wedge (d\xi -id\eta )=-2id\xi d\eta .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Generalized_analytic_functions">Generalized analytic functions</h3></div>

If a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(z)}</annotation>

</semantics>

</math><img src="./4b7703ad73053a9d1e8d6f43d873ac49825b1955.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.34ex; height:2.843ex;" alt="{\displaystyle M(z)}" loading="lazy"> is holomorphic in some complex region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, then

<mfrac>

<mrow>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial M}{\partial {\bar {z}}}}=0,}</annotation>

</semantics>

</math><img src="./ebcc4d8c3fa0ee48ad78d04af8582cdc511cba36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.504ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial M}{\partial {\bar {z}}}}=0,}" loading="lazy"></dd></dl>

in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">. For generalized analytic functions, this equation is replaced by

<mfrac>

<mrow>

</mrow>

<mrow>

<mover>

</mover>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial M}{\partial {\bar {z}}}}=A(z,{\bar {z}})M+B(z,{\bar {z}}){\overline {M}},}</annotation>

</semantics>

</math><img src="./1503334ca6016cfd63349d68883fd036d9463e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.266ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial M}{\partial {\bar {z}}}}=A(z,{\bar {z}})M+B(z,{\bar {z}}){\overline {M}},}" loading="lazy"></dd></dl>

in a region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {M}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {M}}}</annotation>

</semantics>

</math><img src="./c0e2d2cef85c7247cc507120eb8980b6f2714591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.68ex; height:3.009ex;" alt="{\displaystyle {\overline {M}}}" loading="lazy"> is the complex conjugate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(z,{\bar {z}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(z,{\bar {z}})}</annotation>

</semantics>

</math><img src="./7229ad3be736d8c8e38d13a00ce87a2fa20dd525.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.971ex; height:2.843ex;" alt="{\displaystyle A(z,{\bar {z}})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(z,{\bar {z}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B(z,{\bar {z}})}</annotation>

</semantics>

</math><img src="./09fa7b0d908fa3afe54de30cb5d934db026c5226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.992ex; height:2.843ex;" alt="{\displaystyle B(z,{\bar {z}})}" loading="lazy"> are functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}}</annotation>

</semantics>

</math><img src="./52dd0599595d539f7d757ec21da6c6e6ac3ad427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.296ex; height:2.009ex;" alt="{\displaystyle {\bar {z}}}" loading="lazy">.

Generalized analytic functions have applications in <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, in solving certain type of multidimensional <a href="Nonlinear_partial_differential_equation" title="Nonlinear partial differential equation">nonlinear partial differential equations</a> and multidimensional <a href="Inverse_scattering_transform" title="Inverse scattering transform">inverse scattering</a>.<a href="#cite_note-FOOTNOTEAblowitzFokas2003601-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications_to_integrability_theory">Applications to integrability theory</h2></div>

Riemann–Hilbert problems have applications to several related classes of problems.

<dl><dt>A. <a href="Integrable_model" class="mw-redirect" title="Integrable model">Integrable models</a></dt>

<dd>The <a href="Inverse_scattering" class="mw-redirect" title="Inverse scattering">inverse scattering</a> or inverse spectral problem associated to the <a href="Cauchy_problem" title="Cauchy problem">Cauchy problems</a> for 1+1 dimensional <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a> on the line, or to periodic problems, or even to initial-boundary value problems (<a href="#CITEREFFokas2002">Fokas (2002)</a>), can be stated as a Riemann–Hilbert problem. Likewise the inverse <a href="Monodromy" title="Monodromy">monodromy</a> problem for <a href="Painlev%C3%A9_equations" class="mw-redirect" title="Painlevé equations">Painlevé equations</a> can be stated as a Riemann–Hilbert problem.</dd></dl>

<dl><dt>B. <a href="Orthogonal_polynomials" title="Orthogonal polynomials">Orthogonal polynomials</a>, <a href="Random_matrices" class="mw-redirect" title="Random matrices">Random matrices</a></dt>

<dd>Given a weight on a contour, the corresponding orthogonal polynomials can be computed via the solution of a Riemann–Hilbert factorization problem (<a href="#CITEREFFokasItsKitaev1992">Fokas, Its & Kitaev (1992)</a>). Furthermore, the distribution of eigenvalues of random matrices in several classical ensembles is reduced to computations involving orthogonal polynomials (see e.g. <a href="#CITEREFDeift2000">Deift (2000)</a>).</dd></dl>

<dl><dt>C. Combinatorial <a href="Probability" title="Probability">probability</a></dt>

<dd>The most celebrated example is the theorem of <a href="#CITEREFBaikDeiftJohansson1999">Baik, Deift & Johansson (1999)</a> on the distribution of the length of the longest increasing subsequence of a random permutation. Together with the study of B above, it is one of the original rigorous investigations of so-called "integrable probability". But the connection between the theory of integrability and various classical ensembles of random matrices goes back to the work of Dyson (see e.g. <a href="#CITEREFDyson1976">Dyson (1976)</a>).</dd></dl>

<dl><dt>D. Connection to <a href="Donaldson-Thomas_theory" class="mw-redirect" title="Donaldson-Thomas theory">Donaldson-Thomas theory</a></dt>

<dd>The work of Bridgeland <a href="#CITEREFBridgeland2019">Bridgeland (2019)</a> studies a class of Riemann-Hilbert problems coming from Donaldson-Thomas theory and makes connections with Gromov-Witten theory and exact <a href="WKB_approximation" title="WKB approximation">WKB</a>.</dd></dl>

The numerical analysis of Riemann–Hilbert problems can provide an effective way for numerically solving integrable <a href="Partial_differential_equation" title="Partial differential equation">PDEs</a> (see e.g. <a href="#CITEREFTrogdonOlver2016">Trogdon & Olver (2016)</a>).

<div class="mw-heading mw-heading3"><h3 id="Use_for_asymptotics">Use for asymptotics</h3></div>

In particular, Riemann–Hilbert factorization problems are used to extract asymptotic values for the three problems above (say, as time goes to infinity, or as the dispersion coefficient goes to zero, or as the polynomial degree goes to infinity, or as the size of the permutation goes to infinity). There exists a method for extracting the asymptotic behavior of solutions of Riemann–Hilbert problems, analogous to the <a href="Method_of_stationary_phase" class="mw-redirect" title="Method of stationary phase">method of stationary phase</a> and the <a href="Method_of_steepest_descent" title="Method of steepest descent">method of steepest descent</a> applicable to exponential integrals.

By analogy with the classical asymptotic methods, one "deforms" Riemann–Hilbert problems which are not explicitly solvable to problems that are. The so-called "nonlinear" method of stationary phase is due to <a href="#CITEREFDeiftZhou1993">Deift & Zhou (1993)</a>, expanding on a previous idea by <a href="#CITEREFIts1982">Its (1982)</a> and <a href="#CITEREFManakov1974">Manakov (1974)</a> and using technical background results from <a href="#CITEREFBealsCoifman1984">Beals & Coifman (1984)</a> and <a href="#CITEREFZhou1989">Zhou (1989)</a>. A crucial ingredient of the Deift–Zhou analysis is the asymptotic analysis of singular integrals on contours. The relevant kernel is the standard <a href="Cauchy_kernel" class="mw-redirect" title="Cauchy kernel">Cauchy kernel</a> (see <a href="#CITEREFGakhov2001">Gakhov (2001)</a>; also cf. the scalar example below).

An essential extension of the nonlinear method of stationary phase has been the introduction of the so-called finite gap g-function transformation by <a href="#CITEREFDeiftVenakidesZhou1997">Deift, Venakides & Zhou (1997)</a>, which has been crucial in most applications. This was inspired by work of Lax, Levermore and Venakides, who reduced the analysis of the small dispersion limit of the <a href="KdV_equation" class="mw-redirect" title="KdV equation">KdV equation</a> to the analysis of a maximization problem for a logarithmic potential under some external field: a variational problem of "electrostatic" type (see <a href="#CITEREFLaxLevermore1983">Lax & Levermore (1983)</a>). The g-function is the logarithmic transform of the maximizing "equilibrium" measure. The analysis of the small dispersion limit of <a href="KdV_equation" class="mw-redirect" title="KdV equation">KdV equation</a> has in fact provided the basis for the analysis of most of the work concerning "real" orthogonal polynomials (i.e. with the orthogonality condition defined on the real line) and Hermitian random matrices.

Perhaps the most sophisticated extension of the theory so far is the one applied to the "non self-adjoint" case, i.e. when the underlying Lax operator (the first component of the <a href="Lax_pair" title="Lax pair">Lax pair</a>) is not <a href="Self-adjoint" title="Self-adjoint">self-adjoint</a>, by <a href="#CITEREFKamvissisMcLaughlinMiller2003">Kamvissis, McLaughlin & Miller (2003)</a>. In that case, actual "steepest descent contours" are defined and computed. The corresponding variational problem is a max-min problem: one looks for a contour that minimizes the "equilibrium" measure. The study of the variational problem and the proof of existence of a regular solution, under some conditions on the external field, was done in <a href="#CITEREFKamvissisRakhmanov2005">Kamvissis & Rakhmanov (2005)</a>; the contour arising is an "S-curve", as defined and studied in the 1980s by Herbert R. Stahl, Andrei A. Gonchar and Evguenii A Rakhmanov.

An alternative asymptotic analysis of Riemann–Hilbert factorization problems is provided in <a href="#CITEREFMcLaughlinMiller2006">McLaughlin & Miller (2006)</a>, especially convenient when jump matrices do not have analytic extensions. Their method is based on the analysis of d-bar problems, rather than the asymptotic analysis of singular integrals on contours. An alternative way of dealing with jump matrices with no analytic extensions was introduced in <a href="#CITEREFVarzugin1996">Varzugin (1996)</a>.

Another extension of the theory appears in <a href="#CITEREFKamvissisTeschl2012">Kamvissis & Teschl (2012)</a> where the underlying space of the Riemann–Hilbert problem is a compact hyperelliptic <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>. The correct factorization problem is no more holomorphic, but rather <a href="Meromorphic_function" title="Meromorphic function">meromorphic</a>, by reason of the <a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a>. The related singular kernel is not the usual Cauchy kernel, but rather a more general kernel involving meromorphic differentials defined naturally on the surface (see e.g. the appendix in <a href="#CITEREFKamvissisTeschl2012">Kamvissis & Teschl (2012)</a>). The Riemann–Hilbert problem deformation theory is applied to the problem of stability of the infinite periodic <a href="Toda_lattice" title="Toda lattice">Toda lattice</a> under a "short range" perturbation (for example a perturbation of a finite number of particles).

Most Riemann–Hilbert factorization problems studied in the literature are 2-dimensional, i.e., the unknown matrices are of dimension 2. Higher-dimensional problems have been studied by <a href="Arno_Kuijlaars" title="Arno Kuijlaars">Arno Kuijlaars</a> and collaborators, see e.g. <a href="#CITEREFKuijlaarsLópez2015">Kuijlaars & López (2015)</a>.

<ul><li><a href="Riemann%E2%80%93Hilbert_correspondence" title="Riemann–Hilbert correspondence">Riemann–Hilbert correspondence</a></li>

<li><a href="Wiener-Hopf_method" class="mw-redirect" title="Wiener-Hopf method">Wiener-Hopf method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-FOOTNOTEClanceyGohberg1981-1"><a href="#cite_ref-FOOTNOTEClanceyGohberg1981_1-0">^</a> <a href="#CITEREFClanceyGohberg1981">Clancey & Gohberg 1981</a>.

</li>

<li id="cite_note-FOOTNOTEAblowitzFokas200371–72-2"><a href="#cite_ref-FOOTNOTEAblowitzFokas200371–72_2-0">^</a> <a href="#CITEREFAblowitzFokas2003">Ablowitz & Fokas 2003</a>, pp. 71–72.

</li>

<li id="cite_note-FOOTNOTEBitsadze2001-3"><a href="#cite_ref-FOOTNOTEBitsadze2001_3-0">^</a> <a href="#CITEREFBitsadze2001">Bitsadze 2001</a>.

</li>

<li id="cite_note-FOOTNOTEPandey1996-4"><a href="#cite_ref-FOOTNOTEPandey1996_4-0">^</a> <a href="#CITEREFPandey1996">Pandey 1996</a>.

</li>

<li id="cite_note-FOOTNOTEAblowitzFokas2003514-5"><a href="#cite_ref-FOOTNOTEAblowitzFokas2003514_5-0">^</a> <a href="#CITEREFAblowitzFokas2003">Ablowitz & Fokas 2003</a>, pp. 514.

</li>

<li id="cite_note-FOOTNOTEPandey1996§2.2-6"><a href="#cite_ref-FOOTNOTEPandey1996§2.2_6-0">^</a> <a href="#CITEREFPandey1996">Pandey 1996</a>, §2.2.

</li>

<li id="cite_note-FOOTNOTENoble1958§4.2-7"><a href="#cite_ref-FOOTNOTENoble1958§4.2_7-0">^</a> <a href="#CITEREFNoble1958">Noble 1958</a>, §4.2.

</li>

<li id="cite_note-FOOTNOTEAblowitzFokas2003§7.5-8"><a href="#cite_ref-FOOTNOTEAblowitzFokas2003§7.5_8-0">^</a> <a href="#CITEREFAblowitzFokas2003">Ablowitz & Fokas 2003</a>, §7.5.

</li>

<li id="cite_note-FOOTNOTEAblowitzFokas2003598-9"><a href="#cite_ref-FOOTNOTEAblowitzFokas2003598_9-0">^</a> <a href="#CITEREFAblowitzFokas2003">Ablowitz & Fokas 2003</a>, p. 598.

</li>

<li id="cite_note-FOOTNOTEAblowitzFokas2003600-10"><a href="#cite_ref-FOOTNOTEAblowitzFokas2003600_10-0">^</a> <a href="#CITEREFAblowitzFokas2003">Ablowitz & Fokas 2003</a>, p. 600.

</li>

<li id="cite_note-FOOTNOTEVekua2014-11"><a href="#cite_ref-FOOTNOTEVekua2014_11-0">^</a> <a href="#CITEREFVekua2014">Vekua 2014</a>.

</li>

<li id="cite_note-FOOTNOTEAblowitzFokas2003601-12"><a href="#cite_ref-FOOTNOTEAblowitzFokas2003601_12-0">^</a> <a href="#CITEREFAblowitzFokas2003">Ablowitz & Fokas 2003</a>, p. 601.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q5986515#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1217" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q5986515#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1217" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://id.worldcat.org/fast/1097802">FAST</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4317184-9">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85114041">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb122673712">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb122673712">BnF data</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007538989705171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://www.idref.fr/031462359">IdRef</a></li><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/b71d097d-8330-4193-94e2-70ff8fe96f31">Yale LUX</a></li></ul></div></td></tr></tbody></table></div>

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</style><div id="Bernhard_Riemann145" style="font-size:114%;margin:0 4em"><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cauchy%E2%80%93Riemann_equations" title="Cauchy–Riemann equations">Cauchy–Riemann equations</a></li>

<li><a href="Generalized_Riemann_hypothesis" title="Generalized Riemann hypothesis">Generalized Riemann hypothesis</a></li>

<li><a href="Grand_Riemann_hypothesis" title="Grand Riemann hypothesis">Grand Riemann hypothesis</a></li>

<li><a href="Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem" title="Grothendieck–Riemann–Roch theorem">Grothendieck–Hirzebruch–Riemann–Roch theorem</a></li>

<li><a href="Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem" title="Hirzebruch–Riemann–Roch theorem">Hirzebruch–Riemann–Roch theorem</a></li>

<li><a href="Local_zeta_function" title="Local zeta function">Local zeta function</a></li>

<li><a href="Measurable_Riemann_mapping_theorem" title="Measurable Riemann mapping theorem">Measurable Riemann mapping theorem</a></li>

<li><a href="Riemann_(crater)" title="Riemann (crater)">Riemann (crater)</a></li>

<li><a href="Riemann_Xi_function" class="mw-redirect" title="Riemann Xi function">Riemann Xi function</a></li>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Riemann_invariant" title="Riemann invariant">Riemann invariant</a></li>

<li><a href="Riemann_mapping_theorem" title="Riemann mapping theorem">Riemann mapping theorem</a></li>

<li><a href="Riemann_form" title="Riemann form">Riemann form</a></li>

<li><a href="Riemann_problem" title="Riemann problem">Riemann problem</a></li>

<li><a href="Riemann_series_theorem" title="Riemann series theorem">Riemann series theorem</a></li>

<li><a href="Riemann_solver" title="Riemann solver">Riemann solver</a></li>

<li><a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li>

<li><a href="Riemann_sum" title="Riemann sum">Riemann sum</a></li>

<li><a href="Riemann_surface" title="Riemann surface">Riemann surface</a></li>

<li><a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a></li>

<li><a href="Riemann's_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li>

<li><a href="Riemann's_minimal_surface" title="Riemann's minimal surface">Riemann's minimal surface</a></li>

<li><a href="Riemannian_circle" class="mw-redirect" title="Riemannian circle">Riemannian circle</a></li>

<li><a href="Riemannian_connection_on_a_surface" title="Riemannian connection on a surface">Riemannian connection on a surface</a></li>

<li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li>

<li><a href="Riemann%E2%80%93Hilbert_correspondence" title="Riemann–Hilbert correspondence">Riemann–Hilbert correspondence</a></li>

<li><a href="Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue lemma</a></li>

<li><a href="Riemann%E2%80%93Liouville_integral" title="Riemann–Liouville integral">Riemann–Liouville integral</a></li>

<li><a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li>

<li><a href="Riemann%E2%80%93Roch_theorem_for_smooth_manifolds" title="Riemann–Roch theorem for smooth manifolds">Riemann–Roch theorem for smooth manifolds</a></li>

<li><a href="Riemann%E2%80%93Siegel_formula" title="Riemann–Siegel formula">Riemann–Siegel formula</a></li>

<li><a href="Riemann%E2%80%93Siegel_theta_function" title="Riemann–Siegel theta function">Riemann–Siegel theta function</a></li>

<li><a href="Riemann%E2%80%93Silberstein_vector" title="Riemann–Silberstein vector">Riemann–Silberstein vector</a></li>

<li><a href="Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a></li>

<li><a href="Riemann%E2%80%93von_Mangoldt_formula" title="Riemann–von Mangoldt formula">Riemann–von Mangoldt formula</a></li></ul>

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<ul><li> Category</li></ul>

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