Editing Hilbert transform (section)

=== Riemann–Hilbert problem ===
One form of the [[Riemann–Hilbert problem]] seeks to identify pairs of functions {{math|''F''<sub>+</sub>}} and {{math|''F''<sub>−</sub>}} such that {{math|''F''<sub>+</sub>}} is [[holomorphic function|holomorphic]] on the upper half-plane and {{math|''F''<sub>−</sub>}} is holomorphic on the lower half-plane, such that for {{mvar|x}} along the real axis,
<math display="block">F_{+}(x) - F_{-}(x) = f(x)</math>

where {{math|''f''(''x'')}} is some given real-valued function of {{nowrap|<math>x \isin \mathbb{R}</math>.}} The left-hand side of this equation may be understood either as the difference of the limits of {{math|''F''<sub>±</sub>}} from the appropriate half-planes, or as a [[hyperfunction]] distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.

Formally, if {{math|''F''<sub>±</sub>}} solve the Riemann–Hilbert problem
<math display="block">f(x) = F_{+}(x) - F_{-}(x)</math>

then the Hilbert transform of {{math|''f''(''x'')}} is given by{{sfn|Pandey|1996|loc=Chapter 2}}
<math display="block">H(f)(x) = -i \bigl( F_{+}(x) + F_{-}(x) \bigr) .</math>