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The '''
In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '-' subscripts) which are respectively [[analytic function|analytic]] in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also
==
The key step in many
<math>\Phi_+(\alpha) = \frac{1}{2\pi i} \int_{C_1} \Phi(z) \frac{dz}{z-\alpha}</math> and
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==Example==
Let us consider the linear [[partial differential equation]]
:<math>\boldsymbol{L}_{xy}f(x,y)=0,</math>
where <math>\boldsymbol{L}_{xy}</math> is a linear operator which contains
derivatives with respect to <math>x</math> and <math>y</math>,
subject to the mixed conditions on <math>y=0</math>, for some prescribed
function <math>g(x)</math>,
:<math>f=g(x)</math> for <math>x\leq 0, \quad f_{y}=0</math> when <math>x>0</math>.
and decay at infinity i.e. <math>f\rightarrow 0</math>
as <math>\boldsymbol{x}\rightarrow \infty</math>. Taking a [[Fourier transform]] with respect to x results in the following [[
: <math>\boldsymbol{L}_{y}\hat{f}(k,y)-P(k,y)\hat{f}(k,y)=0,</math>
where <math>\boldsymbol{L}_{y}</math> is a linear operator containing
<math>y</math> derivatives only, <math>P(k,y)</math> is a known function
of <math>y</math> and <math>k</math> and
: <math> \hat{f}(k)=\int_{-\infty}^{\infty} f(x,y)e^{-ikx}\textrm{d}x. </math>
If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity
is denoted <math>\hat{F}(k,y)</math>, a general
solution can be written as
: <math>
\hat{f}=C(k)\hat{F}(k,y),
</math>
where <math>C(k)</math> is an unknown function to be determined by the boundary conditions on <math>y=0</math>.
The key idea is to
split <math>\hat{f}</math> into two separate functions, <math>\hat{f}_{+}</math> and <math>\hat{f}_{-}</math> which are analytic in the lower- and upper-halves of the complex plane, respectively
: <math> \hat{f}_{+}(k,y)=\int_{0}^{\infty} f(x,y)e^{-ikx}\textrm{d}x, </math>
:<math> \hat{f}_{
</math>
The boundary conditions then give
: <math>
\hat{f}_{-}(k,0)+\hat{f}_{+}(k,0)
=
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=
C(k)F(k,0)</math>
and, on taking derivatives with respect to <math>y</math>,
: <math> \hat{f}'_{-}(k,0)+\hat{f}'_{+}(k,0) =
\hat{f}'_{-}(k,0)
Eliminating <math>C(k)</math> yields
: <math>
\hat{g}(k)+\hat{f}_{+}(k,0) - \hat{f}'_{-}(k,0)/K(k) = 0,</math>
where
: <math> K(k)=\frac{F'(k,0)}{F(k,0)}.</math>
Now <math>K(k)</math> can be decomposed into the product of functions <math>K^{-}</math> and <math>K^{+}</math> which analytic in the upper-half plane or lower-half plane, respectively
: <math> K(k)=K^{+}(k)K^{-}(k), </math>
: <
\hbox{log}
K^{-}=
\frac{1}{2\pi i}\int_{-\infty}^{\infty} \frac{\hbox{log}(K(z))}{z-k} \textrm{d}z,
\quad
\hbox{Im}k>0,
</math>
: <math>
\hbox{log}
K^{+}=
-\frac{1}{2\pi i}\int_{-\infty}^{\infty} \frac{\hbox{log}(K(z))}{z-k} \textrm{d}z,
\quad
\hbox{Im}k<0.
</math>
Consequently,
: <math>
K_{+}(k)\hat{g}_{+}(k) + K_{+}(k)\hat{f}_{+}(k,0) =
\hat{f}'_{-}(k,0)/K_{-}(k) - K_{+}(k)\hat{g}_{
</math>
where it has been assumed that <math>g</math> can be broken down into functions analytic in the lower-half plane
<math>g_{+}</math> and upper-half plane <math>g_{-}</math>, respectively.
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left- or right-hand sides in their respective half-planes. Furthermore, since it can
be shown that the functions on either side of the above equation decay at large
<math>k</math>, an application of [[Liouville's theorem (complex analysis)|Liouville's theorem]]
:<
\hat{f}_{+}(k,0) = -\hat{g}_{+}(k),
</math>
and so
: <math>
C(k) = \frac{\hat{g}(k)-\hat{g}_{+}(k)}{F(k,0)}.
</math>
== See also ==
* [[Wiener filter]]
==External links==
* {{SpringerEOM |id=W/w097910}}
* [http://www.wikiwaves.org/index.php/Category:Wiener-Hopf Wiener-Hopf method] at Wikiwaves
[[Category:Partial differential equations]]
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