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''harmonically-conjugate functions''
 
''harmonically-conjugate functions''
  
A pair of real harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250402.png" /> which are the real and imaginary parts of some analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250403.png" /> of a complex variable. In the case of one complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250404.png" />, two harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250406.png" /> are conjugate in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250407.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250408.png" /> if and only if they satisfy the Cauchy–Riemann equations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c0250409.png" />:
+
A pair of real harmonic functions $  u $
 +
and $  v $
 +
which are the real and imaginary parts of some analytic function $  f = u + iv $
 +
of a complex variable. In the case of one complex variable $  z = x + iy $,  
 +
two harmonic functions $  u = u ( x, y) $
 +
and $  v = v ( x, y) $
 +
are conjugate in a domain $  D $
 +
of the complex plane $  \mathbf C $
 +
if and only if they satisfy the Cauchy–Riemann equations in $  D $:
 +
 
 +
$$ \tag{1 }
 +
 
 +
\frac{\partial  u }{\partial  x }
 +
  = \
 +
 
 +
\frac{\partial  v }{\partial  y }
 +
,\ \
 +
 
 +
\frac{\partial  u }{\partial  y }
 +
  = -
 +
 
 +
\frac{\partial  v }{\partial  x }
 +
.
 +
$$
 +
 
 +
The roles of  $  u $
 +
and  $  v $
 +
in (1) are not symmetric: $  v $
 +
is a conjugate for  $  u $
 +
but  $  - u $,
 +
and not  $  u $,
 +
is a conjugate for  $  v $.  
 +
Given a harmonic function  $  u = u ( x, y) $,
 +
a local conjugate  $  v = v ( x, y) $
 +
and a local complete analytic function  $  f = u + iv $
 +
are easily determined up to a constant term. This can be done, for example, using the Goursat formula
 +
 
 +
$$ \tag{2 }
 +
f ( z)  =  2u
 +
\left (
 +
{
 +
\frac{z + \overline{z}\; {}  ^ {0} }{2}
 +
} ,\
 +
{
 +
\frac{z - \overline{z}\; {}  ^ {0} }{2i}
 +
}
 +
\right )
 +
- u ( x  ^ {0} , y  ^ {0} ) + ic
 +
$$
 +
 
 +
in a neighbourhood of some point  $  z  ^ {0} = x  ^ {0} + iy  ^ {0} $
 +
in the domain of definition of  $  u $.
 +
 
 +
In the case of several complex variables  $  z = x + iy = ( z _ {1} \dots z _ {n} ) = ( x _ {1} \dots x _ {n} ) + i ( y \dots y _ {n} ) $,
 +
$  n > 1 $,
 +
the Cauchy–Riemann system becomes overdetermined
 +
 
 +
$$ \tag{3 }
 +
 
 +
\frac{\partial  u }{\partial  x _ {k} }
 +
  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{\partial  v }{\partial  y _ {k} }
 +
,\ \
  
The roles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504012.png" /> in (1) are not symmetric: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504013.png" /> is a conjugate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504014.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504015.png" />, and not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504016.png" />, is a conjugate for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504017.png" />. Given a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504018.png" />, a local conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504019.png" /> and a local complete analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504020.png" /> are easily determined up to a constant term. This can be done, for example, using the Goursat formula
+
\frac{\partial  u }{\partial  y _ {k} }
 +
  = -
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\frac{\partial  v }{\partial  x _ {k} }
 +
,\ \
 +
k = 1 \dots n.
 +
$$
  
in a neighbourhood of some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504022.png" /> in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504023.png" />.
+
It follows from (3) that for  $  n > 1 $,
 +
$  u $
 +
can no longer be taken as an arbitrary harmonic function; it must belong to the subclass of pluriharmonic functions (cf. [[Pluriharmonic function|Pluriharmonic function]]). The conjugate pluriharmonic function  $  v $
 +
can then be found using (2).
  
In the case of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504025.png" />, the Cauchy–Riemann system becomes overdetermined
+
There are various analogues of conjugate harmonic functions  $  ( u , v) $
 +
involving a vector function  $  f = ( u _ {1} \dots u _ {m} ) $
 +
whose components  $  u _ {j} = u _ {j} ( x _ {1} \dots x _ {n} ) $
 +
are real functions of real variables  $  x _ {1} \dots x _ {n} $.  
 +
An example is a gradient system  $  f = ( u _ {1} \dots u _ {n} ) $
 +
satisfying the generalized system of Cauchy–Riemann equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{4 }
 +
\sum _ {j = 1 } ^ { n }
  
It follows from (3) that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504028.png" /> can no longer be taken as an arbitrary harmonic function; it must belong to the subclass of pluriharmonic functions (cf. [[Pluriharmonic function|Pluriharmonic function]]). The conjugate pluriharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504029.png" /> can then be found using (2).
+
\frac{\partial  u _ {j} }{\partial  x _ {j} }
 +
  = 0,\ \
  
There are various analogues of conjugate harmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504030.png" /> involving a vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504031.png" /> whose components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504032.png" /> are real functions of real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504033.png" />. An example is a gradient system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504034.png" /> satisfying the generalized system of Cauchy–Riemann equations
+
\frac{\partial  u _ {i} }{\partial  x _ {j} }
 +
  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{\partial  u _ {j} }{\partial  x _ {i} }
 +
,\ \
 +
i, j = 1 \dots n,\
 +
i \neq j,
 +
$$
  
 
which can also be written in abbreviated form:
 
which can also be written in abbreviated form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504036.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div}  f  = 0,\ \
 +
\mathop{\rm curl}  f  = 0.
 +
$$
  
If the conditions (4) hold in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504037.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504038.png" /> homeomorphic to a ball, then there is a harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504041.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504042.png" />, it turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504043.png" /> is an analytic function of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025040/c02504044.png" />. The behaviour of the solutions of (4) is in some respects similar to that of the Cauchy–Riemann system (1), for example in the study of boundary properties (see [[#References|[3]]]).
+
If the conditions (4) hold in a domain $  D $
 +
of a Euclidean space $  \mathbf R  ^ {n} $
 +
homeomorphic to a ball, then there is a harmonic function $  h $
 +
on $  D $
 +
such that $  f = \mathop{\rm grad}  h $.  
 +
When $  n = 2 $,  
 +
it turns out that $  u _ {2} + iu _ {1} $
 +
is an analytic function of the variable $  z = x _ {1} + ix _ {2} $.  
 +
The behaviour of the solutions of (4) is in some respects similar to that of the Cauchy–Riemann system (1), for example in the study of boundary properties (see [[#References|[3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


harmonically-conjugate functions

A pair of real harmonic functions $ u $ and $ v $ which are the real and imaginary parts of some analytic function $ f = u + iv $ of a complex variable. In the case of one complex variable $ z = x + iy $, two harmonic functions $ u = u ( x, y) $ and $ v = v ( x, y) $ are conjugate in a domain $ D $ of the complex plane $ \mathbf C $ if and only if they satisfy the Cauchy–Riemann equations in $ D $:

$$ \tag{1 } \frac{\partial u }{\partial x } = \ \frac{\partial v }{\partial y } ,\ \ \frac{\partial u }{\partial y } = - \frac{\partial v }{\partial x } . $$

The roles of $ u $ and $ v $ in (1) are not symmetric: $ v $ is a conjugate for $ u $ but $ - u $, and not $ u $, is a conjugate for $ v $. Given a harmonic function $ u = u ( x, y) $, a local conjugate $ v = v ( x, y) $ and a local complete analytic function $ f = u + iv $ are easily determined up to a constant term. This can be done, for example, using the Goursat formula

$$ \tag{2 } f ( z) = 2u \left ( { \frac{z + \overline{z}\; {} ^ {0} }{2} } ,\ { \frac{z - \overline{z}\; {} ^ {0} }{2i} } \right ) - u ( x ^ {0} , y ^ {0} ) + ic $$

in a neighbourhood of some point $ z ^ {0} = x ^ {0} + iy ^ {0} $ in the domain of definition of $ u $.

In the case of several complex variables $ z = x + iy = ( z _ {1} \dots z _ {n} ) = ( x _ {1} \dots x _ {n} ) + i ( y \dots y _ {n} ) $, $ n > 1 $, the Cauchy–Riemann system becomes overdetermined

$$ \tag{3 } \frac{\partial u }{\partial x _ {k} } = \ \frac{\partial v }{\partial y _ {k} } ,\ \ \frac{\partial u }{\partial y _ {k} } = - \frac{\partial v }{\partial x _ {k} } ,\ \ k = 1 \dots n. $$

It follows from (3) that for $ n > 1 $, $ u $ can no longer be taken as an arbitrary harmonic function; it must belong to the subclass of pluriharmonic functions (cf. Pluriharmonic function). The conjugate pluriharmonic function $ v $ can then be found using (2).

There are various analogues of conjugate harmonic functions $ ( u , v) $ involving a vector function $ f = ( u _ {1} \dots u _ {m} ) $ whose components $ u _ {j} = u _ {j} ( x _ {1} \dots x _ {n} ) $ are real functions of real variables $ x _ {1} \dots x _ {n} $. An example is a gradient system $ f = ( u _ {1} \dots u _ {n} ) $ satisfying the generalized system of Cauchy–Riemann equations

$$ \tag{4 } \sum _ {j = 1 } ^ { n } \frac{\partial u _ {j} }{\partial x _ {j} } = 0,\ \ \frac{\partial u _ {i} }{\partial x _ {j} } = \ \frac{\partial u _ {j} }{\partial x _ {i} } ,\ \ i, j = 1 \dots n,\ i \neq j, $$

which can also be written in abbreviated form:

$$ \mathop{\rm div} f = 0,\ \ \mathop{\rm curl} f = 0. $$

If the conditions (4) hold in a domain $ D $ of a Euclidean space $ \mathbf R ^ {n} $ homeomorphic to a ball, then there is a harmonic function $ h $ on $ D $ such that $ f = \mathop{\rm grad} h $. When $ n = 2 $, it turns out that $ u _ {2} + iu _ {1} $ is an analytic function of the variable $ z = x _ {1} + ix _ {2} $. The behaviour of the solutions of (4) is in some respects similar to that of the Cauchy–Riemann system (1), for example in the study of boundary properties (see [3]).

References

[1] A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)
[2] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[3] E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)
How to Cite This Entry:
Conjugate harmonic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conjugate_harmonic_functions&oldid=46471
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article