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| − | ''d'Alembert–Euler conditions''
| + | #REDIRECT[[Cauchy-Riemann equations]] |
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| − | Conditions that must be satisfied by the real part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209701.png" /> and the imaginary part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209702.png" /> of a complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209704.png" />, for it to be monogenic and analytic as a function of a complex variable.
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| − | A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209705.png" />, defined in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209706.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209707.png" />-plane, is monogenic at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209708.png" />, i.e. has a derivative at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c0209709.png" /> as a function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097010.png" />, if and only if its real and imaginary parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097012.png" /> are differentiable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097013.png" /> as functions of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097015.png" />, and if, moreover, the Cauchy–Riemann equations hold at that point:
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| − | <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/c020/c020970/c02097016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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| − | If the Cauchy–Riemann equations are satisfied, then the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097017.png" /> can be expressed in any of the following forms:
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| − | <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/c020/c020970/c02097018.png" /></td> </tr></table>
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| − | <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/c020/c020970/c02097019.png" /></td> </tr></table>
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| − | A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097020.png" />, defined and single-valued in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097021.png" />, is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097022.png" /> if and only if its real and imaginary parts are differentiable functions satisfying the Cauchy–Riemann equations throughout <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097023.png" />. Each of the two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097025.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097026.png" /> satisfying the Cauchy–Riemann equations (1) is a harmonic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097028.png" />; the conditions (1) constitute conjugacy conditions of these two harmonic functions: Knowing one of them, the other may be found by integration.
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| − | The conditions (1) are valid for any two orthogonal directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097030.png" />, with the same mutual orientations as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097031.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097032.png" />-axes, in the form:
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| − | <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/c020/c020970/c02097033.png" /></td> </tr></table>
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| − | For example, in polar coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097034.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097035.png" />:
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| − | <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/c020/c020970/c02097036.png" /></td> </tr></table>
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| − | Defining the complex differential operators by
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| − | <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/c020/c020970/c02097037.png" /></td> </tr></table>
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| − | one can rewrite the Cauchy–Riemann equations (1) as
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| − | <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/c020/c020970/c02097038.png" /></td> </tr></table>
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| − | Thus, a differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097039.png" /> of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097041.png" /> is an analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097042.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097043.png" />.
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| − | For analytic functions of several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097046.png" />, the Cauchy–Riemann equations constitute a system of partial differential equations (overdetermined when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097047.png" />) for the functions
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| − | <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/c020/c020970/c02097048.png" /></td> </tr></table>
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| − | <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/c020/c020970/c02097049.png" /></td> </tr></table>
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| − | <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/c020/c020970/c02097050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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| − | or, in terms of the complex differentiation operators:
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| − | <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/c020/c020970/c02097051.png" /></td> </tr></table>
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| − | Each of the two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097053.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097054.png" /> satisfying the conditions (2) is a [[Pluriharmonic function|pluriharmonic function]] of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097057.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097058.png" /> the pluriharmonic functions constitute a proper subclass of the class of harmonic functions. The conditions (2) are conjugacy conditions for two pluriharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020970/c02097060.png" />: Knowing one of them, one can determine the other by integration.
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| − | The conditions (1) apparently occurred for the first time in the works of J. d'Alembert [[#References|[1]]]. Their first appearance as a criterion for analyticity was in a paper of L. Euler, delivered at the Petersburg Academy of Sciences in 1777 [[#References|[2]]]. A.L. Cauchy utilized the conditions (1) to construct the theory of functions, beginning with a memoir presented to the Paris Academy in 1814 (see [[#References|[3]]]). The celebrated dissertation of B. Riemann on the fundamentals of function theory dates from 1851 (see [[#References|[4]]]).
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. d'Alembert, "Essai d'une nouvelle théorie de la résistance des fluides" , Paris (1752)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Euler, ''Nova Acta Acad. Sci. Petrop.'' , '''10''' (1797) pp. 3–19</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.L. Cauchy, "Mémoire sur les intégrales définies" , ''Oeuvres complètes Ser. 1'' , '''1''' , Paris (1882) pp. 319–506</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse" H. Weber (ed.) , ''Riemann's gesammelte math. Werke'' , Dover, reprint (1953) pp. 3–48 (Dover, reprint, 1953)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''1''' , Chelsea (1977) pp. Chapt. 1 (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) pp. 1, Chapt. 1; 2, Chapt. 1 (In Russian)</TD></TR></table>
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| − | ====Comments====
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 24–26</TD></TR></table>
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