Complex analysis is the study of complex numbers together with their derivatives , manipulation, and
other properties. Complex analysis is an extremely powerful tool with an unexpectedly
large number of practical applications to the solution of physical problems. Contour
integration , for example, provides a method of computing difficult integrals
by investigating the singularities of the function in regions of the complex
plane near and between the limits of integration.
The key result in complex analysis is the Cauchy integral theorem , which is the reason that single-variable complex analysis has
so many nice results. A single example of the unexpected power of complex analysis
is Picard's great theorem , which states that
an analytic function assumes every complex
number , with possibly one exception, infinitely often in any neighborhood
of an essential singularity !
A fundamental result of complex analysis is the Cauchy-Riemann equations , which give the conditions a function
must satisfy in order for a complex generalization of the derivative ,
the so-called complex derivative , to exist.
When the complex derivative is defined "everywhere,"
the function is said to be analytic .
See also Analytic Continuation ,
Argument Principle ,
Branch
Cut ,
Branch Point ,
Cauchy
Integral Formula ,
Cauchy Integral Theorem ,
Cauchy Principal Value ,
Cauchy-Riemann
Equations ,
Complex Number ,
Complex
Residue ,
Conformal Mapping ,
Contour
Integration ,
de Moivre's Identity ,
Euler
Formula ,
Inside-Outside Theorem ,
Jordan's Lemma ,
Laurent
Series ,
Liouville's Conformality
Theorem ,
Monogenic Function ,
Morera's
Theorem ,
Permanence of Algebraic Form ,
Picard's Great Theorem ,
Pole ,
Polygenic Function Explore this topic in the MathWorld classroom
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References Arfken, G. "Functions of a Complex Variable I: Analytic Properties, Mapping" and "Functions of a Complex Variable II: Calculus
of Residues." Chs. 6-7 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 352-395
and 396-436, 1985. Boas, R. P. Invitation
to Complex Analysis. New York: Random House, 1987. Churchill,
R. V. and Brown, J. W. Complex
Variables and Applications, 6th ed. New York: McGraw-Hill, 1995. Conway,
J. B. Functions
of One Complex Variable, 2nd ed. New York: Springer-Verlag, 1995. Forsyth,
A. R. Theory
of Functions of a Complex Variable, 3rd ed. Cambridge, England: Cambridge
University Press, 1918. Knopp, K. Theory
of Functions Parts I and II, Two Volumes Bound as One, Part I. New York:
Dover, 1996. Krantz, S. G. Handbook
of Complex Variables. Boston, MA: Birkhäuser, 1999. Lang,
S. Complex
Analysis, 3rd ed. New York: Springer-Verlag, 1993. Mathews, J. H.
and Howell, R. W. Complex
Analysis for Mathematics and Engineering, 5th ed. Sudbury, MA: Jones and
Bartlett, 2006. Mathews, J. H. "Complex
Analysis: Mathematica Notebooks." http://library.wolfram.com/infocenter/MathSource/6099/ .Morse,
P. M. and Feshbach, H. "Functions of a Complex Variable" and "Tabulation
of Properties of Functions of Complex Variables." Ch. 4 in Methods
of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 348-491 and
480-485, 1953. Needham, T. Visual
Complex Analysis. New York: Clarendon Press, 2000. Shaw, W. Complex
Analysis with Mathematica. Cambridge, England: Cambridge University Press,
2006. http://www.mth.kcl.ac.uk/~shaww/web_page/books/complex/ . Silverman,
R. A. Introductory
Complex Analysis. New York: Dover, 1984. Weisstein, E. W.
"Books about Complex Analysis." http://www.ericweisstein.com/encyclopedias/books/ComplexAnalysis.html . Referenced
on Wolfram|Alpha Complex Analysis
Cite this as:
Weisstein, Eric W. "Complex Analysis."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexAnalysis.html
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