Complex Analysis
The study of Complex Numbers, their Derivatives, manipulation, and otherproperties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications tothe solution of physical problems. Contour Integration, for example, provides a method of computing difficultIntegrals by investigating the singularities of the function in regions of the Complex Plane nearand between the limits of integration.
The most fundamental result of complex analysis is the Cauchy-Riemann Equations, which give the conditions aFunction must satisfy in order for a complex generalization of the Derivative, the so-called ComplexDerivative, to exist. When the Complex Derivative is defined ``everywhere,'' the function is said to beAnalytic. A single example of the unexpected power of complex analysis is Picard'sTheorem, which states that an Analytic Function assumes every Complex Number, with possibly one exception,infinitely often in any Neighborhood of an Essential Singularity!
See also Analytic Continuation, Branch Cut, Branch Point, Cauchy Integral Formula, CauchyIntegral Theorem, Cauchy Principal Value, Cauchy-Riemann Equations, Complex Number, ConformalMap, 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 Theorem, Pole, PolygenicFunction, Residue (Complex Analysis)
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. Lang, S. Complex Analysis, 3rd ed. New York: Springer-Verlag, 1993. 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.
Complex Analysis
- Runge-Kutta Method
- Runge-Walsh Theorem
- Running Knot
- Russell's Antinomy
- Russell's Paradox
- Russian Multiplication
- Russian Roulette
- Ryser Formula
- Rényi's Parking Constants
- Rössler Model
- Saalschützian
- Saalschütz's Theorem
- s-Additive Sequence
- Saddle
- Saddle-Node Bifurcation
- Saddle Point (Fixed Point)
- Saddle Point (Function)
- Saddle Point (Game)
- Safarevich Conjecture
- Safe
- Sagitta
- Saint Andrew's Cross
- Saint Anthony's Cross
- Saint Petersburg Paradox
- Sal