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Topic: Complex functions

	Complex Functions and Function Complexities
		Variable names (which are actually pointers), if passed as parameters to functions, will be treated as string literals.
		This mechanism effectively permits script functions to have a "return value" similar to C functions.
		For a function to return a string or array, use a dedicated variable.
	www.tldp.org /LDP/abs/html/complexfunct.html (1497 words)

	Complex analysis - Wikipedia, the free encyclopedia
		Complex analysis is the branch of mathematics investigating functions of complex numbers.
		Complex analysis is particularly concerned with analytic functions of complex variables, known as holomorphic functions.
		Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
	en.wikipedia.org /wiki/Complex_analysis (824 words)

	Learn more about Complex number in the online encyclopedia. (Site not responding. Last check: 2007-11-05)
		The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
		The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics.
		Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
	www.onlineencyclopedia.org /c/co/complex_number.html (1629 words)

	Complex analysis -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		Complex analysis is the branch of (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics investigating (additional info and facts about holomorphic function) holomorphic functions, i.e.
		functions which are defined in some region of the (additional info and facts about complex plane) complex plane, take complex values, and are differentiable as complex functions.
		The function on the larger domain is said to be (additional info and facts about analytically continued) analytically continued from its values on the smaller domain.
	www.absoluteastronomy.com /encyclopedia/c/co/complex_analysis.htm (705 words)

	Functions
		As in C, the function's opening bracket may optionally appear on the second line.
		Variable names (which are actually pointers), if passed as parameters to functions, will be treated as string literals and cannot be dereferenced.
		For a function to return a string or array, use a dedicated variable.
	www.linuxvalley.it /encyclopedia/ldp/guide/abs/functions.html (1317 words)

	30: Functions of a complex variable
		Complex variables studies the effect of assuming differentiability of functions defined on complex numbers.
		Fascinatingly, the effect is markedly different than for real functions; these functions are much more rigidly constrained, and in particular it is possible to make very definite comments about their global behaviour, convergence, and so on.
		Functions in the convex hull of some functions of the form 1/(z-z_j) have roots in the convex hull of the z_j.
	www.math.niu.edu /~rusin/known-math/index/30-XX.html (522 words)

	Analytic Functions, The Magnus Effect, and Wings
		Hence a general complex function f(z) of a complex variable z can be written as two real-valued functions u(x,y) and v(x,y) of two real variables.
		When dealing with a real-valued function of a single real variable, the derivative of the function with respect to that variable is unambiguous, because there is only one way to vary a real number.
		Analytic functions are a subset of all possible functions of a complex variable, because we they satisfy the requirement that the derivative of f(z) is unambiguous.
	www.mathpages.com /home/kmath258/kmath258.htm (3185 words)

	MTH-2C12 : Complex Analysis (Site not responding. Last check: 2007-11-05)
		It introduces complex integration and exposes the remarkable rigidity that the property of differentiability imposes on a complex function.
		Overview: Complex Analysis, in particular the concept of a path integral, was primarily developed by Cauchy in the early 19th century (although under restrictive assumptions) and further contributions were made by Liouville, Laurent and Riemann.
		Weierstrass, in particular, developed the theory of complex functions, though the use of complex variable technique was already widespread among 19th century mathematicians, physicists and engineers (and many of this era combined these roles).
	www.mth.uea.ac.uk /maths/syllabuses/0405/2C1205.html (351 words)

	The evolutionary origin of complex features (Site not responding. Last check: 2007-11-05)
		Populations of digital organisms often evolved the ability to perform complex logic functions requiring the coordinated execution of many genomic instructions.
		Complex functions evolved by building on simpler functions that had evolved earlier, provided that these were also selectively favoured.
		The first genotypes able to perform complex functions differed from their non-performing parents by only one or two mutations, but differed from the ancestor by many mutations that were also crucial to the new functions.
	www.evolutionpages.com /Abstracts/digital_organisms.htm (159 words)

	The Geometer's Sketchpad® - Visualizing Complex Functions
		A great leap forward in the history and development of the complex numbers comes some 250 years after Cardano first proposes them, in the near-simultaneous realizations of Argand, Gauss, and Wessel that a complex number can be interpreted as, and represented by, a geometric point in two dimensions.
		This simple correspondence introduces not only the ability to visualize the complex numbers---or, now, the complex plane---in a simple and straightward sense, but to bring geometric techniques to the interpretation and analysis of the mathematics of that complex plane.
		This paper's purpose is to explore the potential for using these software tools to investigate and model complex numbers and their operations, and to visualize functions defined on the complex plane.
	www.keypress.com /sketchpad/general_resources/recent_talks/complex_ictmt6/index.php (494 words)

	MX31510 - COMPLEX ANALYSIS
		Complex analysis is the study of complex valued functions of a complex variable.
		The important role of complex variables in applied mathematics, for instance, is partly due to the use of the theory of residues in the evaluation of certain real integrals and the application of conformal mapping in hydrodynamics and problems in potential theory.
		The aim of the module is to study the theoretical foundations of complex variable theory and to develop skills in the application of this theory to particular problems.
	www.aber.ac.uk /modules/2002/MX31510.html (182 words)

	Complex variable Analysis (Site not responding. Last check: 2007-11-05)
		Complex variables are often accepted in other parts of analysis when this causes no essential change in the theory; but here we focus on those aspects of analytic behaviour unique to complex functions.
		These functions and those used to describe phenomena in part of mathematical physics both display a considerable degree of regularity not found in general functions of a real variable.
		The effect is markedly different than for real functions: these functions are much more rigidly constrained, and in particular it is possible to make very definite comments about their global behaviour, convergence, and so on.
	www.math.niu.edu /~rusin/known-math/index/tour_cplx.html (276 words)

	Complex Wave functions? - Science Forums and Debate (Site not responding. Last check: 2007-11-05)
		The probabilty is given by the square of the wave function (i.e.
		It is a matter of mathematics that wave functions turn out to be complex, we only apply an interpretory skeletol model to it.
		QM requires wave functions to be complex because there is a need to distinguish between positive and negative frequencies....for eg if we took wave function of the form Acos(kx-wt) and if we have another wave with k1 =-k and w1 = -w then in terms of cosine the wave function essentially remains same -
	www.scienceforums.net /forums/showthread.php?p=89037 (471 words)

	Complex Function Plots (Site not responding. Last check: 2007-11-05)
		It is easy to set up a correspondence between the argument of a complex number and the hue of a color since both are measured as angles.
		I have been using contour lines in saturation and value to represent the modulus of the complex value.
		Moduli between 0 and 1 (or any of these ranges) are essentially wrapped around a rectangle in saturation-value space with one corner at saturation = 1 and value = 1, and the opposite corner at saturation = 0.3 and value = 0.6.
	www.american.edu /academic.depts/cas/mathstat/People/lcrone/ComplexPlot.html (313 words)

	FRACTINT Formula Tutorial (Site not responding. Last check: 2007-11-05)
		The functions fn1(...) to fn4(...) are variable functions - when used, the user is prompted at run time (on the Z screen) to specify one of sin, cos, sinh, cosh, exp, log, sqr, etc. for each required variable function.
		Any particular complex number, therefore, can be plotted as a point on the plane, and any point on the plane has a complex number that corresponds to it.
		Functions can be nested, and the results of the inner function will become the input for the outer function.
	spanky.triumf.ca /www/fractint/frm-tut/frm-tutor.html (15534 words)

	MA3121 Complex Analysis
		The student will be assumed to be familiar with the general notion of continuity of a real function as well as other basic concepts from real analysis, such as differentiability of real functions, power series and integration.
		In this course, we begin with the study of analogues for complex functions of familiar properties of real functions, though differences in the two theories emerge as we proceed.
		Cauchy's theory of complex integration is developed, culminating in a number of remarkable results and strikingly beautiful applications.
	www.mcs.le.ac.uk /Modules/MA-02-03/MA3121.html (400 words)

	Complex Mappings From an Evolutionary Viewpoint
		We watch a given function evolve in time, starting from the unaltered complex plane and ending with the range of the function.
		Mathematically, this deformation of the function is a homotopy, from the identity to the target function.
		This new approach of teaching complex functions appears to lead not only to a much quicker, but also to a much deeper, understanding of complex functions.
	math.nist.gov /mcsd/Seminars/2002/2002-04-02-casey.html (241 words)

	[No title]
		As we have seen, the method of graphical analysis is difficult to use with functions that use complex numbers as their input and output values.
		It is easy to represent the input and output values of a function that uses real numbers for the inputs and outputs (you take a plane and use the input values as the x(values on the plane, and the output values of the function as the y(values of the function.
		Adapting Methods for Complex Functions When the function h(x) = x3 is used to do iterations, and the function uses real numbers as both input and output values, the output values generated to make the orbits were no longer necessarily positive.
	www.math.lsa.umich.edu /mmss/coursesONLINE/chaos/chaos4/chaos4.doc (2666 words)

	Citations: Branch cuts for complex elementary functions - Kahan (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Kahan,"Branch cuts for complex elementary functions ", in The State of the Art in Numerical Analysis: Proceedings of the Joint IMA/SIAM Conference on the State of the Art in Numerical Analysis, University of Birmingham, April 14-18, 1986, M. Powell and A. Iserles, Eds, Oxford University Press.
		One of his proposals was a principle called counter clockwise continuity (CCC) for the determination of the closure of the elementary functions.
		Most importantly, the coincidence of the branch cut for the branch point at 0 with the corresponding branch cut for the logarithm and the fact that both functions are CCC yield nice asymptotic expansions of the branches of W at....
	citeseer.ist.psu.edu /context/4057/0 (2466 words)

	The Complex Exponential and Complex Logarithm (Site not responding. Last check: 2007-11-05)
		Once we understand how to visualize the graphs of relatively uncomplicated complex functions such at the powers of z, we can use the same techniques to investigate other complex functions such as the complex exponential function w = e^z and its inverse relation, the complex logarithm z = ln(w).
		The inverse relation, the complex logarithm, has a quite complicated Riemann surface, with a real part that is a surface of revolution of the graph of the exponential function in the plane and an imaginary part that is a right helicoid, resembling a circular staircase.
		We summarize this investigation of the graph of the complex cube (and its inverse, the complex cube root), by situating the four images at the vertices of a diagram, with arrows between certain vertices.
	www.geom.umn.edu /~banchoff/script/CFGExp.html (167 words)

	Ada 95 RM - G.1.2 Complex Elementary Functions
		The implementation of the Exp function of a complex parameter X is allowed to raise the exception Constraint_Error, signaling overflow, when the real component of X exceeds an unspecified threshold that is approximately log (Complex_Types.Real'Safe_Last).
		For example, many of the complex elementary functions have components that are odd functions of one of the parameter components; in these cases, the result component should have the sign of the parameter component at the origin.
		Other complex elementary functions have zero components whose sign is opposite that of a parameter component at the origin, or is always positive or always negative.
	lglwww.epfl.ch /docs/ada/rm95html-1.0/rm9x-G-01-02.html (1311 words)

	12.5.3. Branch Cuts, Principal Values, and Boundary Conditions in the Complex Plane
		Many of the irrational and transcendental functions are multiply defined in the complex domain; for example, there are in general an infinite number of complex values for the logarithm function.
		The branch cut for the arc sine function is in two pieces: one along the negative real axis to the left of -1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of 1 (inclusive), continuous with quadrant IV.
		Thus the range of the arc tangent function is not identical to that of the arc sine function.
	www.cs.cmu.edu /Groups/AI/html/cltl/clm/node129.html (3105 words)

	Visualization of Complex Functions -- from Mathematica Information Center
		Functions f (x, y) of two variables with values in the field of complex numbers (e.g., analytic functions) are often considered abstract mathematical objects which are difficult to visualize.
		Indeed, the graph of such a function would have to be drawn in a four-dimensional space with coordinates (x, y, Re(f), Im(f)), which cannot be easily done on a sheet of paper.
		In addition to describing the argument of a complex number by the hue of the color, the color map uses the lightness of the color to represent the absolute value.
	library.wolfram.com /infocenter/Articles/2888 (305 words)

	Complex Analysis (Site not responding. Last check: 2007-11-05)
		There are so many books on complex variable theory in existence that there hardly seems room for still another; nevertheless written material is needed for the entertainment of the students.
		Consequently these notes cover some of the why's and wherefore's of complex variables; ranging from the role of the cross ratio and the Schwartz derivative to topics such as the Mandelbrot Set, Elliptic Curves, and spectral densities.
		The derivative of a function of a complex variable
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/complex.html (202 words)

	Math Forum - Ask Dr. Math Archives: High School Imaginary/Complex Numbers
		We know it is possible to look at the graph of a polynomial and tell a great deal about its real roots by looking at the x-intercepts.
		There seem to be some interesting "wiggles" at locations that appear to be related to the "average" of the complex pairs.
		It appears that the "wiggle" of these graphs is always influenced by the complex roots.
	mathforum.org /library/drmath/sets/high_complex.html (680 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-05)
		This seems to be the main difference between real functions and complex functions which seem to have derivatives of every degree.
		The condition of being differentiable is MUCH stronger in the complex case, and one can show that if a function has one derivative, it has derivatives of all orders.
		In the case of complex functions, this limit has to hold, no matter how h goes to zero - along any direction, spiraling in, or whatever.
	www.mathforum.org /library/drmath/view/53404.html (281 words)

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