Functions of a Complex Variable in .NET

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Functions of a Complex Variable
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Ultimately it was realized that to accept numbers that provided solutions to equations such as x2 1 0 was no less meaningful than had been the extension of the real number system to admit a solution for x 1 0, or roots for x2 2 0. The complex number system was in place around 1700, and by the early nineteenth century, mathematicians were comfortable with it. Physical theories took on a completeness not possible without this foundation of complex numbers and the analysis emanating from it. The theorems of the di erential and integral calculus of complex functions introduce mathematical surprises as well as analytic re nement. This chapter is a summary of the basic ideas.
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FUNCTIONS If to each of a set of complex numbers which a variable z may assume there corresponds one or more values of a variable w, then w is called a function of the complex variable z, written w f z . The fundamental operations with complex numbers have already been considered in 1. A function is single-valued if for each value of z there corresponds only one value of w; otherwise it is multiple-valued or many-valued. In general, we can write w f z u x; y iv x; y , where u and v are real functions of x and y.
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EXAMPLE. w z2 x iy 2 x2 y2 2ixy u iv so that u x; y x2 y2 ; v x; y 2xy. called the real and imaginary parts of w z2 respectively. These are
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In complex variables, multiple-valued functions often are replaced by a specially constructed singlevalued function with branches. This idea is discussed in a later paragraph.
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EXAMPLE. Since e2ki 1, the general polar form of z is z  ei  2k . This form and the fact that the logarithm and exponential functions are inverse leads to the following de nition of ln z ln z ln   2k i k 0; 1; 2; . . . ; n . . .
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Each value of k determines a single-valued function from this collection of multiple-valued functions. These are the branches from which (in the realm of complex variables) a single-valued function can be constructed.
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FUNCTIONS OF A COMPLEX VARIABLE
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LIMITS AND CONTINUITY De nitions of limits and continuity for functions of a complex variable are analogous to those for a real variable. Thus, f z is said to have the limit l as z approaches z0 if, given any  > 0, there exists a  > 0 such that j f z lj <  whenever 0 < jz z0 j < . Similarly, f z is said to be continuous at z0 if, given any  > 0, there exists a  > 0 such that j f z f z0 j <  whenever jz z0 j < . Alternatively, f z is continuous at z0 if lim f z f z0 . z!z0 Note: While these de nitions have the same appearance as in the real variable setting, remember that jz z0 j <  means q j x x0 j i y y0 j x x0 2 y y0 2 < : Thus there are two degrees of freedom as x; y ! x0 ; y0 :
DERIVATIVES If f z is single-valued in some region of the z plane the derivative of f z , denoted by f 0 z , is de ned as
z!0
f z z f z z
provided the limit exists independent of the manner in which z ! 0. If the limit (1) exists for z z0 , then f z is called analytic at z0 . If the limit exists for all z in a region r, then f z is called analytic in r. In order to be analytic, f z must be single-valued and continuous. The converse, however, is not necessarily true. We de ne elementary functions of a complex variable by a natural extension of the corresponding functions of a real variable. Where series expansions for real functions f x exists, we can use as de nition the series with x replaced by z. The convergence of such complex series has already been considered in 11.
EXAMPLE 1. We de ne ex 1 z z2 z3 z3 z5 z7 z2 z4 z6 ; sin z z ; cos z 1 . 2! 3! 3! 5! 7! 2! 4! 6! ex cos y i sin y , as well as numerous other relations.
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