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Unformatted text preview: 12 Analytic functions Read: Boas Ch. 14. 12.1 Analytic functions of a complex variable Def. : A function f ( z ) is analytic at z if it has a derivative there f ( z ) = lim z f ( z + z )- f ( z ) z (1) which exists and is independent of the path by which one lets z . Figure 1: Left: function of 1 real variable. Derivative does not exist at x because limit ( f ( x + x )- f ( x )) / x is different from left or from right. Right: possible ways to approach z in the complex plane. To clarify the importance of the path independence, consider the complex func- tion f ( z ) = | z | 2 . This looks smooth enough, since we can write it as f ( x, y ) = x 2 + y 2 . But considered as a complex function it is not analytic, as we can see by applying the definition lim z f ( z + z )- f ( z ) z ( z + z )( z * + z * )- zz * z = z z * + zz * z = ( x + iy )( x- i y ) + ( x- iy )( x + i y ) x + i y = 2 x x + 2 y y x + i y (2) 1 Consider now path 1 approaching z : y = 0, x 0. Then derivative 2 x . On the other hand on path 2 x = 0, y 0, derivative - 2 iy . So this is not an analytic function at any z. In general simple functions of z itself, not | z | , have regions where they are analytic. If a function is analytic and single valued within a given region, we call it regu- lar. If it is multivalued, there are places where the function is not analytic, called branch cuts. Figure 2: Branch cut of w = z 1 / 2 Ex.: consider the function w = z = re i/ 2 . w is a complex number, lets call it w = e i . So we see that = r 1 / 2 and 2 = . We can see that the mapping is not 1-1, since both and + , two different points in the w plane correspond to and...
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This note was uploaded on 04/17/2008 for the course PHZ 3113 taught by Professor Hirshfeld during the Fall '07 term at University of Florida.
- Fall '07