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week12 - 12 Analytic functions Analytic functions of a...

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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 0 ( z ) = lim Δ z 0 f ( z + Δ z ) - f ( z ) Δ z (1) which exists and is independent of the path by which one lets Δ z 0 . Figure 1: Left: function of 1 real variable. Derivative does not exist at x 0 because limit ( f ( x + Δ x ) - f ( x )) / Δ 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 Δ z 0 f ( z + Δ z ) - f ( z ) Δ z ( z + Δ z )( z * + Δ z * ) - zz * Δ z = z Δ z * + Δ zz * Δ z = ( x + iy )(Δ x - i Δ y ) + ( x - iy )(Δ x + i Δ y ) Δ x + i Δ y = 2 x Δ x + 2 y Δ y Δ x + i Δ y (2) 1
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Consider now path 1 approaching z : Δ y = 0, Δ x 0. Then derivative 2 x . On the other hand on path 2 Δ x = 0, Δ 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 iθ/ 2 . w is a complex number, let’s call it w = ρe . So we see that ρ = r 1 / 2 and 2 φ = θ . We can see that the mapping is not 1-1, since both φ and φ + π , two different points in the w plane correspond to θ and θ + 2 π , i.e. the same z . We can define a new function w which is single valued by restricting the value of θ to lie between 0 and 2 π . This is a new complex
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