Introductory Complex and Analysis Applications by William R. Derrick

By William R. Derrick

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Continuing in this fashion one may extract η factors of P ( z ) , thus P{z) has exactly η roots. We next prove one of the most useful theorems in the theory of analytic functions: M a x i m u m P r i n c i p l e If / ( z ) is analytic and nonconstant in a domain G, then | / ( z ) | has no maximum in G. Proof Suppose there is a point Zo in G satisfying | / ( z ) | < | / ( z o ) | for all ζ in G. Since Zo is an interior point there exists a number r > 0 such that | z - Zol

Thus the integrals on the arcs 0= Í „ f(z)dz= \ f{z)dz~ t 43 cancel out and \ f(z)dz. E x a m p l e 1 Since cos ζ is entire, has antiderivative sin z, and ^ is simply connected, we have cos ζ dz = sin ζ = 2 sin i = 2i sinh(l). and along any pwd closed curve γ cos ζ dz = 0. E x a m p l e 2 The function l/z, analytic in ζΦΟ, has antiderivative log ζ. In this case care must be taken to specify the domain G. Suppose G is given by I arg ζ | < π, then for any arc joining — / to / in G ' dz ^ — = Log ζ -i ζ = πι.

By the Cauchy-Goursat theorem f{z)dz= 0 = f(z)dz- f(z)dz, y y-y' y' thus F{z) = f(z) dz = f(z) dz is independent of the choice of path. Let the last line segment of y (γ') be horizontal (vertical) and z^ = + iy^ be the last point of intersection. Then F(z) = i f fix, + it) dt + fit -h i > i ) di + i f fit + iy) fix dt^C -f it) dt + C, 42 C O M P LX E 2 I N T E G R A T I O N where ζ = χ -\- iy and the constant C = f(zi). Taking the partial derivatives, the first equation yields F,(z) = f(z), the second Fy{z) = if(z).

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