Let two functions f and g he as in the statement of

Chapter 0, Problem 7.50

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Let two functions f and g he as in the statement of Rouchc's theorem in Sec. 94. and let the orientation of the contour C there he positive. Then define the function I l /"(::.) + rg'(::.) (f) = - . tl- 2;r i. c f(::) + fg(::.) ~ (f) is never zero on C. This ensures the existence of the integral. (/>) Let rand r0 he any two points in the interval 0::: r ::: 1 and show that It - to I I l r~' - r ~ I l(f) - (Toll = . ' . ' I 2.1 . c (f + r g )( f + fog) ' :: . Then. after pointing out why I fg' - f'g I lfg' - f'gl (f + Tg)(f +tog) ::: (f) - Uo) I .::: A It - To I. Conclude from this inequality that (f) is continuous on the interval 0 .::: r ::: I. (l') By referring to equation (8). Sec. 93. state why the value of the function is. for each value oft in the interval 0 .::: t ::: I. an integer representing the numhcr of zeros off(::)+ tg(::) inside C. Then conclude from the fact that is continuous. as shown in part(/>). that f (::) and/(::) + g(::) have the same number of zeros. counting multiplicities. inside C.