Suppose that a .f11nctio11 f is bou11ded and analytic in

Chapter 0, Problem 6.42

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Suppose that a .f11nctio11 f is bou11ded and analytic in some deleted neighlmrlwod o < 1.:: - .::I < f: rf .::o. rr.r is 1101 wwlytic at .::0 t'1e11 it has a remomble singularity there. To prove this. we assume that .f is not analytic m .:: 0 . As a consequence. the point .::o must be an isolated singularity of .f: and /(.::) is rcprcsenlcd by a Laurenl series ( I ) " -x; b . I (,::) = ' L,_ ll11 (.:: - .::o) II + ' L- _ _ II _ II ""'o ""d L .o) throughoul the deleted neighborhood 0 < I.:: - .::ol < . If C denoles a positively orienled circle I.:: - .:: 0 1 = p. where p < (Fig. 97). we know from Sec. 66 that the cocfficiems "" in expansion (I) can be \\1 rillen b = _1_ ; f(:.) d.:: II ") ( - - ) II I I _;r I (' . - . o (2) (II= J. 2 .... ). Now the boundedness condilion on f lells us that there is a posilive conslanl 1\;/ such lhat I .f (.::)I ~ 1\;/ whenever 0 < I.:: - .::0 1 < E:. Hence it follows from ex press ion ( 2) th al \' () I I I I I I I / \ ' \ ' I ,\;/ lb11 I ~ ~ . I 2:r p = Mp" .!.:7 fJ 11 I (11 = I. 2 .... ) . .... / / ' .... ..... _____ .,... / .\ FIGURE 97 SEC. 84 BEll..\VIOR OF fol'.1\CTIONS !'JEAR ISOLATED S!!'\Gl.:L:\R POll'\TS 257 Si nee chc cocfhcicnts b11 arc conscancs and since p can be chosen arbitrarily small. we may cone Jude char h11 = 0 (11 = I. 2 . ... ) in che Laurene series (I). This tells us chat ::0 is a removable singularicy of .f. and the proof of Theorem 2 is complete. (b) Essential Singular Poi11b, We knov.1 from Example 2 in Sec. 79 chat the behavior of a funccion near an essential singular poinc can be quice ill'egular. The nexc cheorcm. regarding such behavior. is related co Picmd \ cheorcm in th al earlier example and is usually referred lo as the Casorati-lVeierstrass theorem. It stales that in each deleted neighborhood of an essential singular point. a funccion assumes values arbitrarily close to any given number.