And b figure 4 serial of maps from exterior of the

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Unformatted text preview: -plane that g (w) maps one-to-one to the whole z -plane. c) Describe and attempt to draw a Riemann surface on which f (z ) is single-valued and to which g (w) maps one-to-one. Comment on the misleading nature of your picture. d) Identify the branch points of f (z ) and introduce a branch cut to make f (z ) singlevalued. Solution 7 a) z 3 = |z | 3 ei( 1 1 Arg(z ) + 2πk 3 3 ), k = 0, 1, 2 (42) So, f (z ) = z 3 has three values for z = 0. 1 7 ¡£¡¡£¡¡£¡ ¤¤¤ £¤¡¡¡¡¡¡ ¤¡¡£¤¡¡£¤¡¡£¤ ££ £ £ ¡£¤¡¡£¤¡¡£¤¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡¤¡£¡¤¡£¡¤¡£ ¡£¡¤¡£¡¤¡£¡¤ £¤¡¡£¤£¡¡£¤£¡¡£¤£ £ ¤£ ¤£ ¤£ ¤¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡¡£¡¡£¡ ¡¡£¤£¡¡£¤£¡¡£¤£ ¡¦¡¡¦¡¦¡¥ ¥¥¥¥¥ ¡¦¡¡¦¡¦¡ ¦¦¡¡¦¦¡¡¡¦¦¥¡¥ ¦¡¡¦¡¡¡¦¡¥¥ ¡¦¡¡¦¡¦¡¥¡ ¥¥ ¦¥¥ ¥¥ ¦¥¥ ¦¥¥ ¥¥¥¥¥¥ ¦¦¡¡¦¦¡¡¡¦¦¡¥ ¥¡¦¥¡¥¡¦¥¡¦¥¡¡ ¦¡¡¦¡¡¡¦¡¥¥ ¥¡¥¡¥¡¥¡¥¡ ¡¦¡¡¦¡¦¡¥¥ ¨¡¨¡¨¡¨¡¨¡¨¡¡¨¡¨¡¡¨¡¨¡¨¡ §¡¨¡¨¡¨¡¨¡¨¡¡¨¡¨¡¡¨¡¨¡¨¡ §§§§§ §§ §§§ ¡§¡§¡§¡§¡§¡¨¡§¡§¡¨¡§¡§¡§¡¨ §¨¡¨¡¨¡¨¡¨¡¨¡§¡¨¡¨¡§¡¨¡¨¡¨¡§ ¨¡¡¡¡¡¡¨§¨¡¡¡¨§¨¡¡¡¡¨§¨ ¨§ ¨§§ ¨§§ ¨§§ ¨§§ ¨§§ ¨§§ ¨§§ ¨§§ ¨§§ ¨§§ §¡¨¡¨¡¨¡¨¡¨¡§¡¨¡¨¡§¡¨¡¨¡¨¡§ ¨¡§¡§¡§¡§¡§¡¨§¡§¡§¡¨§¡§¡§¡§¡¨§ ¨¡¡¡¡¡¡§¨¡¡¡§¨¡¡¡¡§¨ §§¡§¡§¡§¡§¡§¡¡§§¡¡§¡§¡§¡ ¡§¨¡§¨¡§¨¡§¨¡§¨...
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This note was uploaded on 02/22/2010 for the course ACM 95A taught by Professor Nilesa.pierce during the Fall '06 term at Caltech.

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