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### 2007-02-17

Course: MATH 612, Fall 2008
School: Cornell
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612 homework Math assignment 3 due 2007-02-15 at 3 pm 1. Rudin 10: 2, 4, 5, 16, 17. ^ A circle in the Riemann sphere C is a subset which is either a circle in C or a set of the form l {}, where l is a straight line in C. Equivalently, a circle ^ ^ in C is a subset which under the standard homeomorphism C S2 maps to a 2 ^ = R {} is a circle in C. ^ circle in S . For instance, the extended real line R Mbius...

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612 homework Math assignment 3 due 2007-02-15 at 3 pm 1. Rudin 10: 2, 4, 5, 16, 17. ^ A circle in the Riemann sphere C is a subset which is either a circle in C or a set of the form l {}, where l is a straight line in C. Equivalently, a circle ^ ^ in C is a subset which under the standard homeomorphism C S2 maps to a 2 ^ = R {} is a circle in C. ^ circle in S . For instance, the extended real line R Mbius transformations map circles to circles and for each circle there is a Mbius ^ transformation mapping it to R. 2. Read Rudin 14.3 and do problem 14.30. Let f be a function holomorphic dened in a region (connected open set) C. ~ ~ An analytic extension of f is a holomorphic function f H() dened in a region ~ satisfying f | = f. We say that f cannot be analytically extended beyond ~ if there does not exist an analytic extension of f to any region strictly larger than . 3. Consider the function f ...

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