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MATH 115a: Homework 2

Instructions

• Write your name on each assignment.

• You may (and are encouraged to) work with others, but you must write up solutions in your own words.

• All problem numbers reference the text, “Complex Analysis (Princeton Lectures in Analysis, II)” by Stein and Shakarchi.

Book problems

Chapter 1, Exercises: 5, 6, 7, 8. (For 8, you may want to review the statement of the multivariable chain rule from Calculus.)

Additional problem(s)

1. Find a countable set S and a collection {Di }i∈S  of open discs Di  ⊆ C indexed by S so that every open set U ⊆ C can be written as a union of discs in {Di }i∈S .

2. Carefully show the following claim, which we made in class on 1/26 (and 1/30):      Claim.  Let U ⊆ C be open. Let z0  ∈ U, and f : U → C a function. Then the limit

f\ (z0 ) :=    lim     f(z0 + h) − f(z0 )

h∈C−{0}

exists (i.e. f is holomorphic at z0 ) if and only if there exists a complex number a ∈ C and a function g(z) defined in a neighborhood of z0  so that g(z) → 0 as z → z0  and

f(z) − (f(z0 ) + a(z − z0 )) = (z − z0 )g(z).

[Remark: Of course, if such an a and g exist, then a = f\ (z0 ).]