Exercise 1.
Let
be an open convex set in the plane and let
be a function holomorphic in
Show that, for each pair of points
we can find two points
and
on the segment joining
and
such that
(Hint: Consider the function of a real variable
defined by
=
and apply the Mean Value Theorem to the real and imaginary
parts of
Exercise 2.
Prove the following :
Let a be
a limit point of a subset D of
C.
Suppose that
converges
uniformly to f on D
and
that
Then
converges
and
;
that is,
Exercise 3.
Prove
the following :
Suppose that
is
a sequence of integrable functions on [a,b] which
converges
uniformly to f on [a,b]. Then f is
integrable and
Exercise 4.
Test for uniform convergence in the given region:
a)
b)
c)
Exercise 5.
Show that
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