M417
Fall 1996
hw12.tex due Nov 18, 1996
Let D be a simply connected domain. Let z
0
be a fixed point in D. Suppose that f(z) is analytic in D, that f(z) ≠ 0 in D, and that f(z
0
) = 1.
1.
Show that there is a function H(z) which is analytic in D, H(z
0
) = 0, and
\dfracdHdz = \dfracf
′
f, z ∈ D.
2.
Show that
\dfracd(f exp(−H))dz = 0, z ∈ D.
3.
Show that
f(z) = exp(H(z)), z ∈ D.
4.
Show that " f has a square root defined in D",
i\. e\.
, there is a function g(z), analytic in D, g(z
0
) = 1, such that
(g(z))
2
= f(z), z ∈ D.
The function H(z) is called a logarithm of the nonzero function f. How does the assumption that D is
simply connected
play an essential role?
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On 08 Feb 2017, 16:39.