In July 2020
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If f is analytic in the unit disk Δ(0;1) and |f'(z)-1-i| < √2 for all z belongs to Δ(0;1). Then f is univalent in Δ(0;1).
If f is analytic in a convex domain D such that Re f'(z) is not equal to 0 for all z belongs to D, then f is univalent in D
If f is univalent and analytic in an open set D except for isolated singularities, then f can have at most one singularity and that as a simple pole.
Let f(z) = sin z/z
and f(0) = 0. Explain why f is analytic at z = 0. Find the Maclaurian
series for f(z) and g(z) = ∫ f(ξ)dξ from 0 to z
. Does there exist a function f with an
isolated singularity at 0 and such that |f(z)|~ exp( 1/|z|) near z= 0?
2
, z + z = 2Re(z) and Re(z) ≤ |z|. Hence
show that
i. |z1 + z2|
2 = |z1|
2 + |z2|
2 + 2Re(z1z2),
ii. |z1 + z2| ≤ |z1| + |z2|,
where Re(z) is the real part of z and z the conjugate of z.
2
, z + z = 2Re(z) and Re(z) ≤ |z|. Hence
show that
i. |z1 + z2|
2 = |z1|
2 + |z2|
2 + 2Re(z1z2),
ii. |z1 + z2| ≤ |z1| + |z2|,
where Re(z) is the real part of z and z the conjugate of z. [26 marks]
(b) If z1 = 1 + 2i, find the set of values of z2 for which
(i) |z1 + z2| = |z1| + |z2| (ii) |z1 + z2| = |z1| − |z2|.
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#257634 on May 2020