OO
Z n
(ii) If exp z = E
~.v show that
n=O
exp(a + b) = exp a. exp b,
(iii) If sin z = 1, show that z is a real number.
Va, b c C.
3.
(i) State and prove Taylor's theorem for a function f differentiable in the open disc
D(a,
R), centre a, radius R.
O0
(ii) Find the Taylor expansion of (1

z) 1 of
the form E
c~(z + i) ~, valid in the
n=0
disc
D(i, v/2).
Would the expansion be valid in the disc
D(i,
2)?
(iii) Find the Laurent expansion, about 0, of (z 3  1) 1 valid a) for [z[ < 1,
b) Iz[ > 1.
MATHM211
PLEASE TURN OVER
.
(i) If f is holomorphic and bounded on C, show that f is constant on C.
(ii) Show that if
p(z)
is a nonconstant polynomial with coefficients in C then there
exists w C C, with
p(w) = O.
(iii) If
q(z)
= 7z 7 + 8z 6 + 3z 5 + 4z 4 + 1, show that
sup{Iq(z)l;fz ]
= 2} >
sup{Jq(z)l;Iz I =
1}.
.
(i) Let f be a continuous function defined in the upper half plane Im z ~> 0, where
f(z) , 0
uniformly as Izl ~ c¢.
If m > 0, show that
~re~mZ f(z)dz ~
as
~
oc
0
R
where FR

{Re i°, 0 ~< 8 ~< 7r}.
(ii) Evaluate
jr0
°°
sin
x 1)dx"
x(x 2 +
.
(i) Let f be holomorphic inside and on a convex contour 3, except for a finite
number of poles at al, ..., an inside 3' with residues R1, ..., P~ respectively.
that
(ii) Prove that
n a positive integer.
~ f(z)dz = 27ri(R1 + ... + P~).
cos 2n x + sin 2'~
xdx =
zr;
n
Show
MATHM211
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 Taylor Series, University College London, exp z, best f, Show t h