[lo] 1.
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a) Determine where the function f (a: iy) = ex i e 2 y is differcntiahle and where it is
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b) Let f ( z ) be entire. Prove that f (z) is also entire.
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/[lo] 2. Determine the domain of andyticity of:
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[lo] 3.
a) Find the first three coefficients, ao, al, az, in the Taylor series of
is the radius of convergence? (give a reason for your answer)
b) Let
C,"==,
a,zn
& around zo = 0. What
be the Taylor series of some analytic f (2) at 0. Show that if f is even (i.e.,
= 0 for all odd n.
f (2) = f (-Z) for all z), then a,
[lo] 4. Find two different Laurent series for the function f (z) = Log(z) + l/(z - i) in annuli centered at
zo = 22 and specify the domain of convergence for each series.
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[20] 5. Calculate each of the following contour integrals (assume every closed contour is oriented counter-
clockwise)
Jc z2dz where C the line segment from 0 to 1+ 22.
b) J, -dz
where C is the circle lz - 4i[= 3.
c) Jc sin(3/z)dz where C is the unit circle lzl = 1.
a)
d)
a)
Jc ( z - a ) ( z - O ) dz where C is a counter clockwise circle and cr and ,L? lie strictly inside C.
5zL11=
i)lC[ I +LC)
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[lo] 6. Find
and justify your answer.
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[lo] 7. In each case below, find an examplc: of a function f ( z ) satisfying:
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a) The Taylor series at z = 3 has radius of convergence = 5, and f (2) = 2.
1
b) f ( z ) has an essential singularity at z = 0 , a. pole of order 1 at .z = 1, a pole of order 2 at
z = 2 and is analytic everywhere else.
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c) Jc f (z)dz is equal to i when C is the circle lzl = 1, and it is equal to 1 when C is the circle
I.zl = 3.
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[lo] 8.
a) Let f (z) be an entire function satisfying
lf(z>I 2 1
for all z. Show that f (z)must be a constant function.
b) Let f (z) be an entire function whose image lies above the parabola y = x2 in plane, i.e., f
maps the complex plane into {(x,y) : y > x2}. Show that f (z)must be a constant function.