Transcribed Text

3. Show that z is a real number if and only if Z = z.
4. If Z and w are complex numbers, prove the following equations:
z + w/2 = z² + 2Rezw+ 1w/2.
12WW2 = 
12+W12 + zw² = 2(z²+w²).
o. Let R(z) be a rational function of z. Show that R(z) = R(z) if all the
coefficients in R(z) are real.
3.3
12,+22+...t2,
Also useful is the inequality
3.4
(1211m)512m
Now that we have given a geometric interpretation of the absolute value
let us see what taking a complex conjugate does to a point in the plane.
This is also easy; in fact, z is the point obtained by reflecting z across the
xaxis (i.e., the real axis).
Exercises
1
2. Show that equality occurs in (3.3) if and only if N 0 for any integers
k and I, 1 s k, / = n, for which 21 * 0.
3. Let a € R and c > 0 be fixed. Describe the set of points z satisfying
12a)12+al=2c
for every possible choice of a and c. Now let a be any complex number
and, using a rotation of the plane, describe the locus of points satisfying the
above equation.
2m
5. Let Z = cis for an integer n 2. Show that 1 +z+.
7. If Z e C and Re(z") >0 for every positive integer n, show that Z is a
nonnegative real number.
Exercise
1. Let C be the circle {z: r > 0; let a = ctr cis a and put
LB  I = z: Im (7)0}
where b = cis p. Find necessary and sufficient conditions in terms of B that
LB be tangent to C at a.
2. Which of the following subsets of C are open and which are closed: (a)
{z:z<1} ; (b) the real axis; (c) { Z : = 1 for some integer n > 1); (d)
{ Z E C:z is real and 052<1); (e) { z E C: Z is real and 0<z< 1)?<br="">ball is in fact an onen
4. Show that the union of a finite number of compact sets is compact.
</z<>1. Show that f(z) = ² = x2 + y2 has a derivative only at the origin.
2. Prove that if bn, an are real and positive and 0 < b = lim bn < 80,
a = lim sup an then ab = lim sup Does this remain true if the re
quirement of positivity is dropped?
3. Show that lim n¹/" = 1.
44
Elementary Properties and Examples of Analytic Functions
4. Show that (cos z)' =  sin Z and (sin z)' = cos z.
5. Derive formulas (2.14).
6. Describe the following sets: {z: e² = i), {z: e = 1), {2: e2 =  i),
{z: cos Z = 0}, , {z: sin z = 0}.
7. Prove formulas for cos (z+ w) and sin (z+w).
sin Z
8. Define tan Z =
cos Z
; where is this function defined and analytic?
9. Suppose that 2n, Z € G = C  {z: Z S 0} and 2n = ree , Z = reie where
< < 7. Prove that if 2n Z then On 0 and "n r.
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