Math 347, HW # 5 (due Wed, Sep 26)
1)-6) Chapter 6: problems 6โ11.
7) Find all zโCfor which |z| โ 2z=3โ4i.
8) Solve in Cthe equation (2 +i)z2โ(4 โi)z+1=0.
9) For any z1,z2โCprove the triangle inequalities:
(i) |z1+z2|6|z1|+|z2|.
(ii) ๎
๎
๎|z1| โ |z2|๎
๎
๎6|z1โz2|.
10) Let z1,z2,z3โCsuch that |z1|=|z2|=|z3|=1 and z1+z2+z3=1.
(i) Prove that 1
z1
+1
z2
+1
z3
=1.
(ii) Find z1,z2,z3knowing also that z1z2z3=1.
11) Find a closed formula for the sums
S1=
n
X
k=1
cos kฮธand S2=
n
X
k=1
sin kฮธ.
(Hint: Look at S1+iS 2and at some (complex) geometric series.)
12) Let z0โCwith |z0|<1.
(i) Show that for every zโCwith |z|61 one has ๎
๎
๎
๎
๎
zโz0
1โz0z๎
๎
๎
๎
๎
61.
(ii) Show that if |z|=1, then ๎
๎
๎
๎
๎
zโz0
1โz0z๎
๎
๎
๎
๎
=1.
1