הראה את z + 1 + 1 + z + z ^ 2 + 1 + z ^ 3> = 1?

• ל # | z |> = 1 #

+ z + 2 + z + 1 | - = | (| z + 2 + z + 1) - (z + 1) | = | z ^ 2 | = | z | ^ 2> = 1 #

• ל # | z | <1 #

# | z + 1 | + z + 2 + z + 1 | | | z | z + 1 | + z + 2 + z + 1 | = #

(z + 1 +) + z + 2 + z + 1 | = | z ^ 2 + z | + | z ^ 2 + z + 1 | | = | (| z + 2 + z + 1) (z ^ 2 + z) | = 1 #

לפיכך, # | z + 1 | + 1 + z + z ^ 2 |> = 1 #, # z ## in ## CC #

ו

0 + z + 1 | + 1 + z + z ^ 2 | + 1 + z ^ 3 | = 1 + z + + 1 + z + z ^ 2 |> = 1 #,

'#=#', # z = -1vvz = e ^ ((2k + 1) iπ) #, # k ## in ## ZZ #