❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ❡ ●❡♦❣❡❜r❛
▲❡♦♥❛r❞♦ ❞❡ ▼❛tt♦s ❇❛st♦s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦
✺✶✵✳✵✼ ❇❛st♦s✱ ▲❡♦♥❛r❞♦ ❞❡ ▼❛tt♦s
◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ❡ ●❡♦❣❡❜r❛✴ ▲❡♦♥❛r❞♦ ❞❡ ▼❛tt♦s ❇❛st♦s✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✸✳
✺✼ ❢✳
❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳
❖r✐❡♥t❛❞♦r❛✿ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦
✶✳ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s✳ ✷✳ ●❡♦❣❡❜r❛✳ ✸✳ ❊♥s✐♥♦ ▼é❞✐♦✳ ■✳ ❚ít✉❧♦
❆❣r❛❞❡❝✐♠❡♥t♦s
❊♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛❣r❛❞❡ç♦ ❛ ➶♥❣❡❧❛✱ ❛♠❛❞❛ ❡s♣♦s❛ ❡ ❝♦♠♣❛♥❤❡✐r❛✱ q✉❡ ♠❡ ❛❝♦♠✲ ♣❛♥❤♦✉ ❡ ✐♥❝❡♥t✐✈♦✉ ❞❡s❞❡ ♦ ♣r✐♠❡✐r♦ ❞✐❛ ❞❡st❛ ❥♦r♥❛❞❛✱ ❡♠ ❘✐♦ ❈❧❛r♦ ❡ ❡♠ ♥♦ss❛ ❝❛s❛❀ s✉❛ ♣r❡s❡♥ç❛ ❡ ❡stí♠✉❧♦ sã♦ ❢✉♥❞❛♠❡♥t❛✐s ❡♠ t♦❞❛s ❛s ♠✐♥❤❛s r❡❛❧✐③❛çõ❡s✳
➚ ♥♦ss❛ ♠❛r❛✈✐❧❤♦s❛ ✜❧❤❛ ❆♥❛ ♣❡❧♦ ❡stí♠✉❧♦✱ ❝♦♥✜❛♥ç❛ ❡ r❡✈✐sã♦❀ ❜❡♠ ❝♦♠♦ ❛♦s ❝✉♥❤❛❞♦s✭❛s✮✱ s♦❜r✐♥❤♦s✭❛s✮✱ s♦❣r❛ ❡ ❣❡♥r♦✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳ ❊♠ ♣❛rt✐❝✉❧❛r às ♠✐♥❤❛s q✉❡r✐❞❛s ✐r♠ãs ❊❧✐s❛❜❡t❤ ❡ ❙♦❧❛♥❣❡✱ q✉❡ t❛♥t♦ ❢❛③❡♠ ♣❛r❛ ♠❡ ❛❥✉❞❛r✳
❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♣❛✐✱ ♠❡✉ ❡t❡r♥♦ ▼❡str❡ ❡ ❆♠✐❣♦✱ ♣❡❧♦ ❡①❡♠♣❧♦ ❞❡ ❞✐❣♥✐❞❛❞❡ ❡ ✐❞❡❛❧✐s♠♦✳
▼❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞❛ ❯♥❡s♣ ❘✐♦ ❈❧❛r♦ ❡ t♦❞♦s ♦s ❢✉♥❝✐♦♥ár✐♦s q✉❡ ✈✐❛❜✐❧✐③❛r❛♠ ❡ss❛ ❝♦♥q✉✐st❛✳ ❇❡♠ ❝♦♠♦ à ❈❛♣❡s ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳ ❊♠ ❡s♣❡❝✐❛❧ à ♣r♦❢❡ss♦r❛ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦✱ ❝♦♦r❞❡♥❛❞♦r❛ ❧♦❝❛❧ ❞♦ Pr♦❢♠❛t✱ ♣♦r t❡r ❛❝❡✐t♦ ♦ ❞❡s❛✜♦ ❞❡ t♦r♥❛r r❡❛❧✐❞❛❞❡ ♦ ♣r♦❣r❛♠❛ ❡ t❡r ❛❝♦❧❤✐❞♦ t♦❞♦s ♦s ❛❧✉♥♦s ❝♦♠ t❛♥t♦ ❝❛r✐♥❤♦✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❡stá ✐♥s❡r✐❞♦ ♥♦ ❝♦♥t❡①t♦ ❞♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ❡ tr❛③ ✉♠❛ r❡s❡♥❤❛ ❞❡ ✉♠❛ t❡♦r✐❛ ❛ s❡r ❡♥s✐♥❛❞❛ ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ❣❡r❛❧♠❡♥t❡ ♥❛ t❡r❝❡✐r❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛ s❛❜❡r✱ ❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❆❧é♠ ❞✐ss♦✱ ❡♥❢♦❝❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❞✐✈❡rs✐✜❝❛❞❛s ❛❜♦r❞❛❣❡♥s ❞✐❞át✐❝❛s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❝♦♠ ✉s♦ ❞❡ ♥♦✈❛s ❚❡❝♥♦❧♦❣✐❛s ❞❡ ■♥❢♦r♠❛çã♦ ❡ ❈♦♠✉♥✐❝❛çã♦✱ ♣❛r❛ s✉♣❡r❛r ❛❧❣✉♠❛s ❞✐✜❝✉❧❞❛❞❡s ❝♦♠ ♦ ❡♥s✐♥♦ q✉❡✱ ♠✉✐t❛s ✈❡③❡s✱ ❞❡✲ ❝♦rr❡♠ ❞❡ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❡①❝❡ss✐✈❛♠❡♥t❡ ❢♦r♠❛❧ ❞♦ t❡♠❛✳ ❆ ♣r♦♣♦st❛ ❞❡ ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ❡♠ ❛♠❜✐❡♥t❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡❧❛❜♦r❛❞❛s ❝♦♠ s✉♣♦rt❡ ❞♦ s♦❢t✇❛r❡ ❧✐✈r❡ ✭●❯■✮ ●❡♦❣❡❜r❛ ❡stá ✐♥❝❧✉s❛✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ✐s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ Pr♦❢❡ss✐♦♥❛❧ Pr♦❣r❛♠ ▼❛st❡r ♦❢ ▼❛t❤❡♠❛t✐❝s ✐♥ ◆❛✲ t✐♦♥❛❧ ◆❡t✇♦r❦ ✲ P❘❖❋▼❆❚ ❛♥❞ ❜r✐♥❣s ❛♥ ♦✈❡r✈✐❡✇ ♦❢ t❤❡ t❤❡♦r② t♦ ❜❡ t❛✉❣❤t ✐♥ Pr✐♠❛r② ❙❝❤♦♦❧✱ ✉s✉❛❧❧② ✐♥ t❤❡ t❤✐r❞ ❣r❛❞❡ ♦❢ ❍✐❣❤ ❙❝❤♦♦❧✳ ■t ❛❧s♦ ❢♦❝✉s❡s ♦♥ t❤❡ ♥❡❡❞ ❢♦r ❞✐✈❡rs❡ ❞✐❞❛❝t✐❝ ❛♣♣r♦❛❝❤❡s✱ ❡s♣❡❝✐❛❧❧② ✇✐t❤ t❤❡ ✉s❡ ♦❢ ♥❡✇ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠✲ ♠✉♥✐❝❛t✐♦♥ ❚❡❝❤♥♦❧♦❣✐❡s✱ t♦ ♦✈❡r❝♦♠❡ s♦♠❡ ❞✐✣❝✉❧t✐❡s ✇✐t❤ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ ✇❤✐❝❤ ❛r✐s❡ ♠❛♥② t✐♠❡s st❡♠ ❢r♦♠ ❛♥ ♦✈❡r❧② ❢♦r♠❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t♦♣✐❝✳ ❚❤❡ ♣r♦♣♦s❛❧ ♦❢ ❛❝t✐✈✐t✐❡s ✐♥ s♦♠❡ ❝♦♠♣✉t✐♥❣ ❡♥✈✐r♦♥♠❡♥t ❞❡✈❡❧♦♣❡❞ ✇✐t❤ s✉♣♣♦rt ♦❢ ❢r❡❡ s♦❢t✇❛r❡ ✭●❯■✮ ●❡♦❣❡❜r❛ ✐s ✐♥❝❧✉❞❡❞✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶✸
✶ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s ✶✺
✶✳✶ ◆♦t❛s ❤✐stór✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❛❧❣é❜r✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸ ❘❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡z ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✶✳✹ ❈♦♥❥✉❣❛❞♦ ❡ ✈❛❧♦r ❛❜s♦❧✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✺ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✿❢♦r♠❛ ♣♦❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✳✻ ❋ór♠✉❧❛s ❞❡ ▼♦✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✶✳✼ ❊①tr❛çã♦ ❞❡ r❛í③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✷ ❊♥❢♦q✉❡ ♣❡❞❛❣ó❣✐❝♦ ✸✾
✷✳✶ ❆ ♣r♦♣♦st❛ ❝✉rr✐❝✉❧❛r ❞♦ ❡st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✷ ❆t✐✈✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷✳✶ ❉❡s❝r✐çã♦ ❞❡ ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
■♥tr♦❞✉çã♦
◆♦ ❝❛♣ít✉❧♦ ❱■■■ ❞♦ ❘❡❣✐♠❡♥t♦ ❞♦ Pr♦❢▼❛t✱ ❝♦♥st❛✿
✧❆rt✐❣♦ ✷✽ ✲ ❖ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❞❡✈❡ ✈❡rs❛r s♦❜r❡ t❡♠❛s ❡s♣❡❝í✜❝♦s ♣❡rt✐♥❡♥t❡s ❛♦ ❝✉rrí❝✉❧♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❡ q✉❡ t❡♥❤❛♠ ✐♠♣❛❝t♦ ♥❛ ♣rát✐❝❛ ❞✐❞át✐❝❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳✧
❉❡♥tr♦ ❞❡ss❛s ❞✐r❡tr✐③❡s✱ ❡st❡ tr❛❜❛❧❤♦ ✈❡rs❛ s♦❜r❡ ✉♠ ❝♦♥t❡ú❞♦ q✉❡✱ ♠✉✐t❛s ✈❡✲ ③❡s✱ é ♥❡❣❧✐❣❡♥❝✐❛❞♦ ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❝♦♥❢♦r♠❡ ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ♣❡sq✉✐s❛ ❛t❡st❛♠✿ ♦s ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s✳ ❋✐③❡♠♦s ✉♠❛ r❡✈✐sã♦ t❡ór✐❝❛ ❞♦ t❡♠❛✱ ❞♦s ❛s♣❡❝t♦s ❤✐stór✐❝♦s q✉❡ ♣♦❞❡♠ ❝♦♥t❡①t✉❛❧✐③❛r ♦ s❡✉ ❡♥s✐♥♦✱ ❡ t❛♠❜é♠ ❜✉s❝❛♠♦s r❡❝✉rs♦s ♠❛✐s ❛t✉❛✐s q✉❡ ♣♦ss❛♠ ❢❛❝✐❧✐t❛r s❡✉ ❛♣r❡♥❞✐③❛❞♦✳
◆❛ r❡✈✐sã♦ t❡ór✐❝❛ ❛♣r❡s❡♥t❛❞❛✱ ♠❡r❡❝❡ ❞❡st❛q✉❡ ❛ ♦♣çã♦ ♣❡❧❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ ❝♦♥✲ ❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❛ ♣❛rt✐r ❞❛ ❝♦♥❝❡✐t✉❛çã♦ ❞♦ ❈♦r♣♦ ❞♦s ◆ú♠❡r♦s ❈♦♠✲ ♣❧❡①♦s✳ P❛r❡❝❡✲♥♦s ❝♦♥✈❡♥✐❡♥t❡ ❡✈✐t❛r ❝♦♠❡ç❛r ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❛r❜✐trár✐❛ ❞❡ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛✱ ❝♦♠♦ ❢❛③❡♠ ✈ár✐♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❡ ❛♣♦st✐❧❛s✳ ➱ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ ❛❜♦r✲ ❞❛r t❛❧ ❝❧❛ss❡ ❞❡ ♥ú♠❡r♦s ❝♦♠♦ ✉♠❛ ❡①t❡♥sã♦ ♥❛t✉r❛❧ ❞❛ ❝♦♥str✉çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ❝♦♥❥✉♥t♦s ♥✉♠ér✐❝♦s ❝❛❞❛ ✈❡③ ♠❛✐s ❛♠♣❧♦s✳
❆ s❡❣✉✐r✱ ❞❡♣♦✐s ❞❡ ❛♣r❡s❡♥t❛❞❛s ❛s ♦♣❡r❛çõ❡s ♥❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛✱ ♣❛ss❛♠♦s à ❢♦r♠❛ tr✐❣♦♥♦♠étr✐❝❛✱ ❛♦ ❡st✉❞♦ ❞♦ ❝♦♥❥✉❣❛❞♦✱ ❞♦ ♠ó❞✉❧♦ ❡✱ ✜♥❛❧♠❡♥t❡✱ ❞❛s ❢ór♠✉❧❛s ❞❡ ▼♦✐✈r❡✳
❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣❡❞❛❣ó❣✐❝♦✱ r❡ss❛❧t❛♠♦s q✉❡ ✉♠❛ ❞❛s ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ♥♦ ❡♥s✐♥♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s é ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ♣r♦❢❡ss♦r ❝♦♥t❛♥❞♦ ❛♣❡♥❛s ❝♦♠ ♦ ✉s♦ ❞❡ ❧♦✉s❛ ❡ ❣✐③✱ ❛❧é♠ ❞❡ ❛❧❣✉♠❛s ✐❧✉str❛çõ❡s ✐♠♣r❡ss❛s✱ ♥ã♦ ❝♦♥s❡❣✉❡ ♠♦str❛r ❛❞❡q✉❛✲ ❞❛♠❡♥t❡ ❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❡C✱ ♥❡♠ ❛ ❞✐♥â♠✐❝❛ ❡♥✈♦❧✈✐❞❛ ♥❛ r❡❧❛çã♦ ❡♥tr❡ z✱z¯✱|z|♦✉ ❡♥tr❡ z ❡zn ✭n✐♥t❡✐r♦✮✱ ♣♦r ❡①❡♠♣❧♦✳ ▼✉✐t❛s ✈❡③❡s✱ ❛♣❡♥❛s ♦s
❛s♣❡❝t♦s ❛❧❣é❜r✐❝♦s ❞❛s ♦♣❡r❛çõ❡s ❝♦♠ ♦s ❝♦♠♣❧❡①♦s sã♦ ❡①♣❧♦r❛❞♦s✱ ❧❡✈❛♥❞♦ ♦ ❛❧✉♥♦ ❛ ♥ã♦ ❝♦♥s✐❞❡r❛r ♣r♦✈❡✐t♦s♦ ❞❡❞✐❝❛r✲s❡ ❛ ❡st❡ ❛ss✉♥t♦✱ r❡❧❡❣❛♥❞♦✲♦✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ❛♦ r♦❧ ❞♦s ✧♣♦♥t♦s ♦❜s❝✉r♦s✧ ❞❛ ▼❛t❡♠át✐❝❛✳
❙❡♠ ✉♠❛ ❝♦♥t❡①t✉❛❧✐③❛çã♦ ❤✐stór✐❝❛ ❡ s❡♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❡ ❛❝❡ssí✈❡✐s ♥❛ ❡s❝♦❧❛ ❜ás✐❝❛✱ ♦ ❝❛♠✐♥❤♦ ♠❛✐s ♣r♦♠✐ss♦r ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡st❡ ❝♦♥t❡ú❞♦ ❥✉❧❣❛♠♦s s❡r ❛ ❡①♣❧♦✲ r❛çã♦ ❞❛ r✐❝❛ ✐♥t❡r❛çã♦ ❡♥tr❡ ●❡♦♠❡tr✐❛ ❡ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s r❡♣r❡s❡♥t❛❞♦s ♥♦ ♣❧❛♥♦ ❞❡ ❆r❣❛♥❞✲●❛✉ss✳
❋❡❧✐③♠❡♥t❡✱ ♥♦s ❞✐❛s ❛t✉❛✐s✱ ❥á ❞✐s♣♦♠♦s ❞❡ r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s q✉❡ ♣♦❞❡♠ s❡r ❛❝❡ssí✈❡✐s ❛♦s ♣r♦❢❡ss♦r❡s ❡ ❛♦s ❛❧✉♥♦s ❛tr❛✈és ❞❛ ❲❊❇✱ q✉❡ ❢❛✈♦r❡❝❡♠ ✉♠❛ ❛t✐t✉❞❡
✶✹
✐♥✈❡st✐❣❛t✐✈❛ ❞♦ s✐❣♥✐✜❝❛❞♦ ❣❡♦♠étr✐❝♦ ❞❡ ♠✉✐t❛s ♣r♦♣♦s✐çõ❡s✳ ◆♦ss♦s ❡s❢♦rç♦s s❡ ❝♦♥✲ ❝❡♥tr❛r❛♠✱ ❡♥tã♦✱ ♥❛ ♣r♦♣♦s✐çã♦ ❞❡ s✐t✉❛çõ❡s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ q✉❡ t✐r❛♠ ♣r♦✈❡✐t♦ ❞❛ ❢❛❝✐❧✐❞❛❞❡ q✉❡ ♦s ❥♦✈❡♥s tê♠ ❞❡ ♠❛♥✐♣✉❧❛r r❡❝✉rs♦s ❞❡ ✐♥❢♦r♠át✐❝❛ ❡ ✐♥t❡r♣r❡t❛r ✐♥❢♦r✲ ♠❛çõ❡s ❞✐❣✐t❛✐s✳
✶ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❚♦❞❛s ❛s r❡❢❡rê♥❝✐❛s ♣r❡s❡♥t❡s ❛♦ ✜♥❛❧ ❞♦ t❡①t♦ ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡st❡ ❝❛♣ít✉❧♦ ❞❡st✐♥❛❞♦ ❛♦ ❡st✉❞♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
✶✳✶ ◆♦t❛s ❤✐stór✐❝❛s
❖♣❡r❛çõ❡s ❝♦♠ r❛í③❡s q✉❛❞r❛❞❛s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s ❛♣❛r❡❝❡r❛♠ ❥á ♥♦ sé❝✉❧♦ ❳❱■ ❡♠ r❡❣✐str♦s r❡❧❛❝✐♦♥❛❞♦s ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞❡ t❡r❝❡✐r♦ ❣r❛✉✳
●✐r♦❧❛♠♦ ❈❛r❞❛♥♦✱ ♥❛s❝✐❞♦ ❡♠ P❛✈✐❛ ❡♠ ✶✺✵✶ ❡ ❢❛❧❡❝✐❞♦ ❡♠ ❘♦♠❛ ❡♠ ✶✺✼✻✱ ❢♦✐ ✉♠ ❝✐❡♥t✐st❛ ✈❡rsát✐❧ ❡ ♣♦❧ê♠✐❝♦✳ ➱ ❛✉t♦r ❞♦ ❧✐✈r♦ ▲✐❜❡r ❉❡ ▲✉❞♦ ❆❧❡❛❡ ✭▲✐✈r♦ ❞♦s ❥♦❣♦s ❞❡ ❛③❛r✮✱ ♦♥❞❡ ❜r✐❧❤❛♥t❡♠❡♥t❡ ✐♥tr♦❞✉③✐✉ ❛ ✐❞❡✐❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ q✉❡ s❡ ✉s❛ ♠♦❞❡r♥❛♠❡♥t❡❀ ❛❧✐ t❛♠❜é♠ ❡♥s✐♥♦✉ ♠❛♥❡✐r❛s ❞❡ s❡ tr❛♣❛❝❡❛r ♥♦s ❥♦❣♦s✳
❆♣❡s❛r ❞❡ tr❛ç♦s ♣❡ss♦❛✐s ♥❛❞❛ ❡❞✐✜❝❛♥t❡s✱ ❈❛r❞❛♥♦ ❧❡❣♦✉ à ♣♦st❡r✐❞❛❞❡ ✉♠ ❧✐✈r♦ q✉❡✱ ♥❛ é♣♦❝❛✱ ❡r❛ s❡♠ ❞ú✈✐❞❛ ♦ ♠❛✐♦r ❝♦♠♣ê♥❞✐♦ ❛❧❣é❜r✐❝♦ ❡①✐st❡♥t❡✿ ❛ ❆rt✐s ▼❛❣♥❛❡ ❙✐✈❡ ❞❡ ❘❡❣✉❧✐s ❆❧❣❡❜r❛✐❝✐s✱ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ♣♦r ❆rs ▼❛❣♥❛✱ ♣✉❜❧✐❝❛❞♦ ❡♠ ◆✉r❡♠❜❡r❣✱ ♥❛ ❆❧❡♠❛♥❤❛✱ ❡♠ ✶✺✹✺✳ ➱ ❛tr✐❜✉í❞❛ ❛ ❡❧❡ ❛ ❛✉t♦r✐❛ ❞❡ ✉♠❛ ❢ór♠✉❧❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ t✐♣♦
x3
+px2
+q= 0,
❡♠❜♦r❛ ❡❧❛ t❡♥❤❛ s✐❞♦ ❞❡s❝♦❜❡rt❛ ♣♦r ♦✉tr♦ ♣❡rs♦♥❛❣❡♠ ❝♦♥tr♦✈❡rt✐❞♦ ❞❛ ❤✐stór✐❛✱ ◆✐❝❝♦❧♦ ❋♦♥t❛♥❛✳
◆✐❝❝♦❧♦ ❋♦♥t❛♥❛✱ ❛♣❡❧✐❞❛❞♦ ❚❛rt❛❣❧✐❛ ❡♠ r❛③ã♦ ❞❡ ♣r♦❜❧❡♠❛s ♥❛ ❢❛❧❛✱ r❡s✉❧t❛♥t❡s ❞❡ ❝✐❝❛tr✐③❡s ♥❛ ❜♦❝❛ ❛❞q✉✐r✐❞❛s ♥❛ ✐♥❢â♥❝✐❛✱ ♥❛s❝❡✉ ❡♠ ❇rés❝✐❛✱ ❡♠ ✶✺✵✶✳ ❆♦ ❧♦♥❣♦ ❞❛ ✈✐❞❛✱ ♣✉❜❧✐❝♦✉ ❞✐✈❡rs❛s ♦❜r❛s✱ ✉t✐❧✐③❛♥❞♦ ♦ ❝♦❞♥♦♠❡ ❚❛rt❛❣❧✐❛✱ ❡ ❢♦✐ ♦ ♣r✐♠❡✐r♦✱ ❝❡r❝❛ ❞❡ ✶✵✵ ❛♥♦s ❛♥t❡s ❞❡ ●❛❧✐❧❡✉✱ ❛ r❡❛❧✐③❛r ❝á❧❝✉❧♦s ♥❛ té❝♥✐❝❛ ❞❛ ❛rt✐❧❤❛r✐❛✳ ▼❛s ♦ q✉❡ ♦ ❝♦❧♦❝♦✉ ❞❡✜♥✐t✐✈❛♠❡♥t❡ ♥♦s ❛♥❛✐s ❞❛ ▼❛t❡♠át✐❝❛ ❢♦r❛♠ s✉❛s ❞✐s♣✉t❛s ❝♦♠ ❈❛r❞❛♥♦ s♦❜r❡ ❛s ❡q✉❛çõ❡s ❞♦ t❡r❝❡✐r♦ ❣r❛✉✳
❆♥❛❧✐s❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ❚❛rt❛❣❧✐❛ ♦❜s❡r✈❛♠♦s ❝♦♠♦ s✉r❣✐r❛♠ ❛s ♠❡♥✲ ❝✐♦♥❛❞❛s r❛í③❡s ❞❡ ♥ú♠❡r♦s ♥❡❣❛t✐✈♦s✳ ❋❛r❡♠♦s ✐ss♦ ❛ s❡❣✉✐r✱ ♠❛s ❛♥t❡s ✈❛♠♦s r❡ss❛❧t❛r ✉♠ ❞❡t❛❧❤❡ ❞❛ ♠❛✐♦r ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ♦ ❛❧✉♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✿ ❛ ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦ ✭❚❛rt❛❣❧✐❛✦✮ ♥ã♦ ♥♦s ❞á s♦❧✉çõ❡s ❝♦♠♣❧❡①❛s ❞❛ ❡q✉❛çã♦ ❞♦ t❡r❝❡✐r♦ ❣r❛✉❀ ❡❧❛ ❞❡✈❡ s❡r ❛♣❧✐❝❛❞❛ ♥❛ ❜✉s❝❛ ❞❡ ✉♠❛ r❛✐③ r❡❛❧✳
✶✻ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❈♦♥s✐❞❡r❡♠♦s ❛ ❡q✉❛çã♦ ❣❡r❛❧ ❞♦ t❡r❝❡✐r♦ ❣r❛✉
ax3
+bx2
+cx+d= 0, a6= 0. ✭✶✳✶✮
■♥✐❝✐❛❧♠❡♥t❡✱ ✈❛♠♦s ❛♥✉❧❛r ♦ t❡r♠♦ ❞❡ s❡❣✉♥❞♦ ❣r❛✉✳ ❋❛③❡♥❞♦x=y+m✱ t❡♠♦s✿ a(y+m)3
+b(y+m)2
+c(y+m) +d= 0,
♦✉ s❡❥❛✱
ay3
+ (3am+b)y2
+ (3am2
+ 2bm+c)y+am3
+bm2
+cm+d= 0.
❆❣♦r❛✱ ✈❛♠♦s ❢❛③❡r 3am =−b✱ ✐st♦ é✱ m =− b
3a✳ ◆♦t❡ q✉❡✱ ❝♦♠ ✐ss♦✱ ❛♥✉❧❛♠♦s ♦
t❡r♠♦ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦ry2✱ ♦❜t❡♥❞♦ ❛ ❡q✉❛çã♦
ay3+ 3a
−b
3a
2
+ 2b
−b 3a +c !
y+a
−b 3a 3 +b −b 3a 2 +c −b 3a
+d= 0,
♦✉ s❡❥❛✱
ay3
+
3ac−b2
3a
y+ 2b
3
−9abc+ 27a2
d
27a2 = 0, q✉❡ é ✉♠❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛
y3
+py+q = 0, ✭✶✳✷✮
♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ❢♦r♠❛ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❞❡ t❡r❝❡✐r♦ ❣r❛✉✳
◆♦t❡ q✉❡ s❡ y ❢♦r ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✳✷✮✱ x= y+m s❡rá ✉♠❛ s♦❧✉çã♦
♣❛r❛ ❛ ❡q✉❛çã♦ ✭✶✳✶✮✳ P♦rt❛♥t♦✱ q✉❛♥❞♦ ❡♥❝♦♥tr♦✉ ❛ s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ss❡ t✐♣♦✱ ❚❛rt❛❣❧✐❛ ❞❡✉ ✉♠❛ r❡s♣♦st❛ ❣❡r❛❧ ❡ ♥ã♦ ❛♣❡♥❛s ♣❛rt✐❝✉❧❛r ❛♦ ♣r♦❜❧❡♠❛✱ ♦ q✉❡ ❛✉♠❡♥t❛ ♦ s❡✉ ♠ér✐t♦✳
❆❣♦r❛✱ ✈❛♠♦s ❝♦♥❤❡❝❡r ♦ s❡❣r❡❞♦ ❞❡ q✉❡ ❈❛r❞❛♥♦ s❡ ❛♣r♦♣r✐♦✉✳ ❚♦❞❛s ❛s ❣r❛♥❞❡s ❞❡s❝♦❜❡rt❛s s❡♠♣r❡ ♣❛rt❡♠ ❞❡ ✉♠❛ ✐❞❡✐❛ ❢✉♥❞❛♠❡♥t❛❧✳ ◆♦ ❝❛s♦✱ ❛ ✐❞❡✐❛ ❞❡ ❚❛rt❛❣❧✐❛ ❢♦✐ s✉♣♦r q✉❡ ❛ s♦❧✉çã♦ ♣r♦❝✉r❛❞❛ ❡r❛ ❝♦♠♣♦st❛ ❞❡ ❞✉❛s ♣❛r❝❡❧❛s ❡✱ ❛ss✐♠✱ ❡s❝r❡✈❡✉✿
x =A+B ✳ ❖s ❞♦✐s ❧❛❞♦s ❞❛ ❡q✉❛çã♦ s❡♥❞♦ ✐❣✉❛✐s✱ s❡✉s ❝✉❜♦s t❛♠❜é♠ ♦ s❡rã♦✱ ♣♦r
❝♦♥s❡❣✉✐♥t❡✱ x3
= (A+B)3✱ ♦✉ s❡❥❛✱
x3
=A3
+ 3A2
B+ 3AB2
+B3
=A3
+B3
+ 3AB(A+B).
❈♦♠♦ x=A+B✱ ♦❜t❡♠♦s x3
=A3
+B3
+ 3ABx✱ ♦✉ x3
−3ABx−(A3
+B3
) = 0. ✭✶✳✸✮
❈♦♠♣❛r❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✶✳✷✮ ❡ ✭✶✳✸✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ 3AB =−p ♦✉
A3B3 =−p
3
27. ✭✶✳✹✮
❚❛♠❜é♠ t❡♠♦s✿
A3
+B3
◆♦t❛s ❤✐stór✐❝❛s ✶✼
❆ss✐♠✱ A3 ❡
B3 sã♦ ❞♦✐s ♥ú♠❡r♦s ❞♦s q✉❛✐s ❝♦♥❤❡❝❡♠♦s ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦✳ P♦r ✭✶✳✹✮ ❡ ✭✶✳✺✮✱ t❡♠♦s
A3
(−q−A3
) = −p
3
27 ⇒
A32
+qA3
− p
3
27 = 0.
❆♣❧✐❝❛♥❞♦ ❛ ❋ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛ ✶ ♦❜t❡♠♦s
A3
=
−q±
s q2 −4 −p 3 27 2 ⇓ A3
=−q 2 ±
r
27q2+ 4p3
108 . ✭✶✳✻✮
■♥✐❝✐❛❧♠❡♥t❡✱ ✈❛♠♦s ❡s❝♦❧❤❡r ❛ s♦♠❛✿
A3
=−q 2+
r
27q2
+ 4p3
108 .
❊♥tã♦✱
B3
=−q− −q
2 +
r
27q2
+ 4p3
108
!
⇒B3
=−q 2 −
r
27q2
+ 4p3
108 .
❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♦x=A+B✱ t❡♠♦s
x= 3
s
−q2 +
r
27q2+ 4p3
108 +
3
s
−q2 −
r
27q2+ 4p3
108 . ✭✶✳✼✮
❊s❝♦❧❤❡♥❞♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♠ ✭✶✳✻✮✱ t❡♠♦s A3
=−q 2 −
r
27q2
+ 4p3
108 ❡ ♦❜t❡♠♦s
B3
=−q− −q
2 −
r
27q2 + 4p3
108
!
⇒B3
=−q 2 +
r
27q2+ 4p3
108 .
❡ ♦ r❡s✉❧t❛❞♦ ❞❡ ✭✶✳✼✮✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❋ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦✱ ♥ã♦ s❡ ❛❧t❡r❛✳
P❛r❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ✉♠ ❛❧✉♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ é ♠❛✐s ❛❝♦♥s❡❧❤á✈❡❧ ✐❧✉str❛r ❛ té❝♥✐❝❛ q✉❡ ❧❡✈❛ à ❢ór♠✉❧❛ ❞❡ ❈❛r❞❛♥♦ ❝♦♠ ✉♠ ❡①❡♠♣❧♦ ♥✉♠ér✐❝♦✳ ❙❡❥❛ ❛ ❡q✉❛çã♦
✶❆ ♣r♦✈❛ ❞❡ss❛ ❢ór♠✉❧❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✁✧ ❢ór♠✉❧❛ ❞❡ ❇❤❛s❦❛r❛✁✧ ♥❛ r❡❛❧✐❞❛❞❡ ❢♦✐ ❢❡✐t❛ ♣❡❧♦ ♠❛t❡✲
✶✽ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
x3
−6x−9 = 0✳ ❆q✉✐✱ p=−6 ❡q =−9✳ ▲♦❣♦
x = 3
s
9 2+
r
27.(−9)2+ 4(
−6)3
108 + 3 s 9 2 − r
27(−9)2+ 4(
−6)3
108 . = 3 s 9 2+ r 49 4 + 3 s 9 2− r 49 4 = 3 r
9 + 7 2 +
3
r
9−7 2 = √3
8 +√3
1 = 2 + 1 = 3.
P♦r s✐♠♣❧❡s ✈❡r✐✜❝❛çã♦✱ ❝♦♥st❛t❛✲s❡ q✉❡ 3✱ r❡❛❧♠❡♥t❡✱ é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❛❞❛ ❡
❛ ❢ór♠✉❧❛✱ ❝♦♠♦ ❡s♣❡r❛❞♦✱ ❢✉♥❝✐♦♥♦✉✳
❊♠ ✶✺✹✺✱ ❈❛r❞❛♥♦ ❞❡♣❛r♦✉✲s❡ ❝♦♠ ✉♠ ♣r♦❜❧❡♠❛ s❡♠❡❧❤❛♥t❡ ❛ ❡st❡✿ ✧❯♠ ♠❛r❝❡✲ ♥❡✐r♦ q✉❡r ❝♦♥str✉✐r ❞✉❛s ❝❛✐①❛s✱ ✉♠❛ ❝♦♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛x✱ ♦✉tr❛ ❝♦♠
❛ ❢♦r♠❛ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦♠ ❜❛s❡ r❡t❛♥❣✉❧❛r✱ ❞❡ ❧❛❞♦s3m❡5m✱ ❡ ❞❡ ❛❧t✉r❛ ✐❣✉❛❧
à ❛❧t✉r❛ ❞♦ ❝✉❜♦✳ ❖ ✈❛❧♦r ❞❡x❞❡✈❡ s❡r ❡s❝♦❧❤✐❞♦ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ♦ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦
s❡❥❛ 4m3 ♠❛✐♦r ❞♦ q✉❡ ♦ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦✧✳
❊❧❡ ❛♣❧✐❝♦✉ ❛ ❢ór♠✉❧❛ ♣❛r❛ ❝♦♠♣r♦✈❛r ❛ s♦❧✉çã♦ x= 4 q✉❡ ❡❧❡ ❥á ❝♦♥❤❡❝✐❛ ❡ ♦❜t❡✈❡
✉♠ r❡s✉❧t❛❞♦ s✉r♣r❡❡♥❞❡♥t❡✳ ❱❛♠♦s r❡♣❡t✐r s❡✉s ♣❛ss♦s✳ ❊q✉❛❝✐♦♥❛♥❞♦ ❝♦♠ ❛ ♥♦t❛çã♦ ♠♦❞❡r♥❛✱ t❡♠♦s✿ x3
−15x−4 = 0✳ ❋❛③❡♥❞♦x=A+B✱
♦ r❛❝✐♦❝í♥✐♦ ❞❡ ❚❛rt❛❣❧✐❛ ❧❡✈❛ ❛✿
x3
=A3
+B3
+ 3ABx= 15x+ 4.
❊♥tã♦✱ A3
B3
= 125 ❡A3
+B3
= 4✱ ❞❡ ♦♥❞❡ A3
(4−A3
) = 125✱ ♦✉ s❡❥❛✿
A32
−4A3
+ 125 = 0.
P♦rt❛♥t♦✱ A3
= 2±√−121✳ ❙❡ A3
= 2 +√−121✱ ❡♥tã♦ B3
= 2−√−121✳
❉❡ss❡ ♠♦❞♦✱ t❡♠♦s ❛ s♦❧✉çã♦✿
x= 3
q
2 +√−121 + 3
q
2−√−121.
❙❡ ♦♣t❛r♠♦s ♣♦r A3
= 2−√−121✱ ❝❤❡❣❛r❡♠♦s ❛♦ ♠❡s♠♦ r❡s✉❧t❛❞♦✳
❯♠ ❛❧✉♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ s❛❜❡r✐❛ ❡s❝r❡✈❡r ❛ s♦❧✉çã♦ ❡♥❝♦♥tr❛❞❛ ❝♦♠♦x= √3
2 + 11i
+√3
2−11i ❡✱ ♣♦r t❡♥t❛t✐✈❛✱ ✈❡r✐✜❝❛r q✉❡ (2 +i)3
= (8 + 12i−6−i) = 2 + 11i✱ ❞❡
♠♦❞♦ q✉❡ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ é (2 +i) + (2−i) = 4✱ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ P♦ré♠✱ ❛
◆♦t❛s ❤✐stór✐❝❛s ✶✾
s♦❜r❡ ♥ú♠❡r♦s ✧s♦❢íst✐❝♦s✧✭❝♦♠♦ t❛✐s ♥ú♠❡r♦s ❢♦r❛♠ ❝❤❛♠❛❞♦s ♥❛ é♣♦❝❛✮ ♣r♦❞✉③❡♠ ♥ú♠❡r♦s ❞❡ss❡ t✐♣♦✳
❋♦✐ ❘❡♥è ❉❡s❝❛rt❡s✱ ❡♠ ✶✻✸✼✱ ❝♦♠ ❛ ♦❜r❛ ▲❛ ●é♦♠étr✐❡✱ q✉❡♠ ✐♥tr♦❞✉③✐✉ ♦s t❡r♠♦s ♣❛rt❡ r❡❛❧ ❡ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛✳
❆ r❛✐③ q✉❛❞r❛❞❛ ❞❡ −1 só ♣❛ss♦✉ ❛ s❡r r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ❧❡tr❛ i ❛ ♣❛rt✐r ❞❡ ✶✼✼✼✱
♣❡❧♦ ❣ê♥✐♦ s✉íç♦ ▲❡♦♥❤❛r❞ ❊✉❧❡r✳
❆ ❡①♣r❡ssã♦ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❢♦✐ ✉s❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ●❛✉ss ❡♠ ✶✽✸✶✱ ❡♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ➪❧❣❡❜r❛ ✲ ✧t♦❞❛ ❡q✉❛çã♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ♦✉ ❝♦♠♣❧❡①♦s ❞❡ ❣r❛✉n≥1✱ t❡♠ ♥♦ ❝❛♠♣♦ ❝♦♠♣❧❡①♦✱ ♣❡❧♦ ♠❡♥♦s
✉♠❛ r❛✐③✧ ✲ ❡♠ ✶✼✾✾✱ ❡♠❜♦r❛ ❛ ✐❞❡✐❛ ❥á t✐✈❡r❛ s✐❞♦ ❝♦♥❥❡❝t✉r❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ♣♦r ●✐r❛r❞✱ ❉❡s❝❛rt❡s ❡ ❏❡❛♥ ▲❡ ❘♦♥❞ ❉✬❆❧❡♠❜❡rt✳
❏❡❛♥ ❘♦❜❡rt ❆r❣❛♥❞ ❡ ❈❛s♣❛r ❲❡ss❡❧✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ♠♦t✐✈❛❞♦s ♣❡❧❛ ❣❡♦♠❡✲ tr✐❛ ❡ ♣❡❧❛ t♦♣♦❣r❛✜❛✱ ❞❡ ♠❛♥❡✐r❛ ✐♥t✉✐t✐✈❛ ❡ ♣rát✐❝❛✱ r❡♣r❡s❡♥t❛r❛♠ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ ♣♦♥t♦s ✭❡ ❝♦♠♦ ✈❡t♦r❡s✮ ♥✉♠ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳ ❊ ●❛✉ss ❞❡✜♥✐✉ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♥❛ ❢♦r♠❛ a+bi✱ ♦♥❞❡ a ❡ b sã♦ ♥ú♠❡r♦s r❡❛✐s ❡ i2
=−1✳
❖ ♣❛ss♦ ❞❡❝✐s✐✈♦ ♥♦ s❡♥t✐❞♦ ❞❡ ❢♦r♠❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ❢♦✐ ❛ r❡✲ ♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ss❡s ♥ú♠❡r♦s ❝♦♠♦ ♣♦♥t♦s ❞♦ ♣❧❛♥♦✳ ❖ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ❛ t❡r ✉♠❛ ✈✐sã♦ ❝❧❛r❛ ❞❡ t❛❧ r❡♣r❡s❡♥t❛çã♦ ❡ ❡①♣❧♦rá✲❧❛ ❡♠ s✉❛s ✐♥✈❡st✐❣❛çõ❡s ❢♦✐ ●❛✉ss✱ ❝♦♥❢♦r♠❡ ✜❝❛ ❝❧❛r♦✱ ❡♠❜♦r❛ ❞❡ ♠♦❞♦ ✐♠♣❧í❝✐t♦✱ ❡♠ s✉❛ t❡s❡ ❡s❝r✐t❛ ❡♠ ✶✼✾✾✳ ❚♦❞❛✈✐❛✱ ●❛✉ss só ❡①♣ôs ❛♦ ♣ú❜❧✐❝♦ s✉❛s ✐❞❡✐❛s ❛ ❡ss❡ r❡s♣❡✐t♦ ❡♠ ✶✽✸✶✱ ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ ✐♥tr♦❞✉③✐r ♦s ✐♥t❡✐r♦s ❣❛✉ss✐❛♥♦s✳
❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s C ❢♦✐ ✜♥❛❧♠❡♥t❡ ❞❡✜♥✐❞♦ ❞❡ ♠♦❞♦ r✐❣♦r♦s♦ ♣♦r
❍❛♠✐❧t♦♥ ❡♠ ✶✽✸✼✳ ✷
❉❡✈❡✲s❡ ❛ ❏❡❛♥✲❘♦❜❡rt ❆r❣❛♥❞ ✭✶✼✻✽✲✶✽✷✷✮ ✉♠❛ ♣✉❜❧✐❝❛çã♦✱ ❡♠ ✶✽✵✻✱ ❝♦♠ ❛ ❛t✉❛❧ ❡ ❞❡✜♥✐t✐✈❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❉✐❛❣r❛♠❛ ❞❡ ❆r❣❛♥❞✳
❖ ♣❧❛♥♦ ❞❡ ❆r❣❛♥❞✲●❛✉ss é ♠✉✐t♦ út✐❧✱ ♣♦✐s ❛tr❛✈és ❞❡❧❡ ♣♦❞❡♠♦s ✧❛❧❣❡❜r✐③❛r✧ ✈❡t♦✲ r❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s✱ ♦ q✉❡ t❡♠ ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s ❡♠ ❞✐✈❡rs♦s ❝❛♠♣♦s ❞❛ ▼❛t❡♠át✐❝❛✱ ❞❛ ❊♥❣❡♥❤❛r✐❛ ❡ ❞❛ ❋ís✐❝❛✳ ❚❛♠❜é♠ é ✐♠♣♦rt❛♥t❡ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣❡❞❛❣ó❣✐❝♦ ♣♦r ❢♦r♥❡❝❡r ✉♠ s✐❣♥✐✜❝❛❞♦ ❝♦♥❝r❡t♦ ❛♦ t❡♠❛✱ ❛❧é♠ ❞❛ ♠❡r❛ ♠❛♥✐♣✉❧❛çã♦ ❛❧❣é❜r✐❝❛✳
✷❏á ❡♠ ✶✽✸✸✱ ❍❛♠✐❧t♦♥ ❛♣r❡s❡♥t♦✉ ✉♠ ❛rt✐❣♦ à ❆❝❛❞❡♠✐❛ ■r❧❛♥❞❡s❛ ✧❡♠ q✉❡ ✐♥tr♦❞✉③✐✉ ✉♠❛ á❧❣❡❜r❛
✷✵ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
Plano de Argand em: http://upload.wikimedia.org/wikipedia/commons/1/13/Argandgaussplane.gif
✶✳✷ ❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❛ r❡♣r❡s❡♥t❛çã♦
❛❧❣é❜r✐❝❛
❉❡✜♥✐çã♦ ✶✳✶✳ ❉❡✜♥✐♠♦s ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦
C={(x, y) :x∈R ❡ y∈R},
❝♦♠ ❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✿ s❡ z = (x, y) ❡ w = (a, b)
♣❡rt❡♥❝❡r❡♠ ❛ C✱ ❡♥tã♦
z+w= (x+a, y+b) ❡ zw = (xa−yb, xb+ya). ✭✶✳✽✮
❖s ❡❧❡♠❡♥t♦s ❞❡ C sã♦ ❞❡♥♦♠✐♥❛❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳
❉❡♥♦t❛r❡♠♦s ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ (0,0) s✐♠♣❧❡s♠❡♥t❡ ♣♦r 0 ❡ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ (1,0) s✐♠♣❧❡s♠❡♥t❡ ♣♦r 1✳
❉❡✜♥✐çã♦ ✶✳✷✳ P❛r❛ ❝❛❞❛ z = (x, y)∈C, ❞❡✜♥✐♠♦s
−z = (−x,−y) ❡ s❡ z 6= 0 z−1
=
x x2+y2,
−y x2+y2
. ✭✶✳✾✮
❖ ♥ú♠❡r♦ z−1 t❛♠❜é♠ é ❞❡♥♦t❛❞♦ ♣♦r 1
z ♦✉1/z.
❉❡❝♦rr❡♥t❡s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ C✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧✲ t❛❞♦✳
Pr♦♣♦s✐çã♦ ✶✳✶✳ ❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s s❡ ✈❡r✐✜❝❛♠ ♣❛r❛ q✉❛✐sq✉❡r z, w, t∈C ✿ ✭❛✮ z+ (w+t) = (z+w) +t ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❛❞✐çã♦✮✳
❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❛❧❣é❜r✐❝❛ ✷✶
✭❝✮ 0 +z =z ✭❡❧❡♠❡♥t♦ ♥❡✉tr♦✮✳
✭❞✮ z+ (−z) = 0 ✭❡❧❡♠❡♥t♦ ♦♣♦st♦✮✳
✭❡✮ z(wt) = (zw)t ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✮✳
✭❢✮ zw =wz ✭❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✮✳
✭❣✮ 1z =z ✭❡❧❡♠❡♥t♦ ✉♥✐❞❛❞❡✮✳
✭❤✮ zz−1
= 1 s❡ z 6= 0 ✭❡❧❡♠❡♥t♦ ✐♥✈❡rs♦✮✳
✭✐✮ z(w+t) =zw+zt ✭❞✐str✐❜✉t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ r❡❧❛çã♦ à ❛❞✐çã♦✮✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ z= (x, y), w = (a, b)❡ t= (c, d)✳ ❊♥tã♦✿
✭❛✮ ❯s❛♥❞♦ ❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s✿
z+ (w+t) = (x, y) + (a+c, b+d) = (x+ (a+c), y+ (b+d)) = ((x+a) +c,(y+b) +d) = (x+a, y+b) + (c, d) = (z+w) +t.
✭❜✮ ❯s❛♥❞♦ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❛ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s✿
z+w = (x, y) + (a, b) = (x+a, y+b) = (a+x, b+y) = (a, b) + (x, y) = w+z.
✭❝✮ ❯s❛♥❞♦ ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❛ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s✿
z+ 0 = (x, y) + (0,0) = (x+ 0, y+ 0) = (x, y)
= z.
✭❞✮ ❯s❛♥❞♦ ❛ ❉❡✜♥✐çã♦ ✶✳✷ ❡ ♦ ❡❧❡♠❡♥t♦ ♦♣♦st♦ ❞❛ ❛❞✐çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s✿
z+ (−z) = (x, y) + (−x,−y) = (x−x, y−y) = (0,0)
✷✷ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
✭❡✮ ❈♦♠♦wt= (a, b)(c, d) = (ac−bd, ad+bc)❡zw= (x, y)(a, c) = (xa−yb, xb+ya)✱
t❡♠♦s✿
z(wt) = (x, y)(ac−bd, ad+bc)
= [x(ac−bd)−y(ad+bc), x(ad+bc) +y(ac−bd)] = (xac−xbd−yad−ybc, xad+xbc+yac−ybd) = (xac−ybc−xbd−yad, xbc+yac+xad−ybd) = [(xa−yb)c−(xb+ya)d, (xb+ya)c+ (xa−by)d] = (zw)t,
♦♥❞❡ ✉s❛♠♦s ❛ ♣r♦♣r✐❡❞❛❞❡ ❛ss♦❝✐❛t✐✈❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ ✭❢✮ ❯s❛♥❞♦ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s✿
zw = (x, y)(a, b)
= (xa−yb, xb+ya) = (ax−by, ay+bx) = wz.
✭❣✮ ❯s❛♥❞♦ ♦ ❡❧❡♠❡♥t♦ ✉♥✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ t❡♠♦s✿
1z = (1,0)(x, y)
= (1x−0y,1y+ 0x) = (1x,1y)
= (x, y) = z.
✭❤✮ ❈♦♠♦ z−1
= x x2+y2, −
y x2+y2
s❡z 6= 0 ✭✈❡❥❛ ❉❡✜♥✐çã♦ ✶✳✷✮✱ t❡♠♦s✿
zz−1
= (x, y)
x x2+y2,
−y x2+y2
= x x x2+y2
−y
−y x2+y2
, x
−y x2+y2
+y
x x2+y2
=
x2
+y2
x2+y2,
−xy+yx x2+y2
= (1,0) = 1.
❖ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❛❧❣é❜r✐❝❛ ✷✸
t❡♠♦s✿
z(w+t) = (x, y)[(a, b) + (c, d)] = (x, y)(a+c, b+d) = (x(a+c)−y(b+d), x(b+d) +y(a+c)) = (xa+xc−yb−yd, xb+xd+ya+yc)
= [(xa−yb) + (xc−yd),(xb+ya) + (xd+yc)] = (xa−yb, xb+ya) + (xc−yd, xd+yc)
= (x, y)(a, b) + (x, y)(c, d) = zw+zt.
❖❜s❡r✈❛çã♦ ✶✳✶✳ ❉❡♥♦♠✐♥❛✲s❡ ❝♦r♣♦ ✉♠ ❝♦♥❥✉♥t♦ ♥♦ q✉❛❧ ❡stã♦ ❞❡✜♥✐❞❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ q✉❡ s❛t✐s❢❛③❡♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ♠❡♥❝✐♦♥❛❞❛s ♥❛ Pr♦♣♦s✐çã♦ ✶✳✶✳ P♦r ❡st❛ r❛③ã♦ é q✉❡ ❝❤❛♠❛♠♦sC❞❡ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❛ss✐♠ ❝♦♠♦ ♥♦s r❡❢❡r✐♠♦s ❛ R ❝♦♠♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❡ ❛ Q❝♦♠♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ❚❡♥❞♦ ❞❡✜♥✐❞♦ ❛s ♦♣❡r❛çõ❡s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠C✱ ❞❡✜♥✐♠♦s ❛s ♦♣❡r❛çõ❡s ❞❡ s✉❜tr❛çã♦ ❡ ❞✐✈✐sã♦ ❞❛ ♠❛♥❡✐r❛ ✉s✉❛❧✿
❉❡✜♥✐çã♦ ✶✳✸✳ ❉❛❞♦s z, w ∈C,
z−w=z+ (−w) ❡ z
w =zw
−1
, s❡ w6= 0.
❆❧é♠ ❞✐ss♦✱ ❛ ♣♦t❡♥❝✐❛çã♦ t❛♠❜é♠ é ❞❡✜♥✐❞❛ ❞❛ ♠❛♥❡✐r❛ ✉s✉❛❧✿ ❉❡✜♥✐çã♦ ✶✳✹✳ ❉❛❞♦ z ∈C,
z0
= 1, zn=z· · ·z
| {z }
n vezes
❡ z−n=z−1
· · ·z−1
| {z }
n vezes
, s❡ z 6= 0 (n ≥1).
❉❡❝♦rr❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶ q✉❡ ❞✐✈❡rs❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ♦♣❡r❛çõ❡s ❛r✐t♠ét✐❝❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s sã♦ ✈á❧✐❞❛s ♣❛r❛ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ P♦r ❡①❡♠♣❧♦✱ ❛ s♦♠❛ ❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞✉❛s ❢r❛çõ❡s z1
w1 ❡ z2
w2
❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ♣❡❧❛s ❢ór♠✉❧❛s✿
z1
w1
+ z2
w2
= z1w2+z2w1
w1w2
❡ z1
w1
z2
w2
= z1z2
w1w2
. ✭✶✳✶✵✮
❆❧é♠ ❞✐ss♦✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❝❧✉sã♦✿ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ R é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ C✱ ♦✉ s❡❥❛✱ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✳ ■st♦ ♦❝♦rr❡ q✉❛♥❞♦ ❞❡♥♦t❛♠♦s q✉❛❧q✉❡r ♥ú♠❡r♦ ❝♦♠♣❧❡①♦(x,0)✱ ❝♦♠x∈R✱ s✐♠♣❧❡s♠❡♥t❡ ♣♦r x✳ ❖✉ s❡❥❛✱ ❛tr❛✈és ❞❡ss❛ ❝♦♥✈❡♥çã♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ R é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ C ❞❛ ❢♦r♠❛ (x,0)✳ ❊st❛♠♦s ❡st❡♥❞❡♥❞♦✱ ♣♦rt❛♥t♦✱ ❛ ✐❞❡✐❛ ❥á ❡①♣♦st❛ ❡♠
✷✹ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❖❜s❡r✈❛çã♦ ✶✳✷✳ ❆ s♦♠❛ ❞♦s ♥ú♠❡r♦s r❡❛✐sx ❡ a ♦✉ ❛ s♦♠❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s x ❡ a ❧❡✈❛ ❛♦ ♠❡s♠♦ r❡s✉❧t❛❞♦❀ ❛♥❛❧♦❣❛♠❡♥t❡ ❝♦♠ ♦ ♣r♦❞✉t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s x ❡ a
❡ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s x ❡ a✳
(x,0) + (a,0) = (x+a,0) =x+a ❡
(x,0)(a,0) = (xa−0.0, x.0 + 0.a) = (xa,0) =xa.
P♦r ❝♦♥s❡❣✉✐♥t❡✱ ♥ã♦ ❡①✐st❡ ❛♠❜✐❣✉✐❞❛❞❡ ♥❛s ♥♦t❛çõ❡s x+a❡ xa✱ ❝♦♠ ❛ ❛❞♦çã♦ ❞❛
✐♥❝❧✉sã♦ ❛♣r❡s❡♥t❛❞❛ ❛❝✐♠❛✳
●❡r❛❧♠❡♥t❡✱ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s sã♦ ❞❡✜♥✐❞♦s ❝♦♠♦ s❡♥❞♦ ♦s ♥ú♠❡r♦s ❞❛ ❢♦r♠❛x+yi✱ ♦♥❞❡x❡y sã♦ ♥ú♠❡r♦s r❡❛✐s ❡i é ✉♠ ❛❧❣❛r✐s♠♦ ✐♠❛❣✐♥ár✐♦✱
q✉❡ s❛t✐s❢❛③ i2
= −1✳ ❆ s❡❣✉✐r✱ ✈❡r❡♠♦s q✉❡ ❡ss❛ ❞❡✜♥✐çã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
✈✐st❛ ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❞❡✜♥✐çã♦ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s ❛♣r❡s❡♥t❛❞❛ ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✳
◆♦t❡ q✉❡
(0,1)2
= (0,1)(0,1) = (0.0−1.1,0.1 + 1.0) = (−1,0) =−1,
♦✉ s❡❥❛✱ ♦ ♥ú♠❡r♦ −1 ♣♦ss✉✐ ✉♠❛ ✧r❛✐③ q✉❛❞r❛❞❛✧ ❡♠ C✦
❉❡✜♥✐çã♦ ✶✳✺✳ ❈❤❛♠❛r❡♠♦s ❞❡ ❛❧❣❛r✐s♠♦ ✐♠❛❣✐♥ár✐♦ ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ (0,1) ❡
♦ ❞❡♥♦t❛r❡♠♦s ♣♦r i✳
❚❡♠♦s✱ ❡♥tã♦✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛ ❞♦ ❛❧❣❛r✐s♠♦ ✐♠❛❣✐♥ár✐♦✿
i2 =−1.
❊✱ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ q✉❛❧q✉❡rz = (x, y)✱ t❡♠♦s✿
z = (x, y) = (x,0) + (0, y) = (x,0) + (y,0)(0,1),
♦✉ s❡❥❛✱
z =x+yi.
▲♦❣♦✱ ♦ ♣❛r (x, y) ❡ ❛ ❡①♣r❡ssã♦ x+yi r❡♣r❡s❡♥t❛♠ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✳
❆ ❡①♣r❡ssã♦ x+yi é ❞❡♥♦♠✐♥❛❞❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❞❡z✳
❖❜s❡r✈❛çã♦ ✶✳✸✳ ◗✉❛♥❞♦ ✉s❛♠♦s ❛ ❢♦r♠❛ ❛❧❣é❜r✐❝❛ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❛s ❞❡✜✲ ♥✐çõ❡s ❞❡ z+w ❡ zw ❞❛❞❛s ❡♠ ✭✶✳✽✮ t♦r♥❛♠✲s❡ ❞❡ s✐♠♣❧❡s ❡♥t❡♥❞✐♠❡♥t♦ ❛ ♣❛rt✐r ❞❡
❛❧❣✉♠❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❛❞✐çã♦ ❡ ❞❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ C ❥á ❛♣r❡s❡♥t❛❞❛s✳ ❈♦♠ ❡❢❡✐t♦✱ s❡ z =x+yi ❡ w=a+bi ❢♦r❡♠ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ ❡♥tã♦
z+w= (x+yi) + (a+bi) = x+a+yi+bi= (x+a) + (y+b)i
❡
zw= (x+yi)(a+bi) =xa+yia+xbi+ybi2
❘❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡ z ✷✺
❉❡✜♥✐çã♦ ✶✳✻✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z = x+yi✱ ❞❡✜♥✐♠♦s ❛ ♣❛rt❡ r❡❛❧ ❡ ❛
♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ z ♣♦r
❘❡z =x ❡ ■♠z =y,
r❡s♣❡❝t✐✈❛♠❡♥t❡✳
◗✉❛♥❞♦ ❘❡z = 0, ❞✐r❡♠♦s q✉❡ z é ✐♠❛❣✐♥ár✐♦ ♣✉r♦✳
❘❡ss❛❧t❛♠♦s q✉❡✱ ♥❛ ❡①♣r❡ssã♦ ❞❡ q✉❛❧q✉❡r ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✱ s✉❛s ♣❛rt❡s r❡❛❧ ❡ ✐♠❛❣✐♥ár✐❛ sã♦ ♥ú♠❡r♦s r❡❛✐s✳
✶✳✸ ❘❡♣r❡s❡♥t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡
z
❆s ❝♦♦r❞❡♥❛❞❛s ❞❡ ✉♠ ♣♦♥t♦ ❞❡ R2✱ ♣♦r ❡①❡♠♣❧♦
(x, y)✱ ❝♦♥st✐t✉❡♠ ❛ ♣❛rt❡ r❡❛❧
❡ ❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r✿
❉❡✜♥✐çã♦ ✶✳✼✳ ❆ ✐♠❛❣❡♠ ❞♦ ❝♦♠♣❧❡①♦ z = x+yi é ♦ ♣♦♥t♦ (x, y) ❞♦ ♣❧❛♥♦ ❝❛rt❡✲
s✐❛♥♦✳ P♦❞❡♠♦s✱ t❛♠❜é♠✱ ❝❤❛♠❛r ♦ s❡❣♠❡♥t♦ ♦r✐❡♥t❛❞♦ q✉❡ ❧✐❣❛ ❛ ♦r✐❣❡♠ ❞♦ ♣❧❛♥♦ à ✐♠❛❣❡♠ ❞❡ z ❞❡ ✈❡t♦r r❡♣r❡s❡♥t❛t✐✈♦ ❞❡ss❡ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦❀ ✈❡t♦r ❡ss❡✱ −→Oz✱ ❝♦♠
❝♦♠♣♦♥❡♥t❡s x ❡ y✳
◆❡st❡ ❝♦♥t❡①t♦✱ ❝❤❛♠❛r❡♠♦s ♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❞❡ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❞❡ ❡✐①♦ r❡❛❧ ❡ ♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ❞❡ ❡✐①♦ ✐♠❛❣✐♥ár✐♦✳
✷✻ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
✶✳✹ ❈♦♥❥✉❣❛❞♦ ❡ ✈❛❧♦r ❛❜s♦❧✉t♦
❉❡✜♥✐çã♦ ✶✳✽✳ ❖ ❝♦♥❥✉❣❛❞♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦z =x+yié ♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z =x−yi✳
●r❛✜❝❛♠❡♥t❡✱ z é ♦ ♣♦♥t♦ ❞♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ♦❜t✐❞♦ ❛tr❛✈és ❞❛ r❡✢❡①ã♦ ❞❡ z ❡♠
r❡❧❛çã♦ ❛♦ ❡✐①♦ r❡❛❧✳
❈♦♥❥✉❣❛❞♦ ❡ ✈❛❧♦r ❛❜s♦❧✉t♦ ✷✼
Pr♦♣♦s✐çã♦ ✶✳✷✳ ❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s s❡ ✈❡r✐✜❝❛♠ ♣❛r❛ q✉❛✐sq✉❡r z, w ∈C✿ ✭❛✮ z =z, z±w=z±w ❡z w =z w.
✭❜✮ z+z = 2 ❘❡z ❡ z−z = 2i ■♠z.
✭❝✮ z ∈R s❡✱ ❡ s♦♠❡♥t❡ s❡✱z =z.
✭❞✮ z s❡rá ✐♠❛❣✐♥ár✐♦ ♣✉r♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ z =−z.
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠z =x+yi❡w=a+bi✳ ❊♥tã♦z=x−yi❡w=a−bi✳ P♦rt❛♥t♦✿
✭❛✮ z =x−yi=x+yi=z❀
z+w= (x+a) + (y+b)i= (x+a)−(y+b)i= (x−yi) + (a−bi) =z+w❀ z−w= (x−a) + (y−b)i= (x−a)−(y−b)i= (x−yi)−(a−bi) =z−w❀ z w = (xa−yb) + (xb−ya)i= (xa−yb)−(xb+ya)i=z w✳
✭❜✮ z+z = (x+yi) + (x−yi) = (x+x) + (y−y)i= 2x+ 0i= 2❘❡z❀ z−z = (x+yi)−(x−yi) = (x−x) + (y+y)i= 0x+ 2yi= 2i■♠z✳
✭❝✮ s❡ z ∈ R✱ ❡♥tã♦ ■♠z = 0✱ ♦✉ s❡❥❛✱ y = 0 ❡ z = x+ 0i ❡✱ ♣♦rt❛♥t♦✱ z = x =
x−0i=z✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡z =z✱ ♦✉ s❡❥❛✱ s❡x+yi=x−yi✱ ❡♥tã♦ 2yi= 0✱
❞♦♥❞❡ y= 0 ❡ z =x∈R✳
✭❞✮ s❡ z ❢♦r ✐♠❛❣✐♥ár✐♦ ♣✉r♦✱ ❡♥tã♦ ❘❡z = 0✱ ♦✉ s❡❥❛✱x= 0 ❡z = 0 +yi ❡✱ ♣♦rt❛♥t♦✱ z =yi=−(−yi) = −z✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡z =−z✱ ♦✉ s❡❥❛✱ s❡x+yi=−(x−yi)✱
❡♥tã♦ 2x= 0✱ ❞♦♥❞❡x= 0 ❡ z =yi✳
❆tr❛✈és ❞❛ ♥♦çã♦ ❞❡ ❝♦♥❥✉❣❛❞♦✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r ❛ ❡①♣r❡ssã♦ ❞♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z =x+yi6= 0 ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
z−1
=
1
x+yi
=
1
x+yi
x+yi x+yi
=
1
x+yi
x−yi x−yi
= x−yi
x2+y2
= x
x2+y2 +
−y x2+y2i.
❉❡✜♥✐çã♦ ✶✳✾✳ ❉❡✜♥✐r❡♠♦s ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ✭♦✉ ♠ó❞✉❧♦✮ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦
z =x+yi ♣♦r
✷✽ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
●r❛✜❝❛♠❡♥t❡✱ ♦ ♥ú♠❡r♦ r❡❛❧ |z| ♥♦s ❞á ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡−→Oz.
❆❧é♠ ❞✐ss♦✱ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ |z−w| é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛s ✐♠❛❣❡♥s ❞❡ z ❡ w✳
❆ ✜❣✉r❛ ❛ s❡❣✉✐r ♥♦s ♠♦str❛ ❛ ✐♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞❛ s♦♠❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s z = x+yi ❡ w = a +bi✳ ❖ ✈❡t♦r s♦❜r❡ ❛ ❞✐❛❣♦♥❛❧ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦
r❡♣r❡s❡♥t❛ ❛ s♦♠❛ z+w✳
❋✐❣✉r❛ ✶✳✸✿ ❙♦♠❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
Pr♦♣♦s✐çã♦ ✶✳✸✳ ❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s s❡ ✈❡r✐✜❝❛♠ ♣❛r❛ q✉❛✐sq✉❡r z, w∈C:
✭❛✮ ❘❡z ≤ |❘❡z| ≤ |z| ❡ ■♠z ≤ |■♠z| ≤ |z|.
✭❜✮ |z|2
=zz, |z|=|z| ❡|zw|=|z||w|.
✭❝✮ |z+w| ≤ |z|+|w|.
✭❞✮ |z+w| ≥ ||z| − |w||.
❆ ❞❡s✐❣✉❛❧❞❛❞❡ (c) é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✳
❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❙❡ z = x+yi✱ ❡♥tã♦ |z| = px2+y2
≥ √x2 =
|❘❡z| ≥ ❘❡z✳
❆♥❛❧♦❣❛♠❡♥t❡✱ |z|=px2+y2
≥py2 =
|■♠z| ≥■♠z✳
✭❜✮ ❙❡z =x+yi✱ ❡♥tã♦
|z|2
= (px2 +y2)2
=x2
+y2
. ✭✶✳✶✶✮
P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦z =x−yi✱ t❡♠♦s
zz = (x+yi)(x−yi) = x2
+y2
. ✭✶✳✶✷✮
P♦r ✭✶✳✶✶✮ ❡ ✭✶✳✶✷✮✱ ❝♦♥❝❧✉í♠♦s q✉❡|z|2
=zz✳
❖❜s❡r✈❡✱ t❛♠❜é♠✱ q✉❡ |z|=px2 + (
−y)2 =px2+y2 =
❈♦♥❥✉❣❛❞♦ ❡ ✈❛❧♦r ❛❜s♦❧✉t♦ ✷✾
❆❣♦r❛✱ s❡ w=a+bi✱ ❡♥tã♦ zw = (xa−yb) + (xb+ya)i✳ P♦rt❛♥t♦✱
|zw| = p(xa−yb)2+ (xb+ya)2
= px2a2
−2xayb+y2b2+x2b2+ 2xbya+y2a2
= px2a2+y2b2+x2b2+y2a2
= px2(a2 +b2) +y2(a2+b2)
= p(x2+y2)(a2+b2)
= p(x2+y2)p(a2+b2)
= |z||w|.
✭❝✮ ❆✜r♠❛♠♦s q✉❡✿
|z+w|2
=|z|2
+ 2❘❡zw+|w|2
.
❈♦♠ ❡❢❡✐t♦✱
|z+w|2
= (z+w)(z+w) = (z+w)(z+w) =zz+zw+wz+ww.
❱❡❥❛ q✉❡
wz =w z=w z =z w,
♣♦rt❛♥t♦✱
|z+w|2 =|z|2+ (zw+z w) +|w|2 =|z|2+ 2❘❡zw+|w|2.
❈♦♠♦
|z|2
+ 2❘❡ (zw) +|w|2
≤ |z|2
+ 2|zw|+|w|2
=|z|2
+ 2|z||w|+|w|2
= (|z|+|w|)2
=|z|2
+ 2|z||w|+|w|2
,
s❡❣✉❡ q✉❡
|z+w|2
≤(|z|+|w|)2
.
❊①tr❛✐♥❞♦ ❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡s❡❥❛❞❛✳
✭❞✮ ❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ♦❜t✐❞❛ ♥♦ ✐t❡♠(c)✱ t❡♠♦s
|z|=|(z+w)−w| ≤ |z+w|+| −w|=|z+w|+|w|, ❞♦♥❞❡ |z+w| ≥ |z| − |w|.
❚r♦❝❛♥❞♦ ♦s ♣❛♣é✐s ❞❡ z ❡ w ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ♦❜t❡♠♦s
|z+w| ≥ |w| − |z|.
❈♦♠♦||z| − |w||=|z| − |w|s❡|z| ≥ |w|❡||z| − |w||=|w| − |z|s❡|w| ≥ |z|,✈❡♠♦s
q✉❡✱ ❡♠ q✉❛❧q✉❡r ❝❛s♦✱
✸✵ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❙❡ z 6= 0✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❛ ♣r✐♠❡✐r❛ ✐❣✉❛❧❞❛❞❡ ❞♦ ✐t❡♠ (b)❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
|z|2
= z
z−1, ♦✉ s❡❥❛ ,
z−1
= z
|z|2.
❊ss❛ ✐❞❡♥t✐❞❛❞❡ ♠♦str❛ q✉❡✱ r❡♣r❡s❡♥t❛♥❞♦ ❣r❛✜❝❛♠❡♥t❡ z✱z−1 ❛♣♦♥t❛ ♥❛ ❞✐r❡çã♦ ❞❡
z
❡ t❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦1/|z|✳
❋✐❣✉r❛ ✶✳✹✿ ■♠❛❣❡♠ ❞♦ ✐♥✈❡rs♦ ❞❡ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦
✶✳✺ ❚r✐❣♦♥♦♠❡tr✐❛ ❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✿❢♦r♠❛ ♣♦❧❛r
❘❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡❧❡♠❡♥t❛r❡s ♥♦s ♣❡r♠✐t❡♠ ❡st❛❜❡❧❡❝❡r ✉♠❛ ♥♦✈❛ r❡♣r❡s❡♥✲ t❛çã♦ ♣❛r❛ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦z✳
❉❡✜♥✐çã♦ ✶✳✶✵✳ ❆ r❡♣r❡s❡♥t❛çã♦ tr✐❣♦♥♦♠étr✐❝❛ ❞❡ z é✿ z =|z|(cosθ+is❡♥θ).
❙❡ θ ∈R s❛t✐s✜③❡r ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ ❞✐r❡♠♦s q✉❡ θ é ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ z✳
❈♦♥s✐❞❡r❡♠♦s ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ z = x+yi 6= 0✳ ❙❡❥❛ α ♦ â♥❣✉❧♦ q✉❡ ♦ ❡✐①♦
r❡❛❧ ♣♦s✐t✐✈♦ ❢♦r♠❛ ❝♦♠ ♦ ✈❡t♦r ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ z ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✳
❈♦♠♦ cosα= x
|z| ❡ s❡♥α = y
|z|, t❡♠♦s✿
❚r✐❣♦♥♦♠❡tr✐❛ ❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✿❢♦r♠❛ ♣♦❧❛r ✸✶
❋✐❣✉r❛ ✶✳✺✿ ❋♦r♠❛ ♣♦❧❛r ❞❡ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦
❆ss✐♠✱ é s❡♠♣r❡ ♣♦ssí✈❡❧ ♣❛ss❛r ❞❛ r❡♣r❡s❡♥t❛çã♦ ❛❧❣é❜r✐❝❛ ♣❛r❛ ❛ ♣♦❧❛r✱ ❡ ✈✐❝❡✲ ✈❡rs❛✳
P❡❧❛ ❞❡✜♥✐çã♦✱ α é ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ z✳ ❊♥tr❡t❛♥t♦✱ q✉❛❧q✉❡r α ❞❛ ❢♦r♠❛ α + 2kπ, ❝♦♠ k ∈Z,t❛♠❜é♠ s❛t✐s❢❛③ ❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ ✉♠❛ q✉❡ ✈❡③ q✉❡ ❛s ❢✉♥çõ❡s s❡♥♦
❡ ❝♦ss❡♥♦ sã♦ ✷ ♣❡r✐ó❞✐❝❛s ❝♦♠ ♣❡rí♦❞♦2π✳
❈♦♥❝❧✉í♠♦s✱ ❡♥tã♦✱ q✉❡ z ♣♦ss✉✐ ✐♥✜♥✐t♦s ❛r❣✉♠❡♥t♦s✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ θ ❢♦r
t❛❧ q✉❡ z = |z|(cosθ+is❡♥θ), ❡♥tã♦ cosθ = cosα ❡ s❡♥θ = s❡♥α, ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ θ =α+ 2kπ ♣❛r❛ ❛❧❣✉♠ k ∈Z✳
❆ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ argz ❞❡ t♦❞♦s ♦s ❛r❣✉♠❡♥t♦s ❞❡ z é ❞❛❞♦ ♣♦r✿
argz ={α+ 2kπ:k ∈Z}.
❖ ú♥✐❝♦ ❛r❣✉♠❡♥t♦ ❞❡ z q✉❡ ♣❡rt❡♥❝❡ ❛♦ ✐♥t❡r✈❛❧♦ (−π, π] é ❞❡♥♦♠✐♥❛❞♦ ❛r❣✉✲
♠❡♥t♦ ♣r✐♥❝✐♣❛❧ ❞❡ z ❡ é ❞❡♥♦t❛❞♦ ♣♦r ❆r❣z.
❉❡✜♥✐çã♦ ✶✳✶✶✳ ❆ ✐❞❡♥t✐❞❛❞❡
z =|z|[cos(❆r❣z) +is❡♥(❆r❣z)]
é ❞❡♥♦♠✐♥❛❞❛ ❛ ❢♦r♠❛ ♣♦❧❛r ❞❡ z✳
❆❣♦r❛✱ s❡❥❛♠ z1 = |z1|(cosθ1 +is❡♥θ1)❡ z2 = |z2|(cosθ2 +is❡♥θ2) ❞♦✐s ♥ú♠❡r♦s
✷▲❡♠❜r❛♠♦s q✉❡✿ ✉♠❛ ❢✉♥çã♦ f :R→Ré ♣❡r✐ó❞✐❝❛ ❝♦♠ ♣❡rí♦❞♦ τ s❡f(t+τ) =f(t)✱ ♣❛r❛ t♦❞♦
✸✷ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❝♦♠♣❧❡①♦s ♥ã♦ ♥✉❧♦s✳ ❱❛♠♦s ♦❜t❡r ❛ r❡♣r❡s❡♥t❛çã♦ ♣♦❧❛r ♣❛r❛z1z2✳ ❱❡❥❛ q✉❡✿
z1z2 = |z1|(cosθ1+is❡♥θ1)|z2|(cosθ2+is❡♥θ2)
= |z1||z2|(cosθ1+is❡♥θ1)(cosθ2+is❡♥θ2)
= |z1||z2|[cosθ1(cosθ2+is❡♥θ2) +is❡♥θ1(cosθ2+is❡♥θ2)]
= |z1||z2|[cosθ1cosθ2+ cosθ1is❡♥θ2+is❡♥θ1cosθ2+is❡♥θ1is❡♥θ2]
= |z1||z2|[(cosθ1cosθ2−s❡♥θ1s❡♥θ2) +i(cos θ1s❡♥θ2+s❡♥θ1cosθ2)], ❞❡ ♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡
z1z2 =✸|z1||z2|[cos(θ1+θ2) +is❡♥(θ1+θ2)].
❊st❛ ✐❣✉❛❧❞❛❞❡ ♥♦s ❞á ❛ ✐♥t❡r♣r❡t❛çã♦ ❣rá✜❝❛ ❞♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠♣❧❡✲ ①♦s✳ ❱❡❥❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳
❋✐❣✉r❛ ✶✳✻✿ Pr♦❞✉t♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
❋ór♠✉❧❛s ❞❡ ▼♦✐✈r❡ ✸✸
❊♥tã♦✱ z1z2 t❡♠ ✈❛❧♦r ❛❜s♦❧✉t♦ |z1||z2|❡ t❡♠ θ1+θ2 ❝♦♠♦ ❛r❣✉♠❡♥t♦✳
❙❡❥❛z =|z|[cos(θ) +is❡♥(θ)] ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ ♥ã♦ ♥✉❧♦✳ ❱❛♠♦s✱ ❛❣♦r❛✱ ♦❜t❡r
❛ r❡♣r❡s❡♥t❛çã♦ ♣♦❧❛r ♣❛r❛ z−1 ✳ ❙❛❜❡♠♦s q✉❡
cos(−θ) = cos(θ) ❡ s❡♥(−θ) =−s❡♥(θ)✳
P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡ z =|z|[cos(−θ) +is❡♥(−θ)]✳
❚❛♠❜é♠ ❥á ✈✐♠♦s q✉❡z−1
= z
|z|2; ❡♥tã♦✿ z
−1
= |z|[cos(−θ) +is❡♥(−θ)]
|z|2 .
❙❡♥❞♦ ❛ss✐♠✱ ❛ r❡♣r❡s❡♥t❛çã♦ ♣♦❧❛r ❞❡z−1 é✿
z−1
=|z|−1
[cos(−θ) +is❡♥(−θ)].
❆ ♣❛rt✐r ❞❛s r❡♣r❡s❡♥t❛çõ❡s ♣♦❧❛r❡s ❡♥❝♦♥tr❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ❝♦♥❝❧✉í♠♦s q✉❡
z1z− 1
2 =|z1| |z− 1
2 |[cos(θ1+ (−θ2)) +s❡♥(θ1+ (−θ2))], ♦ q✉❡ ♥♦s ❞á ❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛ ❞❡ ❞✐✈✐sã♦✿
z1
z2
= |z1|
|z2|
[cos(θ1−θ2) +s❡♥(θ1−θ2)].
❋✐❣✉r❛ ✶✳✼✿ ◗✉♦❝✐❡♥t❡ ❞❡ ❞♦✐s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s
✶✳✻ ❋ór♠✉❧❛s ❞❡ ▼♦✐✈r❡
❙❡❥❛z ∈C t❛❧ q✉❡ |z|=ρ6= 0✳ P❛r❛n ∈Z✱ ✈❛❧❡✿
✸✹ ◆ú♠❡r♦s ❈♦♠♣❧❡①♦s
❋✐❣✉r❛ ✶✳✽✿ P♦tê♥❝✐❛s ❞❡ z,|z|= 1✳
❊①tr❛çã♦ ❞❡ r❛í③❡s ✸✺
◆❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❋ór♠✉❧❛ ❞❡ ▼♦✐✈r❡✱ ✉t✐❧✐③❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛✲ t❡♠át✐❝❛✳ ❙❡♥❞♦ ❛ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛ ✉♠ ♠ét♦❞♦ ❛✐♥❞❛ r❡str✐t♦ ❛♦ ❡♥s✐♥♦ s✉♣❡r✐♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ✐♥❢♦r♠❛♠♦s q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ♣r♦✈❛ ❞❡st❛ ❢ór♠✉❧❛ ❛♣❡♥❛s ♣❛r❛ ❛✉①✐❧✐❛r ♦ ♣r♦❢❡ss♦r ✲ ❧❡✐t♦r ✲ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❥á ❢❛♠✐❧✐❛r✐③❛❞♦ ❝♦♠ t❛❧ ♠ét♦❞♦✱ ♥❛ r❡✈✐sã♦ ❞❛ t❡♦r✐❛ ❞❡ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ P❛r❛ ♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ✭r❡✮✈❡r ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✱ ✐♥❞✐❝❛♠♦s ❛ r❡❢❡rê♥❝✐❛ ❬✻❪✱ ❈❛♣ít✉❧♦ ■■■✳
❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♣r♦✈❛r q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ✭✶✳✶✸✮ é ✈á❧✐❞❛ ♣❛r❛n ∈N✱ ✉s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✳
✭■✮ ❙❡ n= 0✱z0
= 1 = ρ0
(cos 0 +is❡♥ 0)✳
✭■■✮ ❆❞♠✐t❛♠♦s q✉❡ ❛ ❢ór♠✉❧❛ s❡❥❛ ✈á❧✐❞❛ ♣❛r❛n=k✿ zk=ρk[cos(k.θ)+is❡♥(k.θ)]✳
✭■■■✮ ❆❣♦r❛ ✈❛♠♦s ♣r♦✈❛r q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ n = k+ 1✳ ❈♦♠ ❡❢❡✐t♦✱
✉s❛♥❞♦ ♦ ✐t❡♠ ❛♥t❡r✐♦r✱ t❡♠♦s✿
zk.z = [ρk(cos(k.θ) +is❡♥(k.θ)].[ρ(cos(θ) +is❡♥(θ)]
= (ρk.ρ)[cos(k.θ).cos(θ) +i2 s❡♥(k.θ)s❡♥(θ)] +i[(cos(k.θ).s❡♥(θ) +s❡♥(k.θ) cos(θ)] = (ρk.ρ)[cos(kθ+θ)] +i[s❡♥(kθ+θ)]
= ρk+1
[cos((k+ 1)θ) +is❡♥((k+ 1)θ)] =zk+1
.
❋✐♥❛❧♠❡♥t❡✱ ✈❛♠♦s ❝♦♥st❛t❛r q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ✭✶✳✶✸✮ é ✈á❧✐❞❛ ♣❛r❛ n∈Z✳
❈♦♥s✐❞❡r❛♥❞♦ ♦ ❝❛s♦ ❞❡ ✐♥t❡r❡ss❡ n < 0✱ ♣♦❞❡♠♦s t♦♠❛r m = −n✳ ❊♥tã♦✱ zn =
z−m = 1
zm✳
❈♦♠♦ −n = m ∈ N✱ ♣❛r❛ m✱ ❛ ✐❣✉❛❧❞❛❞❡ ✭✶✳✶✸✮ é ✈❡r❞❛❞❡✐r❛✱ ♣❡❧♦ ♦ q✉❡ ❢♦✐ ✈✐st♦
❛❝✐♠❛✱ ♥♦s ❧❡✈❛♥❞♦ ❛✿
zn = 1
ρm[(cos(m.θ) +is❡♥(m.θ)]
= 1
ρm
[cos(m.θ)−is❡♥(m.θ)]
[cos(m.θ) +is❡♥(m.θ)][cos(m.θ)−is❡♥(m.θ)] = 1
ρm
[cos(m.θ)−is❡♥(m.θ)] cos2(m.θ) +s❡♥2(m.θ)
= ρ−m[cos(m.θ)
−is❡♥(m.θ)] ✶ = ρ−m[cos((−m).θ) +is❡♥((−m).θ)] = ρn[cos(nθ) +is❡♥(n.θ)].
✶✳✼ ❊①tr❛çã♦ ❞❡ r❛í③❡s
❉❛❞♦s ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ w ❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n ≥ 1✱ ❞✐r❡♠♦s q✉❡ z ∈ C é ✉♠❛ r❛✐③ ♥✲és✐♠❛ ❞❡ w s❡
