Open As Funções Hiperbólicas e suas Aplicações

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❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❡ s✉❛s

❆♣❧✐❝❛çõ❡s

♣♦r

▼❛r✐❛ ❞♦ ❇♦♠ ❈♦♥s❡❧❤♦ ❞❛ ❙✐❧✈❛ ❇❡s❡rr❛ ❋r❡✐t❛s

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❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❡ s✉❛s

❆♣❧✐❝❛çõ❡s

†

♣♦r

▼❛r✐❛ ❞♦ ❇♦♠ ❈♦♥s❡❧❤♦ ❞❛ ❙✐❧✈❛ ❇❡s❡rr❛ ❋r❡✐t❛s

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ r❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚ ✲ ❉▼ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

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F866f Freitas, Maria do Bom Conselho da Silva Beserra.

As funções hiperbólicas e suas aplicações / Maria do Bom Conselho da Silva Beserra Freitas.-- João Pessoa, 2015. 60f. : il.

Orientador: Bruno Henrique Carvalho Ribeiro

Dissertação (Mestrado) – UFPB/CCEN

1. Matemática. 2. Funções hiperbólicas. 3. Ângulos hiperbólicos. 4. Catenária.

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❉❡❞✐❝❛tór✐❛

❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣♦r s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♠✳ ❊♠ ❡s♣❡❝✐❛❧ ♠❡✉ ❡s♣♦s♦ ❘♦❜ér✐♦✱ s❡✉ ❝❛r✐♥❤♦✱ ❛♣♦✐♦ ❡ ❞❡❞✐❝❛çã♦ ❢♦✐ q✉❡♠ ❞❡✉✱ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s✱ ❢♦rç❛ ♣❛r❛ ❝♦♥t✐♥✉❛r❀ s✉❛ ♣r❡s❡♥ç❛ ❡ s❡✉ ❛♠♦r ❝♦♥st❛♥t❡ ♣❡r♠✐t✐r❛♠ q✉❡ ♥♦ss❛s ✜❧❤❛s ♥ã♦ s❡♥t✐ss❡♠ t❛♥t♦ ♠✐♥❤❛ ❛✉sê♥❝✐❛✳ ❊ss❛ ✈✐tór✐❛ ♥ã♦ é ♠✐♥❤❛✱ é ♥♦ss❛✦

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣♦r t♦❞❛s ❛s ✈✐tór✐❛s ❝♦♥❝❡❞✐❞❛s ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ♣❡❧❛ ❢♦rç❛ q✉❡ ♠❡ ❞❡✉ ♣❛r❛ ✈❡♥❝❡r t♦❞♦s ♦s ❞❡s❛✜♦s ♥❡ss❡s ❞♦✐s ❛♥♦s ❞❡ ❝✉rs♦✳

❆♦ ♠❡✉ ❡s♣♦s♦ ❘♦❜ér✐♦✱ ♣♦r t♦❞❛ ❝♦♠♣r❡❡♥sã♦✱ ❞❡❞✐❝❛çã♦ ❡ ✐♥❝❡♥t✐✈♦✳ ❆ ♠✐♥❤❛ ♠ã❡ ■r❡♥✐❝❡✱ ♣♦r t♦❞❛ ♦r❛çã♦✱ s❡♠♣r❡ ♣❡❞✐♥❞♦ ♣❡❧♦ ♠❡✉ s✉❝❡ss♦✳ ❆s ♠✐♥❤❛s ✜❧❤❛s ❘✐❛♥❡✱ ❘❡♥❛❧❡ ❡ ❘❛❢❛❡❧❛✱ q✉❡ ❛♣❡s❛r ❞❛ ♣♦✉❝❛ ✐❞❛❞❡ ♠❡ ❛♣♦✐❛r❛♠ ❡ ❝♦♠♣r❡❡♥❞❡r❛♠ ♠✐♥❤❛ ❛✉sê♥❝✐❛✱ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ❞❡ s✉❛s ✈✐❞❛s✱ ♥❡ss❡s ❞♦✐s ❛♥♦s✳

❆♦ ♠❡✉ ♣r♦❢❡ss♦r ♦r✐❡♥t❛❞♦r ❉r✳ ❇r✉♥♦ ❘✐❜❡✐r♦✱ ♣❡❧♦ s✉♣♦rt❡ q✉❡ ♠❡ ❞❡✉ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

❆s ♠✐♥❤❛s ❛♠✐❣❛s ●✐③❡❧❡ ▼❛rt✐♥s✱ q✉❡ ❛❧é♠ ❞❡ ♠❡ ❛❝♦❧❤❡r ❡♠ s✉❛ ❝❛s❛✱ ❛❥✉❞♦✉ ♠✉✐t♦ ♥❛s ❧♦♥❣❛s ♠❛❞r✉❣❛❞❛s q✉❡ ❡st✉❞❛♠♦s ❥✉♥t❛s✱ ❙♦❝♦rr♦ ❙♦✉s❛✱ ❆♥tô♥✐❛ ❡ ❙♦❝♦rr♦ ❙❛♥t♦s✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✱ ❛♣♦✐♦ ❡ ❛❥✉❞❛ s❡♠♣r❡ q✉❡ ♣r❡❝✐s❡✐✳

❆ ❡st❛ ✉♥✐✈❡rs✐❞❛❞❡✱ s❡✉ ❝♦r♣♦ ❞♦❝❡♥t❡✱ ❞✐r❡çã♦ ❡ ❛❞♠✐♥✐str❛çã♦ q✉❡ ♦♣♦rt✉♥✐③❛r❛♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ ❝✉rs♦✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡ss❡ ❝✉rs♦✱ ♣♦r ♠❡ ♣r♦♣♦r❝✐♦♥❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✳

❆ ❈❛♣❡s✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✱ s❡♠ ♦ q✉❛❧ ♥ã♦ t❡r✐❛ ❝♦♥❞✐çõ❡s ❞❡ r❡❛❧✐③❛r ♦ P❘❖❋▼❆❚✳

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❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

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❚❛❧✈❡③ ♥ã♦ t❡♥❤❛ ❝♦♥s❡❣✉✐❞♦ ❢❛③❡r ♦ ♠❡❧❤♦r✱ ♠❛s ❧✉t❡✐ ♣❛r❛ q✉❡ ♦ ♠❡❧❤♦r

❢♦ss❡ ❢❡✐t♦✳ ◆ã♦ s♦✉ ♦ q✉❡ ❞❡✈❡r✐❛ s❡r✱ ♠❛s ●r❛ç❛s ❛ ❉❡✉s✱ ♥ã♦ s♦✉ ♦

q✉❡ ❡r❛ ❛♥t❡s✳

✭▼❛rt❤✐♥ ▲✉t❤❡r ❑✐♥❣✮✳

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❘❊❙❯▼❖

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❡st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❡ s✉❛s ❛♣❧✐❝❛çõ❡s✳ ■♥✐❝✐❛♠♦s ❝♦♠ ✉♠❛ ❛♥á❧✐s❡ ❞❡ ❝♦♠♦ ❡ss❛s ❢✉♥çõ❡s sã♦ ❛❜♦r❞❛❞❛s ❡♠ ❛❧❣✉♥s ❧✐✈r♦s ❞❡ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❝♦♠✉♠❡♥t❡ ✉s❛❞♦s ♥♦s ❝✉rs♦s ❞❡ ❣r❛❞✉❛çã♦ ♥❛ ár❡❛ ❞❡ ❡①❛t❛s✱ ❝♦♥st❛t❛♥❞♦ q✉❡ sã♦ ❢❡✐t❛s ❛tr❛✈és ❞❡ s✉❛ ❞❡✜♥✐çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❊♠ s❡❣✉✐❞❛ ❡①♣✉s❡♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ ✉t✐❧✐③❛♥❞♦✲s❡ ❞❛ ❤✐♣ér❜♦❧❡ ❝♦♠♦ ❝✉r✈❛ ❣❡r❛tr✐③ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦ ❞❡ â♥❣✉❧♦s ❤✐♣❡r❜ó❧✐❝♦s✳ ❆s ❞❡✜♥✐çõ❡s s❡ ❞❡r❛♠ ♣❛r❛❧❡❧❛♠❡♥t❡ à ❝♦♥str✉çã♦ ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✐r❝✉❧❛r❡s✱ ❛♥❛❧✐s❛♥❞♦ s✉❛s s❡♠❡❧❤❛♥ç❛s ❡ ❞✐❢❡r❡♥ç❛s✳ P♦r ✜♠ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s✱ ❡♠ ❡s♣❡❝✐❛❧ ❡ ❞❡ ❢♦r♠❛ ♠❛✐s ❞❡t❛❧❤❛❞❛ ❛ ❝❛t❡♥ár✐❛✳

P❛❧✈r❛s✲❝❤❛✈❡✿ ❋✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ ➶♥❣✉❧♦s ❤✐♣❡r❜ó❧✐❝♦s✱ ❈❛t❡♥ár✐❛✳

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❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t ❛ st✉❞② ❛❜♦✉t t❤❡ ❍②♣❡r❜♦❧✐❝ ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✳ ❲❡ st❛rt ✇✐t❤ ❛♥❛❧②s✐s ♦❢ ❤♦✇ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ ❛♣♣r♦❛❝❤❡❞ ✐♥ s♦♠❡ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❜♦♦❦s ❝♦♠♠♦♥❧② ✉s❡❞ ✐♥ ❣r❛❞✉❛t❡ ❝♦✉rs❡s ✐♥ ❡①❛❝t s❝✐❡♥❝❡s✱ ♥♦t✐♥❣ t❤❛t ❛r❡ ♠❛❞❡ t❤r♦✉❣❤ ✐ts ❡①♣♦♥❡♥t✐❛❧ s❡tt✐♥❣✳ ❚❤❡♥ ✇❡ ❡①♣♦s❡❞ ❛♥ ❛♣♣r♦❛❝❤ ✉s✐♥❣ ❤②♣❡r❜♦❧❡ ❛s ❣❡♥❡r❛t✐♥❣ ❝✉r✈❡ ❢r♦♠ t❤❡ st✉❞② ♦❢ ❤②♣❡r❜♦❧✐❝ ❛♥❣❧❡s✳ ❚❤❡ ❞❡✜♥✐t✐♦♥s ❣✐✈❡♥ ✐t ✐♥ ♣❛r❛❧❧❡❧ ✇✐t❤ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❝✐r❝✉❧❛r tr✐❣♦♥♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✱ ❛♥❛❧②③✐♥❣ t❤❡✐r s✐♠✐❧❛r✐t✐❡s ❛♥❞ ❞✐✛❡r❡♥❝❡s✳ ❋✐♥❛❧❧② ✇❡ ♣r❡s❡♥t s♦♠❡ ♦❢ ✐ts ❛♣♣❧✐❝❛t✐♦♥s✱ ✐♥ ♣❛rt✐❝✉❧❛r ❛♥❞ ✐♥ ♠♦r❡ ❞❡t❛✐❧ t❤❡ ❝❛t❡♥❛r② s❤❛♣❡✳

❑❡②✇♦r❞s✿ ❍②♣❡r❜♦❧✐❝ ❢✉♥❝t✐♦♥s✱ ❍②♣❡r❜♦❧✐❝ ❛♥❣❧❡s✱ ❝❛t❡♥❛r②✳

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❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✸

✶✳✶ ❘❡s✉♠♦ ❞♦ ❝♦♥t❡ú❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ▲✐✈r♦ ❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ▲✐✈r♦ ❇ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✸ ▲✐✈r♦ ❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✹ ▲✐✈r♦ ❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✶✳✺ ▲✐✈r♦ ❊ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❆♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✶

✷✳✶ ❆ ❤✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✶ ❆ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❤✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✷ ❆ssí♥t♦t❛s ❞❛ ❤✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ➶♥❣✉❧♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✶ ●rá✜❝♦s ❞♦ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✸ ❆♣❧✐❝❛çõ❡s ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✸✸

✸✳✶ ❆ ❈❛t❡♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

(12)

✸✳✶✳✶ ❖ ❙t✳ ▲♦✉✐s ●❛t❡✇❛② ❆r❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✶✳✷ ▲✐♥❤❛s ❞❡ ❚r❛♥s♠✐ssã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✽

(13)

■♥tr♦❞✉çã♦

❆ ❛❜♦r❞❛❣❡♠ ❞❡ ✉♠ ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦ ❞❡✈❡ ❛♣♦✐❛✲s❡ ♥❛ ❝♦♥❝❡✐t✉❛çã♦✱ ♠❛♥✐♣✉❧❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s✱ ♦✉ s❡❥❛✱ ❞❡✜♥✐r ♦ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦ ❡♠ ❡st✉❞♦✱ ❡①❡r❝✐t❛r ♦ q✉❡ ❢♦✐ ❞❡✜♥✐❞♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♥❝♦♥tr❛r ❛♣❧✐❝❛çõ❡s r❡❛✐s q✉❡ ❡st✐♠✉❧❡♠ ❡ ❥✉st✐✜q✉❡♠ ♦ ♣♦rq✉ê ❞❡ s❡ ❡st✉❞❛r t❛❧ ❝♦♥t❡ú❞♦✳

❆♦ ❧❡r♠♦s ♦ t❡①t♦ ❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✭r❡❢❡rê♥❝✐❛❬✺❪✮ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❛ ❢r❛s❡ ✧❛❝❤❡✐ ♠✉✐t♦ ❡♥❣r❛ç❛❞♦ ♣❡♥s❛r✲s❡ ❡♠ ❥✉♥t❛r ❞✉❛s ♣❛❧❛✈r❛s ❝♦♠♦ s❡♥♦ ❡ ❤✐♣ér❜♦❧❡ ♣❛r❛ ♥♦♠❡❛r ✉♠

ex

−e−x

2 ”

✭❙ô♥✐❛✱ ✶✾✽✽✮✳

P❡r❝❡❜❡♠♦s q✉❡ ♥ã♦ s❡ tr❛t❛ ❞❡ ✉♠ s❡♥t✐♠❡♥t♦ ✐s♦❧❛❞♦ ❡ q✉❡ ❛♣❡s❛r ❞❡ t❛♥t♦ t❡♠♣♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❛❝♦♥t❡❝❡ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ ♥♦s ❝✉rs♦s ❞❡ ▼❛t❡♠át✐❝❛ ♥❛s ❯♥✐✈❡rs✐❞❛❞❡s✳ ❊✱ ♦ ❞❡s❡❥♦ ❞❡ ❛♣r♦❢✉♥❞❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ss❡ ❝♦♥t❡ú❞♦✱ t♦r♥❛♥❞♦✲♦ ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦ ❢♦✐ ♦ ♠♦t✐✈♦ ❞❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦str❛♠♦s q✉❡ ❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s sã♦ ❛❜♦r❞❛❞❛s ♥♦s ❧✐✈r♦s ❞❡ ❈á❧❝✉❧♦ ■ ❞❡ ❢♦r♠❛ ✐❣✉❛❧✐tár✐❛ ❛ ♣❛rt✐r ❞❛s ❡①♣♦♥❡♥❝✐❛✐sex _❡_e−x_{✱ ❡ss❛ ❛❜♦r❞❛❣❡♠} ♥ã♦ ♣r♦♣♦r❝✐♦♥❛ ✉♠❛ ❝♦♠♣r❡❡♥sã♦ ❝❧❛r❛ ❞❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ❛s ❞❡✜♥✐çõ❡s ❞❛s ❢✉♥çõ❡s ❝♦♠ ❛ ❤✐♣ér❜♦❧❡✳

❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s s✉r❣✐r❛♠ ❞❛ ❝♦♠♣❛r❛çã♦ ❞❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣♦r ✉♠❛ ❤✐♣ér❜♦❧❡✱ ❞❛♥❞♦ ♦r✐❣❡♠ ❛s ❞❡✜♥✐çõ❡s ❡ ✐❞❡♥t✐❞❛❞❡s✳ ❖ ♠❛t❡♠át✐❝♦ s✉íç♦ ❏♦❤❛♥♥ ❍❡✐♥r✐❝❤ ▲❛♠❜❡rt ✭✶✼✷✽✲✶✼✼✼✮ ❢♦✐ ❛ ♣r✐♠❡✐r❛ ♣❡ss♦❛ ❛ ❡st✉❞❛r ❡ss❛s ❢✉♥çõ❡s✳

(14)

❱✐s❛♥❞♦ ❛♠♣❧✐❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ❝♦♥❝❡✐t✉❛çã♦ ❞❡ss❛s ❢✉♥çõ❡s✱ ✈❛♠♦s ❢❛③❡r ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛ ♣❛rt✐r ❞❛ ❤✐♣ér❜♦❧❡ ❝♦♠♦ ❝✉r✈❛ ❣❡r❛tr✐③✱ ❞❡✜♥✐♥❞♦ â♥❣✉❧♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ ❢❛③❡♥❞♦ ✉♠ ♣❛r❛❧❡❧♦ ❝♦♠ ❛s ❞❡✜♥✐çõ❡s ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✐r❝✉❧❛r❡s ♣❛r❛ ❞❡✜♥✐r ❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✳

P♦r ✜♠✱ ❞❡s❡❥❛♠♦s ✐♥st✐❣❛r ♦ ❡st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✱ ♠♦str❛♥❞♦ ❛❧❣✉♠❛s ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s✳ ❈♦♠♦ ❡ss❛s ❢✉♥çõ❡s s✉r❣❡♠ ❢♦rt❡♠❡♥t❡ ♥❛s ár❡❛s ❞❡ ❡♥❣❡♥❤❛r✐❛ ❡ ❛rq✉✐t❡t✉r❛✱ ♣♦✐s✱ tr❛③❡♠ ❝♦♥s✐❣♦ ♦ ❡st✉❞♦ ❞❛ ❝❛t❡♥ár✐❛✱ q✉❡ ❞❡s❝r❡✈❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s ♣❧❛♥❛s s❡♠❡❧❤❛♥t❡s às q✉❡ s❡r✐❛♠ ❣❡r❛❞❛s ♣♦r ✉♠❛ ❝♦r❞❛ s✉s♣❡♥s❛ ♣❡❧❛s s✉❛s ❡①tr❡♠✐❞❛❞❡s ❡ s✉❥❡✐t❛s ❛ ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✱ ❛ t❡♥sã♦ ✐♥t❡r♥❛ ❢♦r♠❛❞❛ ❡♥tr❡ ♦s ❞♦✐s ♣♦♥t♦s ❡①tr❡♠♦s ❞á ❝♦♥❞✐çõ❡s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✈ár✐❛s ♦❜r❛s ✐♠♣♦rt❛♥t❡s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❛ ♣♦♥t❡ ❏✉s❝❡❧✐♥♦ ❑✉❜✐ts❝❤❡❦✱ ❡♠ ❇r❛sí❧✐❛✱ ♥♦ ❇r❛s✐❧ ❡ ♦ ●❛t❡✇❛② ❆r❝❤✱ ❡♠ ❙t✳ ▲♦✉✐s✳

❈♦♠ ✐ss♦ q✉❡r❡♠♦s ❝♦♥tr✐❜✉✐r ♣❛r❛ q✉❡ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ♣♦ss❛♠ ❝♦♠♣r❡❡♥❞❡r✱ ❡❢❡t✉❛r ❡ ❝♦♥tr♦❧❛r ♦s ♣r♦❝❡ss♦s ♠❛t❡♠át✐❝♦s ❡♥✈♦❧✈✐❞♦s ♥❛ ❝♦♥❝❡✐t✉❛çã♦✱ ♠❛♥✐♣✉❧❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✳

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❈❛♣ít✉❧♦ ✶

❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s

❍✐♣❡r❜ó❧✐❝❛s ♥♦s ❧✐✈r♦s ❞❡ ❈á❧❝✉❧♦

❆ ❛❜♦r❞❛❣❡♠ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦ ❞❡✈❡ ❢✉♥❞❛♠❡♥t❛r✲s❡ ❡♠ três ❡t❛♣❛s✿ ❝♦♥❝❡✐t✉❛çã♦✱ ♠❛♥✐♣✉❧❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s✳

◆❡st❡ ❈❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❛♥á❧✐s❡ ❝rít✐❝❛ ❞❡ ❝♦♠♦ ♦ ❝♦♥t❡ú❞♦ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s é ❛❜♦r❞❛❞♦ ❡♠ ❧✐✈r♦s ❞❡ ❈á❧❝✉❧♦ ✳

❚❡♥❞♦ ❡♠ ✈✐st❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❞♦s r❡s✉❧t❛❞♦s ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦

senhx= e

x

−e−x

2

❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦

coshx = e

x₊_e−x

2 ,

✈♦❧t❛♠♦s ❛ ♥♦ss❛ ❛t❡♥çã♦ ♣❛r❛ ♦s ❧✐✈r♦s ❞❡ ❈á❧❝✉❧♦ ✱ t❡♥❞♦ ❝♦♠♦ ❢♦❝♦ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ss❡ ❝♦♥t❡ú❞♦ ♥♦s ❡①❡♠♣❧❛r❡s✳ ❋♦r❛♠ ❛♥❛❧✐s❛❞❛s s❡✐s ♦❜r❛s ❞❡♥♦♠✐♥❛❞❛s ❆✱ ❇✱ ❈✱ ❉✱ ❊ ❡ ❋ t❛❜❡❧❛✶✳✶✱ q✉❡ sã♦ ✉t✐❧✐③❛❞❛s ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♣❡❧❛s ❯♥✐✈❡rs✐❞❛❞❡s ♥♦s ❝✉rs♦s ❞❡ ▼❛t❡♠át✐❝❛✳

(16)

❈❛♣ít✉❧♦ ✶✳ ❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✳✶✳ ❘❡s✉♠♦ ❞♦ ❝♦♥t❡ú❞♦

❚❛❜❡❧❛ ✶✳✶✿ ▲✐✈r♦s ❛♥❛❧✐s❛❞♦s

▲■❱❘❖ ❖❇❘❆ ❊❉■❚❖❘❆ ❆❯❚❖❘❊❙

❆ ❈á❧❝✉❧♦s ❞❛s ❋✉♥çõ❡s ❞❡ ✉♠❛ ❱❛r✐á✈❡❧ ▲❚❈ ●❡r❛❧❞♦ ➪✈✐❧❛

❇ ❈á❧❝✉❧♦ ▲❚❈ ▼✉st❛❢❛ ❆✳ ▼✉♥❡♠ ❡

❉❛✈✐❞ ❏✳ ❋♦✉❧✐s

❈ ❈á❧❝✉❧♦ ❆❇❉❘ ●❡♦r❣❡ ❇✳ ❚❤♦♠❛s

❉ ❈á❧❝✉❧♦ ❈❊◆●❆●❊ ❏❛♠❡s ❙t❡✇❛rt

❊ ❈á❧❝✉❧♦ ❆ P❊❆❘❙❖◆ ❉✐✈❛ ▼✳ ❋❧❛♠♠✐♥❣ ❡

▼✐r✐❛♥ ❇✉ss ●♦♥ç❛❧✈❡s ❋ ❯♠ ❈✉rs♦ ❞❡ ❈á❧❝✉❧♦ ▲❚❈ ❍❛♠✐❧t♦♥ ▲✳ ●✉✐❞♦r✐③③✐

✶✳✶ ❘❡s✉♠♦ ❞♦ ❝♦♥t❡ú❞♦ ♥❛s ♦❜r❛s

✶✳✶✳✶ ▲✐✈r♦ ❆

❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s

❆s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✿ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ ❝♦✲s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ t❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛ ❡ ❝♦✲t❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✱ ❞❡s✐❣♥❛❞❛s ♣❡❧♦s sí♠❜♦❧♦ssenh✱ cosh✱ tanh❡ coth✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ ❛ss✐♠ ❞❡✜♥✐❞❛s✿

senhx= e

x₋_e−x

2 ✱ cosh=

ex₊_e−x

2 ✱

tanhx= senhx coshx =

ex₋_e−x ex₊_e−x✱

cothx= coshx senhx =

ex₊_e−x ex₋_e−x✳

✶✳✶✳✷ ▲✐✈r♦ ❇

❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s

❈❡rt❛s ❝♦♠❜✐♥❛çõ❡s ❞❡ ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s✱ q✉❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ✉♠❛ ❤✐♣ér❜♦❧❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ ❝♦♠ q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♦ ❝ír❝✉❧♦✱ ♣r♦✈❛r❛♠ s❡r ✐♠♣♦rt❛♥t❡s ❡♠ ♠❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞❛✳ ❊ss❛s ❢✉♥çõ❡s sã♦ ❝❤❛♠❛❞❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❡ s✉❛s s❡♠❡❧❤❛♥ç❛s ❝♦♠ ❛s ❢✉♥çõ❡s

(17)

❈❛♣ít✉❧♦ ✶✳ ❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✳✶✳ ❘❡s✉♠♦ ❞♦ ❝♦♥t❡ú❞♦

tr✐❣♦♥♦♠étr✐❝❛s sã♦ ❡♥❢❛t✐③❛❞❛s ❝❤❛♠❛♥❞♦✲❛s ❞❡ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ ❝♦✲s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ t❛♥❣❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ❊❧❛s sã♦ ❞❡✜♥✐❞❛s ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

❉❡✜♥✐çã♦ ✶✿ ❆s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s (i) senhx= e

x₋_e−x

2 (ii) cosh=

ex₊_e−x

2

(iii) tanhx= senhx coshx =

ex

−e−x

ex₊_e−x (iv) cothx=

coshx senhx =

ex₊_e−x ex₋_e−x

(v)sechx= 1 coshx ❂

2

ex₊_e−x (vi)cschx =

1 senhx ❂

2 ex₋_e−x✳

✶✳✶✳✸ ▲✐✈r♦ ❈

❋✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s

❆s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s sã♦ ❢♦r♠❛❞❛s ❛ ♣❛rt✐r ❞❡ ❝♦♠❜✐♥❛çõ❡s ❞❡ ❞✉❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ex _❡ _e−x_{✳ ❆s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s s✐♠♣❧✐✜❝❛♠ ♠✉✐t❛s ❡①♣r❡ssõ❡s} ♠❛t❡♠át✐❝❛s ❡ sã♦ ✐♠♣♦rt❛♥t❡s ❡♠ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✳ ❙ã♦ ✉s❛❞❛s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ♣r♦❜❧❡♠❛s t❛✐s ❝♦♠♦ ❝❛❧❝✉❧❛r ❛ t❡♥sã♦ ❡♠ ✉♠ ❝❛❜♦ s✉s♣❡♥s♦ ♣❡❧❛s ❡①tr❡♠✐❞❛❞❡s✱ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ❡❧étr✐❝❛✱ ♣♦r ❡①❡♠♣❧♦✳ ❚❛♠❜é♠ tê♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ◆❡st❛ s❡çã♦✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ❛♣r❡s❡♥t❛çã♦ ❞❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ s❡✉s ❣rá✜❝♦s✱ ❝♦♠♦ s✉❛s ❞❡r✐✈❛❞❛s sã♦ ❝❛❧❝✉❧❛❞❛s ❡ ♣♦r q✉❡ ❡❧❛s sã♦ ❝♦♥s✐❞❡r❛❞❛s ♣r✐♠✐t✐✈❛s ✐♠♣♦rt❛♥t❡s✳

❆s ♣❛rt❡s ♣❛r ❡ í♠♣❛r ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧

❘❡❝♦r❞❡ ❛s ❞❡✜♥✐çõ❡s ♣❛r❡s ❡ í♠♣❛r❡s ✈✐st❛s ♥❛ ❙❡çã♦ ✶✳✷ ❡ ❛ s✐♠❡tr✐❛ ❞❡ s❡✉s ❣rá✜❝♦s✳ ❯♠❛ ❢✉♥çã♦ ♣❛rf s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦f(₋x) = f(x)✱ ❡♥q✉❛♥t♦ ✉♠❛ ❢✉♥çã♦ í♠♣❛r s❛t✐s❢❛③ f(₋x) = ₋f(x)✳ ❚♦❞❛ ❢✉♥çã♦ f q✉❡ s❡❥❛ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✐♥t❡r✈❛❧♦ ❝❡♥tr❛❞♦ ♥❛ ♦r✐❣❡♠ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ú♥✐❝❛ ❝♦♠♦ ❛ s♦♠❛ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣❛r ❡ ❞❡ ✉♠❛ ❢✉♥çã♦ í♠♣❛r✳ ❆ ❞❡❝♦♠♣♦s✐çã♦ é

f(x) = f(x) +f(−x)

2 +

f(x)₋f(₋x)

2 ✳

(18)

❈❛♣ít✉❧♦ ✶✳ ❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✳✶✳ ❘❡s✉♠♦ ❞♦ ❝♦♥t❡ú❞♦

❙❡ ❡s❝r❡✈❡r♠♦s ex _{❞❡ss❛ ♠❛♥❡✐r❛✱ t❡r❡♠♦s} ex ₌ e

x₊_e−x

2 +

ex₋_e−x

2 ✳

❆s ♣❛rt❡s ♣❛r ❡ í♠♣❛r ❞❡ex_{✱ ❞❡♥♦♠✐♥❛❞❛ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦} ❞❡ ①✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ sã♦ út❡✐s à s✉❛ ♠❛♥❡✐r❛✳ ❊❧❛s ❞❡s❝r❡✈❡♠ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♦♥❞❛s ❡♠ só❧✐❞♦s ❡❧ást✐❝♦s ❡ ❛ ❢♦r♠❛ ❞♦s ✜♦s s✉s♣❡♥s♦s ❞❛ r❡❞❡ ❡❧étr✐❝❛✳ ❆ ❧✐♥❤❛ ❝❡♥tr❛❧ ❞♦ P♦rt❛❧ ❞♦ ❆r❝♦ ❞♦ ❖❡st❡ ❡♠ ❙t✳ ▲♦✉✐s é ✉♠❛ ❝✉r✈❛ ♣♦♥❞❡r❛❞❛ ❞❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✳

❉❡✜♥✐çã♦ ❡ ✐❞❡♥t✐❞❛❞❡s

❆s ❢✉♥çõ❡s ❞❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ sã♦ ❞❡✜♥✐❞❛s ♣❡❧❛s ❞✉❛s ♣r✐♠❡✐r❛s ❡q✉❛çõ❡s ❞❛ ❚❛❜❡❧❛ ✼✳✸✳ ❊ss❛ t❛❜❡❧❛ t❛♠❜é♠ ❛♣r❡s❡♥t❛ ❛s ❞❡✜♥✐çõ❡s ❞❡ t❛♥❣❡♥t❡✱ ❝♦t❛♥❣❡♥t❡✱ s❡❝❛♥t❡ ❡ ❝♦ss❡❝❛♥t❡ ❤✐♣❡r❜ó❧✐❝♦s✳ ❈♦♠♦ ✈❡r❡♠♦s✱ ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ♣♦ss✉❡♠ ✉♠❛ sér✐❡ ❞❡ s✐♠✐❧❛r✐❞❛❞❡s ❝♦♠ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❞❛s q✉❛✐s s❡✉s ♥♦♠❡s ❞❡r✐✈❛♠✳

❋✐❣✉r❛ ✶✳✶✿ ❚❛❜❡❧❛ ✼✳✸

(19)

❈❛♣ít✉❧♦ ✶✳ ❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✳✶✳ ❘❡s✉♠♦ ❞♦ ❝♦♥t❡ú❞♦

✶✳✶✳✹ ▲✐✈r♦ ❉

❋❯◆➬Õ❊❙ ❍■P❊❘❇Ó▲■❈❆❙

❈❡rt❛s ❝♦♠❜✐♥❛çõ❡s ❞❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ex _❡ _e₋x _{s✉r❣❡♠ ❢r❡q✉❡♥t❡♠❡♥t❡} ❡♠ ♠❛t❡♠át✐❝❛ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❡✱ ♣♦r ✐ss♦✱ ♠❡r❡❝❡♠ ♥♦♠❡s ❡s♣❡❝✐❛✐s✳ ❊❧❛s sã♦ ❛♥á❧♦❣❛s✱ ❞❡ ♠✉✐t❛s ♠❛♥❡✐r❛s✱ às ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ r❡❧❛çã♦ ❝♦♠ ❛ ❤✐♣ér❜♦❧❡ q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s tê♠ ❝♦♠ ♦ ❝ír❝✉❧♦✳ P♦r ❡ss❛ r❛③ã♦ sã♦ ❝❤❛♠❛❞❛s ❝♦❧❡t✐✈❛♠❡♥t❡ ❞❡ ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s✱ ❡✱ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✱ ❞❡ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳

❉❊❋■◆■➬Õ❊❙ ❉❆❙ ❋❯◆➬Õ❊❙ ❍■P❊❘❇Ó▲■❈❆❙ senhx= e

x₋_e−x

2 cossechx= 1 senhx coshx= e

x₊_e−x

2 sechx= 1 coshx tghx = senhx

coshx cotghx =

coshx senhx

❖s ❣rá✜❝♦s ❞♦ s❡♥♦ ❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ♣♦❞❡♠ s❡r ❡s❜♦ç❛❞♦s ✉s❛♥❞♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❣rá✜❝❛✳

✶✳✶✳✺ ▲✐✈r♦ ❊

❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❆s ❡①♣r❡ssõ❡s ❡①♣♦♥❡♥❝✐❛✐s ex

−e−x

2 ❡

ex₊_e−x

2

♦❝♦rr❡♠ ❢r❡q✉❡♥t❡♠❡♥t❡ ♥❛ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✳

❊st❛s ❡①♣r❡ssõ❡s ❞❡✜♥❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛s ❢✉♥çõ❡s s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❞❡ x ❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❞❡ x✳

❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ss❛s ❢✉♥çõ❡s ♥♦s ❧❡✈❛ ❛ ❢❛③❡r ✉♠❛ ❛♥❛❧♦❣✐❛ ❝♦♠ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✳

(20)

❈❛♣ít✉❧♦ ✶✳ ❆❜♦r❞❛❣❡♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✶✳✷✳ ❆♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛

❙❊◆❖ ❍■P❊❘❇Ó▲■❈❖ ❊ ❈❖❙❙❊◆❖ ❍■P❊❘❇Ó▲■❈❖

❆ ❢✉♥çã♦ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ ❞❡♥♦t❛❞❛ ♣♦r senh✱ ❡ ❛ ❢✉♥çã♦ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ ❞❡♥♦t❛❞❛ ♣♦r cosh✱ sã♦ ❞❡✜♥✐❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦r✿

senhx= e

x₋_e−x

2

❡

coshx = e

x₊_e−x

2 .

❖ ❞♦♠í♥✐♦ ❡ ❛ ✐♠❛❣❡♠ ❞❛s ❢✉♥çõ❡ssenh ❡cosh sã♦✿

D(senh) = (_−∞,+_∞)✱

D(cosh) = (_−∞,+_∞)✱

Im(senh) = (_−∞,+_∞)❡

Im(cosh) = (1,+_∞)✳

✶✳✷ ❆♥á❧✐s❡ q✉❛❧✐t❛t✐✈❛ ❞♦s s❡✐s ❡①❡♠♣❧❛r❡s

❖s ❧✐✈r♦s ❆✱ ❇✱ ❈✱ ❉ ❡ ❊ tr❛t❛♠ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦ s❡♥♦ ❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦s ❞❡ ❢♦r♠❛ ✐❣✉❛❧✐tár✐❛✳ ❚♦❞♦s ✉s❛♠ ❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ♣❛r❛ ❞❡✜♥✐✲❧❛s✱ ❛ ♣❛rt✐r ❞♦s r❡s✉❧t❛❞♦s ❞❛ s♦♠❛ ❡ s✉❜tr❛çã♦ ❞❡ ✉♠❛ ❡①♣♦♥❡♥❝✐❛❧ ❝r❡s❝❡♥t❡ ex _{❡ ✉♠❛ ❡①♣♦♥❡♥❝✐❛❧ ❞❡❝r❡s❝❡♥t❡} _e−x_{✱ ❞❡ ❢♦r♠❛ ❞✐r❡t❛✱ s❡♠ ❞❡♠♦♥str❛çõ❡s✱ ❝♦♠♦} ✈✐st♦ ♥♦s r❡s✉♠♦s✳ ❉❡✐①❛♥❞♦ ❛❧❣✉♠❛s ✐♥t❡rr♦❣❛çõ❡s ❡ ❛té ✐♥❞❛❣❛çõ❡s ✧❛❝❤❡✐ ♠✉✐t♦ ❡♥❣r❛ç❛❞♦ ♣❡♥s❛r✲s❡ ❡♠ ❥✉♥t❛r ❞✉❛s ♣❛❧❛✈r❛s ❝♦♠♦ s❡♥♦ ❡ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦s ♣❛r❛ ♥♦♠❡❛r ✉♠

ex₋_e−x

2 ”

✭❙ô♥✐❛ P✐♥t♦ ❞❡ ❈❛r✈❛❧❤♦✮❬✶❪✳

◗✉❛♥❞♦ ❡st✉❞❛♠♦s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ♣♦r ✉♠ ❞❡ss❡s ❡①❡♠♣❧❛r❡s ♥ã♦ ✜❝❛ ❝❧❛r❛ ❛ ❛ss♦❝✐❛çã♦ ❞❡ss❛s ❢✉♥çõ❡s ❝♦♠ ❛ ❤✐♣ér❜♦❧❡ ❡ ♣♦r ✐ss♦ ❛❝❤❛♠♦s ❡str❛♥❤♦ ❡ ❛té s❡♠

(21)

s❡♥t✐❞♦ ♦ ♥♦♠❡ ❤✐♣❡r❜ó❧✐❝♦✳ ❚♦❞♦s r❡❧❛t❛♠ ❛ s❡♠❡❧❤❛♥ç❛ q✉❡ ❡①✐st❡ ❡♥tr❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s ❡ q✉❡ ❛ ♠❡s♠❛ r❡❧❛çã♦ q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s t❡♠ ❝♦♠ ♦ ❝ír❝✉❧♦ ❛ ❤✐♣❡r❜ó❧✐❝❛ t❡♠ ❝♦♠ ❛ ❤✐♣ér❜♦❧❡✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ ❡♥❝♦♥tr❛♠♦s ❡ss❛s s❡♠❡❧❤❛♥ç❛s✱ ♦✉ s❡❥❛✱ ❛ ú♥✐❝❛ s❡♠❡❧❤❛♥ç❛ sã♦ ♦s ♥♦♠❡s ❞❛❞♦s ❛❝r❡s❝❡♥t❛❞♦s ❞❡ ❤✐♣❡r❜ó❧✐❝♦✳

❖s ❧✐✈r♦s ❆✱ ❉ ❡ ❊ tr❛③❡♠ ❛ ❥✉st✐✜❝❛t✐✈❛ ❞❛ ♥♦♠❡♥❝❧❛t✉r❛ ❤✐♣❡r❜ó❧✐❝❛ ❛ ♣❛rt✐r ❞❛ ✐❞❡♥t✐❞❛❞❡ cosh2

x₋senh2

x = 1✳ ✧❆ ✐❞❡♥t✐❞❛❞❡ ❞❡♠♦♥str❛❞❛ ♥♦ ❊①❡♠♣❧♦ ✶✭❛✮ ❢♦r♥❡❝❡ ✉♠ ✐♥❞í❝✐♦ ♣❛r❛ ❛ r❛③ã♦ ❞♦ ♥♦♠❡ ❢✉♥çã♦ ❤✐♣❡r❜ó❧✐❝❛✧✭❈á❧❝✉❧♦ ■ ✲ ❏❛♠❡s ❙t❡✇❛rt✳ ♣✳✷✸✽✮✱ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ s❡ ❢❛③ ❛ ♣❛rt✐r ❞❛ s✉❜st✐t✉✐çã♦ ❞♦s ✈❛❧♦r❡s ❞❛❞♦s ❛ senhx= e

x

−e−x

2 ❡coshx =

ex₊_e−x

2 ✳

❉❡♠♦♥str❛çã♦✿

cosh2

x₋senh2

x= (e

x₊_e−x

2 )

2 −(e

x

−e−x

2 )

2

= e

2x_{+ 2 +}_e−2x

4 −

e2x

−2 +e−2x

4 =

4

4 = 1

❆ ♣❛rt✐r ❞❡ss❛ ✐❞❡♥t✐❞❛❞❡ ♦s ❧✐✈r♦s ❢❛③❡♠ ✉♠❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ x2

+y2

= 1 ❡ ❛ ❤✐♣ér❜♦❧❡ x2 −y2

= 1✱ ❥✉st✐✜❝❛♥❞♦ ♦ ♥♦♠❡ ❤✐♣❡r❜ó❧✐❝❛✳ ◆❛ s❡q✉ê♥❝✐❛✱ ❢♦r❛♠ ❛♥❛❧✐s❛❞❛s ❛s ❛♣❧✐❝❛çõ❡s ❞❡ss❛s ❢✉♥çõ❡s ♥❛ ♥❛t✉r❡③❛✳

❖s ❧✐✈r♦s ❆ ❡ ❇ ♥ã♦ ♠❡♥❝✐♦♥❛♠ ♥❡♥❤✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ss❛s ❢✉♥çõ❡s✱ ♦✉ s❡❥❛✱ ♣♦r q✉❡ ❡ ♣❛r❛ q✉❡ ❡st✉❞á✲❧❛s❄

❈ ❡ ❊ tr❛③❡♠ ❞❡ ❢♦r♠❛ ❜❛st❛♥t❡ r❡s✉♠✐❞❛ ❛❧❣✉♠❛s ❞❛s ❛♣❧✐❝❛çõ❡s ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✱ ❝♦♠♦ ❛ ❝❛t❡♥ár✐❛ ❡ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♥❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ❉ t❡♠ ❛ ♣r❡♦❝✉♣❛çã♦ ❞❡ ♠♦str❛r q✉❡ ❛❧❣✉♥s ❢❡♥ô♠❡♥♦s✱ ❝♦♠♦ ♥❛ ❝✐ê♥❝✐❛ ❡ ♥❛ ❡♥❣❡♥❤❛r✐❛✱ q✉❡ ♦ ❞❡❝❛✐♠❡♥t♦ ❞❡ ✉♠❛ ❡♥t✐❞❛❞❡ ❝♦♠♦ ❛ ❧✉③✱ ❛ ✈❡❧♦❝✐❞❛❞❡✱ ❛ ❡❧❡tr✐❝✐❞❛❞❡ ♦✉ ❛ r❛❞✐♦❛t✐✈✐❞❛❞❡ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s✳

❖ ❧✐✈r♦ ❋✱ ♥ã♦ ♠❡♥❝✐♦♥❛❞♦ ❛té ❛q✉✐✱ ♥ã♦ tr❛③ ❡ss❡ ❝♦♥t❡ú❞♦✳ ❖ q✉❡ ❝♦♥s✐❞❡r❛♠♦s

(22)

✉♠❛ ❢❛❧❤❛✱ ♣♦✐s✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✸ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s tê♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✱ ♦ q✉❡ t♦r♥❛ s❡✉ ❡st✉❞♦ ✐♠♣♦rt❛♥t❡✳

✧❆s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s sã♦ ❛♥á❧♦❣❛s✱ ❞❡ ♠✉✐t❛s ♠❛♥❡✐r❛s✱ às ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ r❡❧❛çã♦ ❝♦♠ ❛ ❤✐♣ér❜♦❧❡ q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s tê♠ ❝♦♠ ♦ ❝ír❝✉❧♦✧✭❈á❧❝✉❧♦ ✲ ❏❛♠❡s st❡✇❛rt✳ ♣✳✷✸✻✮✳

❊♥tã♦ ♣♦r q✉❡ ♥ã♦ ♦❜tê✲❧❛s ✉t✐❧✐③❛♥❞♦✲s❡ ❞❛ ❤✐♣ér❜♦❧❡ ❝♦♠♦ ❝✉r✈❛ ❣❡r❛tr✐③ ❡ ❢❛③❡r ❛ ❛♥á❧✐s❡ ❝♦♠♦ é ❢❡✐t❛ ❝♦♠ ♦ ❝ír❝✉❧♦ ♥❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ♣r♦♣♦r❝✐♦♥❛♥❞♦ ❛♦s ❡st✉❞❛♥t❡s ✐❞❡✐❛s ❡ ♠ét♦❞♦s q✉❡ ❧❤❡ ♣❡r♠✐t❛♠ ❛♣r❡❝✐❛r ♦ ✈❛❧♦r ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ❡ ❛ s✉❛ ♥❛t✉r❡③❛❄

(23)

❈❛♣ít✉❧♦ ✷

❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s

✷✳✶ ❆ ❤✐♣ér❜♦❧❡

❉❡✜♥✐çã♦ ✶ ❙❡❥❛♠ F1 ❡ F2 ♣♦♥t♦s ❞✐st✐♥t♦s ❞♦ ♣❧❛♥♦✱ c = 1

2d(F1, F2) ❡ ♦ ♥ú♠❡r♦

a t❛❧ q✉❡ 0 < a < c✳ ❈❤❛♠❛✲s❡ ❤✐♣ér❜♦❧❡ ❛ ❝✉r✈❛ ❞♦ ♣❧❛♥♦ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦♥t♦s P ❞♦ ♣❧❛♥♦ q✉❡ s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦

|d(P, F1)−d(P, F2)|= 2a. ✭✷✳✶✮

F1 ❡ F2 sã♦ ♣♦♥t♦s ❞❛ ❤✐♣ér❜♦❧❡ ❞❡♥♦♠✐♥❛❞♦s ❢♦❝♦s ❡ ❛s ❞✐stâ♥❝✐❛sd1 =d(P, F1)

❡ d2 = d(P, F2) sã♦ r❛✐♦s ❢♦❝❛✐s ❞❡ ♣♦♥t♦ P✳ ❆ r❡t❛ q✉❡ ❝♦♥té♠ ♦s ❢♦❝♦s ❝❤❛♠❛✲s❡

❡✐①♦ ❢♦❝❛❧✳ ❖ ❝❡♥tr♦ ❞❛ ❤✐♣ér❜♦❧❡ é ♦ ♣♦♥t♦ ♠é❞✐♦ ❡♥tr❡ F1 ❡ F2✳ ❆ ♠❡❞✐❞❛ cé ❛

❞✐stâ♥❝✐❛ ❢♦❝❛❧ ❞❛ ❤✐♣ér❜♦❧❡✱ ♦✉ s❡❥❛✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ ❝❛❞❛ ❢♦❝♦ ❛♦ ❝❡♥tr♦✳ ❆ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ❡✐①♦ ❢♦❝❛❧ q✉❡ ✐♥t❡rs❡❝t❛ ♦ ❝❡♥tr♦ ❞❛ ❤✐♣ér❜♦❧❡ é ❞❡♥♦♠✐♥❛❞❛ ❡✐①♦ ♥♦r♠❛❧✳ ❖s ♣♦♥t♦s q✉❡ ✐♥t❡rs❡❝t❛♠ ♦ ❡✐①♦ ❢♦❝❛❧ sã♦ ♦s ✈ért✐❝❡s V1 ❡ V2✳

❱❡❥❛ q✉❡ ❛ ❊q✉❛çã♦ ✭✷✳✶✮ ❡q✉✐✈❛❧❡ ❛

d(P, F1)−d(P, F2) =±2a✱

(24)

❈❛♣ít✉❧♦ ✷✳ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✷✳✶✳ ❆ ❤✐♣ér❜♦❧❡

❧♦❣♦✱ ❝♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡

d(P, F1) = d(P, F2) + 2a. ✭✷✳✷✮

◆❡st❡ ❝❛s♦ ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛♦ ❢♦❝♦F1 é ♠❛✐♦r q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛♦ ❢♦❝♦ F2✱

d(P, F1)> d(P, F2)✱

♦✉ ❡♥tã♦ ♣♦❞❡♠♦s t❡r

d(P, F2) = d(P, F1) + 2a. ✭✷✳✸✮

❆❣♦r❛✱ t❡♠✲s❡ ♦ ❝♦♥trár✐♦✱ ❛ ❞✐stâ♥❝✐❛ ❞❡ P ❛♦ ❢♦❝♦ F1 é ♠❡♥♦r q✉❡ ❛ ❞✐stâ♥❝✐❛ ❞❡

P ❛♦ ❢♦❝♦F2✱

d(P, F1)< d(P, F2)✳

❯♠ ♣♦♥t♦ P✱ ♥ã♦ ♣♦❞❡ s❛t✐s❢❛③❡r s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛s ❊q✉❛çõ❡s ✭✷✳✷✮ ❡ ✭✷✳✸✮✳ ■st♦ é✱ ❛ ❤✐♣ér❜♦❧❡ é ❢♦r♠❛❞❛ ♣♦r ❞♦✐s ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s ❞❡ ♣♦♥t♦s ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✶✳

❋✐❣✉r❛ ✷✳✶✿ ❍✐♣ér❜♦❧❡

(25)

❈❛♣ít✉❧♦ ✷✳ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✷✳✶✳ ❆ ❤✐♣ér❜♦❧❡

❊ss❡s ❝♦♥❥✉♥t♦s ❞❡ ♣♦♥t♦s sã♦ ♦s r❛♠♦s ❞❛ ❤✐♣ér❜♦❧❡✳ ❖s ♣♦♥t♦s s♦❜r❡ ♦ r❛♠♦ ❞❛ ❞✐r❡✐t❛ s❛t✐s❢❛③❡♠ ❛ ❊q✉❛çã♦ ✭✷✳✷✮ ❡♥q✉❛♥t♦✱ ♦s ♣♦♥t♦s s♦❜r❡ ♦ r❛♠♦ ❞❛ ❡sq✉❡r❞❛ s❛t✐s❢❛③❡♠ ❛ ❊q✉❛çã♦ ✭✷✳✸✮✳

❱❡❥❛ q✉❡ ❛ ✜❣✉r❛ ✭✷✳✶✮ ♠♦str❛ ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞❛ ❤✐♣ér❜♦❧❡✱ ❝✉❥♦s ❡✐①♦s ❢♦❝❛❧ ❡ ♥♦r♠❛❧ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦s ❡✐①♦s ❝❛rt❡s✐❛♥♦sOx❡ Oy✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❧♦❣♦ ♦ ❝❡♥tr♦ ❞❛ ❤✐♣ér❜♦❧❡ é ❛ ♦r✐❣❡♠ O ❞♦s ❡✐①♦s✳

✷✳✶✳✶ ❆ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❤✐♣ér❜♦❧❡

❱❛♠♦s ❞❡t❡r♠✐♥❛r ❛ ❡q✉❛çã♦ ❞❛ ❤✐♣ér❜♦❧❡ ❡♠ r❡❧❛çã♦ ❛ ✉♠ s✐st❡♠❛ ❞❡ ❡✐①♦s OXY

❍✐♣ér❜♦❧❡ ❝♦♠ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ♦ ❡✐①♦ OX ❈♦♥s✐❞❡r❛♥❞♦ ❛ ❤✐♣ér❜♦❧❡ ❞❛ ✜❣✉r❛ ✷✳✶ ❝✉❥♦ ❡✐①♦ ❢♦❝❛❧ ❡ ♥♦r♠❛❧ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦s ❡✐①♦s ❝❛rt❡s✐❛♥♦sOx❡Oyr❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ ❝❡♥tr♦ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ♦r✐❣❡♠O = (0,0) ❞♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✱ ♦s ❢♦❝♦s t❡rã♦ ❝♦♦r❞❡♥❛❞❛s F1 = (−c,0)❡ F2 = (c,0)✱c >0✳

❙❡♥❞♦ ♦ ♣♦♥t♦ P = (x, y) ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ ❤✐♣ér❜♦❧❡✱ ❞❡s❡♥✈♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ t❡♠♦s✿

p

(x+c)2₊_y2

−p(x₋c)2₊_y2 ₌ ±2a.

❉❡♣♦✐s ❞❡ ❡❧✐♠✐♥❛r♠♦s ♦s r❛❞✐❝❛✐s✱ ❡❧❡✈❛♥❞♦ ❛♦ q✉❛❞r❛❞♦✱ ♦❜t❡♠♦s✿

(x+c)2

+y2

= (x₋c)2

+y2

±4a+p(x₋c)2₊_y2

♦✉ s❡❥❛✱

4xc₋4a2

=_±4ap(x₋c)2₊_y2

(26)

❈❛♣ít✉❧♦ ✷✳ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✷✳✶✳ ❆ ❤✐♣ér❜♦❧❡

❝♦♥s❡q✉❡♥t❡♠❡♥t❡

cx₋a2

=_±ap(x₋c)2₊_y2_.

❊❧❡✈❛♥❞♦ ♦s ♠❡♠❜r♦s ❞❡ss❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❛♦ q✉❛❞r❛❞♦ ✈❡♠✱

c2

x2

−2cxa2

+a4

=a2

(x2

−2xc+c2

+y2

)

♣♦rt❛♥t♦

c2

x2

+a4

=a2

(x2

+c2

+y2

),

❡♥tã♦✿

x2

(c2 −a2

)₋a2

y2

=a2

(c2 −a2

)

❝♦♠♦ c > a✱ ❞❡✜♥❛b=√c2

−a2✱ ❧♦❣♦✱

b2

x2 −a2

y2

=b2

a2

❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r b2

a2✱ t❡♠♦s✿

x2

a2 −

y2

b2 = 1. ✭✷✳✹✮

❆ ❊q✉❛çã♦ ✭✷✳✹✮ é ❞❡♥♦♠✐♥❛❞❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❤✐♣ér❜♦❧❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ♦ ❡✐①♦ OX✳

❍✐♣ér❜♦❧❡ ❝♦♠ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ♦ ❡✐①♦ OY ❖♥❞❡2b é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ✈ért✐❝❡s ❡a é t❛❧ q✉❡ c2

−b2

=a2_✳

◆❡st❡ ❝❛s♦✱ ♦ ❡✐①♦ ❢♦❝❛❧ ❡ ♥♦r♠❛❧ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦s ❡✐①♦s OY ❡ OX r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s F1 = (0,−c), F2 = (0, c), A1 = (0,−a), A2 = (0, a), B1 =

(₋b,0) ❡ B2 = (b,0)✳

(27)

❈❛♣ít✉❧♦ ✷✳ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✷✳✶✳ ❆ ❤✐♣ér❜♦❧❡

❋❛③❡♥❞♦ ❝♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ❡♥❝♦♥tr❛♠♦s ❛ ❡q✉❛çã♦

y2

a2 −

x2

b2 = 1. ✭✷✳✺✮

❆ ❊q✉❛çã♦ ✭✷✳✺✮ é ❞❡♥♦♠✐♥❛❞❛ ❡q✉❛çã♦ ❝❛♥ô♥✐❝❛ ❞❛ ❤✐♣ér❜♦❧❡ ❞❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❡ r❡t❛ ❢♦❝❛❧ ❝♦✐♥❝✐❞❡♥t❡ ❝♦♠ ♦ ❡✐①♦ OY✳

✷✳✶✳✷ ❆ssí♥t♦t❛s ❞❛ ❤✐♣ér❜♦❧❡

❖ r❡tâ♥❣✉❧♦ ABCD ❛❜❛✐①♦ ❞❡ ❧❛❞♦s 2a ❡ 2b ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✷✳✷ ❡ t❛♥❣❡♥t❡ à ❤✐♣ér❜♦❧❡ ♥♦s ✈ért✐❝❡s V1 ❡ V2 é ❞❡♥♦♠✐♥❛❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❜❛s❡ ❞❛ ❤✐♣ér❜♦❧❡✳

❋✐❣✉r❛ ✷✳✷✿ ❆ssí♥t♦t❛s

❆s r❡t❛s q✉❡ ❝♦♥té♠ ❛s ❞✐❛❣♦♥❛✐s ❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❜❛s❡ sã♦ ❛s ❛ssí♥t♦t❛s ❞❛ ❤✐♣ér❜♦❧❡✱ ❡ tê♠ ✐♥❝❧✐♥❛çã♦ ±_ab ❡♠ r❡❧❛çã♦ ❛ r❡t❛ ❢♦❝❛❧✳ ❊♠ ❡s♣❡❝✐❛❧✱ q✉❛♥❞♦a=b✱ ❛s ❛ssí♥t♦t❛s sã♦ ♣❡r♣❡♥❞✐❝✉❧❛r❡s ❡♥tr❡ s✐✳ ◆❡ss❡ ❝❛s♦✱ ❛ ❤✐♣ér❜♦❧❡ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❡q✉✐❧át❡r❛✳

▲♦❣♦✱ ❛s ❛ssí♥t♦t❛s ❞❛ ❤✐♣ér❜♦❧❡ sã♦ ❛s r❡t❛sr1 :y=

b

ax ❡r2 :y=− b ax✳

(28)

❈❛♣ít✉❧♦ ✷✳ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✷✳✷✳ ➶♥❣✉❧♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡

❚❡♦r❡♠❛ ✶ ❆s ❛ssí♥t♦t❛s ♥ã♦ ✐♥t❡r❝❡♣t❛♠ ❛ ❤✐♣ér❜♦❧❡✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❛♠♦s ❛ ❛ssí♥t♦t❛ r1 :y =

b

ax ❞❛ ❤✐♣ér❜♦❧❡ H✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ ♣♦♥t♦ Q= (x0, y0) t❛❧ q✉❡ r1∩H =Q✳

❊♥tã♦ ❛s ❝♦♦r❞❡♥❛❞❛s ❞❡Q ❞❡✈❡ s❛t✐s❢❛③❡r ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ r1✱ ✐st♦ é✱

y0 =

b ax0

❡✱ t❛♠❜ê♠ ❛ ❡q✉❛çã♦ ❞❛ ❤✐♣ér❜♦❧❡

x0 2

a2 −

y0 2

b2 = 1.

❙✉❜st✐t✉✐♥❞♦ y0 = _abx0 ♥❛ ❡q✉❛çã♦ ❞❛ ❤✐♣ér❜♦❧❡✱ t❡♠♦s✱

x02

a2 −

(b

ax0)

2

b2 = 1,

♦ q✉❡ r❡s✉❧t❛rá

1 = x0

2

a2 −

x02

a2 = 0

q✉❡ é ✉♠ ❛❜s✉r❞♦✳

❖✉ s❡❥❛✱ ♥ã♦ ♣♦❞❡ ❡①✐st✐r ✉♠ ♣♦♥t♦ ❝♦♠✉♠ à ❛ssí♥t♦t❛ ❡ à ❤✐♣ér❜♦❧❡✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ r2✳

❊①✐st❡♠ ♣♦♥t♦s ❞❛ ❤✐♣ér❜♦❧❡ tã♦ ♣ró①✐♠♦s ❞❛s ❛ssí♥t♦t❛s q✉❛♥t♦ q✉❡✐r❛♠♦s✱ ♠❛s✱ ❥❛♠❛✐s ❛s ❛ssí♥t♦t❛s ✐♥t❡r❝❡♣t❛rã♦ ❛ ❤✐♣ér❜♦❧❡✳

✷✳✷

➶♥❣✉❧♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡

❈♦♥s✐❞❡r❡♠♦s ♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦ ❝♦♠ ♦s ❡✐①♦s x ❡ y ❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ r♦t❛çã♦ ❞❡ π

4 ❞♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✱ ♦❜t❡♠♦s ♠❛✐s ❞♦✐s ❡✐①♦s X ❡ Y✳ ❙❡❥❛M ✉♠ ♣♦♥t♦ q✉❡ ❡stá s♦❜r❡ ♦ ❣rá✜❝♦xy= 1

(29)

x=OP, y=OQ, X =OD ❡ Y =DM✳ ❆ss✐♠✱

x=OP =OE₋P E=ODcosπ

4 −DM sen π 4

=

√

2

2 (X−Y)

❡✱

y =OF +F Q=ODcosπ

4 +M Dsen π 4

=

√

2

2 (X+Y)

❊♥tã♦✿

xy=

√

2

2 (X−Y)(X+Y)

√ 2 2 = 1 2 2

4(X

2 −Y2

) = 1

2

X2 −Y2

= 1.

❈♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ✷✳✸✳

❋✐❣✉r❛ ✷✳✸✿ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛ ❡q✉❛çã♦ xy= 1 2

❖✉ s❡❥❛✱ ♦ ❣rá✜❝♦ ❞❛ ❡q✉❛çã♦xy= 1

2 é ✉♠❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛ ❝♦♠ a=b = 1✳ ❖ â♥❣✉❧♦ ❤✐♣❡r❜ó❧✐❝♦ é ✉♠❛ ✜❣✉r❛ ❣❡♦♠étr✐❝❛ q✉❡ ❞✐✈✐❞❡ ❛ ❤✐♣ér❜♦❧❡✱ t❡♥❞♦ ✉♠❛ r❡❧❛çã♦ ❝♦♠✉♠ ❝♦♠ ♦ â♥❣✉❧♦ ♥♦ ❝ír❝✉❧♦✳

(30)

P♦r ✐ss♦ ♣❛r❛ ❞❡✜♥✐r♠♦s ♦ â♥❣✉❧♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ ✈❛♠♦s ❢❛③ê✲❧♦ ♣❛r❛❧❡❧♦ ❛♦ â♥❣✉❧♦ ❞❡✜♥✐❞♦ ♥♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ 1✳

❆ ♠❡❞✐❞❛ ❞❡ ✉♠ â♥❣✉❧♦ ♥✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦1✱ M ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ ❡♠ r❛❞✐❛♥♦s✱ ♠❡❞❡θ r❛❞✐❛♥♦s s❡ ♦ ❛r❝♦ X2

−Y2

= 1 q✉❡ ❞❡✜♥❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r s✉❜t❡♥❞✐❞♦ ❡♥tr❡ ❡❧❡ ♠❡❞❡θ ✉♥✐❞❛❞❡s ❤✐♣❡r❜ó❧✐❝♦ AOM ❡ ✉♠ â♥❣✉❧♦

❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ❆Ô▼ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛✳

❙❛❜❡♠♦s t❛♠❜é♠ q✉❡ ✉♠ â♥❣✉❧♦θ r❛❞✐❛♥♦s ❊♥tã♦ ♦ â♥❣✉❧♦ ❆Ô▼ ♠❡❞❡ θ ♣r❡ss✉♣õ❡ ✉♠ s❡t♦r ❝✐r❝✉❧❛r ❞❡ ár❡❛ θ

2✱ ♥♦ ❝ír❝✉❧♦ q✉❛♥❞♦ ❛ ár❡❛ ❞❡ s❡t♦r AOM ❞❡ r❛✐♦1✳ ❆ss✐♠ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ✉♠ â♥❣✉❧♦ ✈❛❧❡ θ

2 ✉♥✐❞❛❞❡s ❞❡ ár❡❛✳ ♠❡❞❡θ r❛❞✐❛♥♦s s❡ ♦ s❡t♦r s✉❜t❡♥❞✐❞♦ ❡♥tr❡ ❡❧❡

♠❡❞❡ θ

2 ✉♥✐❞❛❞❡s ❞❡ ár❡❛✳ ❱❛♠♦s ✈❡r ❞❡ ❢♦r♠❛ ❞❡t❛❧❤❛❞❛✳

❘❡t♦r♥❛♥❞♦ ❛♦s ❡✐①♦s x ❡ y✱ ❝♦♠ ❛ ❤✐♣ér❜♦❧❡ xy = 1

2✱ ✜❣✉r❛ ✷✳✹✱ t♦♠❡ M ❡ N ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r ♥♦ ♠❡s♠♦ r❛♠♦ ❞❛ ❤✐♣ér❜♦❧❡✳

❖ ♣♦♥t♦ M t❡♠ ❝♦♦r❞❡♥❛❞❛s x=OP ❡y =OQ✳ ❖ ♣♦♥t♦ N t❡♠ ❝♦♦r❞❡♥❛❞❛s x=OR ❡ y=OS✳

❆ ár❡❛ ❞♦s r❡tâ♥❣✉❧♦s OP M Q ❡ ORN S ❞❡♥♦t❛r❡♠♦s ♣♦r✿

(31)

AOP M Q =OP ×OQ=xy=

1

2✱

❡

AORN S =OR×OS =xy =

1 2

r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❋✐❣✉r❛ ✷✳✹✿ ❘❛♠♦ ❞✐r❡✐t♦ ❞❛ ❤✐♣ér❜♦❧❡xy= 1 2

▲♦❣♦✱ AOP M Q =AORN S✱ ♦ q✉❡ ✐♠♣❧✐❝❛AST M Q =AP RN T✳

P❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ s❡t♦rOM N✱ ✈❛♠♦s ❣✐r❛r ❛ ✜❣✉r❛ ✭✷✳✹✮ ❞❡ π

4 ❡ r❡t♦r♥❛r ❛ ❤✐♣ér❜♦❧❡ X2

−Y2

= 1✱ ✜❣✉r❛ ✷✳✺✳ ❱❡❥❛ q✉❡

AOP M =

1

2AOP M Q =

1

2AORN S

❡

(32)

AORN M =AOP M +AP RN M =AORN +AP RN M✳

P♦r ♦✉tr♦ ❧❛❞♦

AORN M =AORN +AON M✳

▲♦❣♦✱

AON M =AP RN M✳

❊♠ ✉♠ r❛❝✐♦❝í♥✐♦ ❛♥á❧♦❣♦ ✈❡♠♦s q✉❡ ❛AON M =AQSN M =AP RN M✳

❋✐❣✉r❛ ✷✳✺✿ ❘❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❛ ❤✐♣ér❜♦❧❡ x2 −y2

= 1✳

P♦rt❛♥t♦ ♦ q✉❡ ♣r❡❝✐s❛♠♦s ❞❡✜♥✐r é ❛ ár❡❛ P RN M q✉❡ é ❜❡♠ ♠❛✐s ♣rát✐❝♦ ♥❛ ❤✐♣ér❜♦❧❡ xy = 1

2 ♥♦s ❡✐①♦s x ❡ y ✜❣✉r❛ ✭✷✳✹✮✳ ❆ ár❡❛ P RN M é ❛ ár❡❛ s♦❜r❡ ♦ ❣rá✜❝♦ ❞❡ y= 1

2x✱ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ x=OP ❡x=OR✳ ▲♦❣♦✿

(33)

AP RN M =

Z OR OP 1

2xdx

= 1

2|lnOR−lnOP|

❂1 2 lnOR OP ✳

❙❡M ❡stá ❛ ❡sq✉❡r❞❛ ❞❡ N ❡♥tã♦

AP RN M =

1

2ln

OR OP✳ ❊ s❡ M ❡stá ❛ ❞✐r❡✐t❛ ❞❡ N✱ ❡♥tã♦

AP RN M =

1

2ln

OP OR✳

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ASN M Q ✐♥t❡❣r❛♥❞♦ ❛ ❢✉♥çã♦ x =

1

2y✱

♦❜t❡♥❞♦

ASN M Q =

1

2ln

OS OQ✳

❖❜s❡r✈❡ q✉❡ s❡M =N ❡♥tã♦ AP RN M = 0 ❡ s❡ M 6=N ❡♥tã♦ AP RN M >0✳ ◗✉❛♥❞♦M s❡ ❛❢❛st❛ ❞❡N ♣❡❧❛ ❞✐r❡✐t❛✱ ♦ s❡❣✉✐♠❡♥t♦OP ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳ ❆ss✐♠✱ ❝♦♠♦ ♦ t❛♠❛♥❤♦ OR ❡stá ✜①♦✱ AP RN M =

1

2(lnOP − lnOR) ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳

◗✉❛♥❞♦ M s❡ ❛❢❛st❛ ❞❡ N ♣❡❧❛ ❡sq✉❡r❞❛✱ ♦ s❡❣✉✐♠❡♥t♦ OP t❡♥❞❡ ❛ ③❡r♦ ❡ lnOP ❞❡❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳ ❊♥tã♦✱ AP RN M =

1

2(lnOR − lnOP) ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✳

▲♦❣♦✱ AON M =AP RN M ✈❛r✐❛ ❞❡ 0 ❛+∞✳ ❈♦♥✈❡♥❝✐♦♥❛♥❞♦✱ t❡♠♦s✿

•❙❡ ♦ ♣♦♥t♦ M ❡stá ❛❝✐♠❛ ❞♦ ❡✐①♦ ❞♦s X′_s_{✱ ♦ â♥❣✉❧♦ q✉❡ ❡❧❡ ❞❡✜♥❡ t❡♠ ♠❡❞✐❞❛}

♣♦s✐t✐✈❛✳

•❙❡ ♦ ♣♦♥t♦M ❡stá ❛❜❛✐①♦ ❞♦ ❡✐①♦ ❞♦sX′_s_{✱ ♦ â♥❣✉❧♦ q✉❡ ❡❧❡ ❞❡✜♥❡ t❡♠ ♠❡❞✐❞❛}

♥❡❣❛t✐✈❛✳

(34)

❈❛♣ít✉❧♦ ✷✳ ❊st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✷✳✸✳ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s

▲❡♠❜r❡♠♦s q✉❡ ✐ss♦ ♥ã♦ ♦❝♦rr❡ ♥♦ ❝ír❝✉❧♦✱ ❡ss❡s â♥❣✉❧♦s ♠❡❞✐❞♦s ♥♦ ❝ír❝✉❧♦✱ s❡✉s ✈❛❧♦r❡s ❡st❛r✐❛♠ ❡♥tr❡ −π₄ ❡ +π

4✳

❆♣❡s❛r ❞❡ ✈ár✐❛s s❡♠❡❧❤❛♥ç❛s✱ ❡①✐st❡♠ ❛❧❣✉♠❛s ❞✐❢❡r❡♥ç❛s ✐♥t❡r❡ss❛♥t❡s ❞♦ â♥✲ ❣✉❧♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ ❡ ♦ â♥❣✉❧♦ s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳

◆❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♦ â♥❣✉❧♦ é ♣❡r✐ó❞✐❝♦✱ ◆❛ ❤✐♣ér❜♦❧❡ ♦ â♥❣✉❧♦ ♥ã♦ é ♣❡r✐ó❞✐❝♦✱ ❞❡ ♣❡rí♦❞♦ 2π r❛❞✐❛♥♦s✱ ♣♦✐s✱ ❛ ✉♠❛ ✈❡③ q✉❡ ❛ ❝✉r✈❛t✉r❛ ♥ã♦ é ❝♦♥st❛♥t❡✱ ❝✐r❝✉♥❢❡rê♥❝✐❛ t❡♠ ❝✉r✈❛t✉r❛ ❝♦♥t❛♥t❡✳ ❛ ár❡❛ ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ❧♦❣♦ ♦ ❧♦❣♦ ♦ â♥❣✉❧♦ é ❧✐♠✐t❛❞♦✳ â♥❣✉❧♦ ❤✐♣❡r❜ó❧✐❝♦ é ✐❧✐♠✐t❛❞♦✳

✷✳✸ ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s

❆s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s sã♦ ❞❡✜♥✐❞❛s ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s✱ ❧♦❣♦ ✈❛♠♦s ❢❛③❡r ✉♠ ❡st✉❞♦ ❞❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ✉s❛♥❞♦ ❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛ ❝♦♠♦ ❝✉r✈❛ ❣❡r❛tr✐③✱ ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ❛s tr✐❣♦♥♦♠étr✐❝❛s ❝♦♠ ♦ ❝ír❝✉❧♦✱ ♦✉ s❡❥❛✱ ❞❛r❡♠♦s ❛s ❋✉♥çõ❡s ❍✐♣❡r❜ó❧✐❝❛s ♦ ♠❡s♠♦ tr❛t❛♠❡♥t♦ q✉❡ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ❞❡s❡♥✈♦❧✈❡r❡♠♦s ❛❧❣✉♠❛s r❡❧❛çõ❡s✳

P❛r❛ ✜❝❛r ♠❛✐s ❝❧❛r❛ ❡ss❛ s❡♠❡❧❤❛♥ç❛✱ ✈❛♠♦s ❢❛③❡r ♦ ❡st✉❞♦ ❞❡ ❢♦r♠❛ ♣❛r❛❧❡❧❛✳

(35)

❈♦❧♦❝❛♥❞♦ ♦s r❡s✉❧t❛❞♦s s♦❜r❡ ♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦X2

+Y2

= 1 à ❡sq✉❡r❞❛ ❡ ♦s r❡s✉❧✲ t❛❞♦s s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ ❡q✉✐❧át❡r❛ X2

−Y2

= 1 ♥❛ ❝♦❧✉♥❛ ❞❛ ❞✐r❡✐t❛✳ X2

+Y2

= 1 X2

−Y2

= 1

❙❡❥❛ M ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❙❡❥❛M ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ ❝✉r✈❛ ❞❡ ♠♦❞♦ ❞❡ r❛✐♦ 1❡ ❝❡♥tr♦ ♥❛ ♦r✐❣❡♠ ❞♦s ❡✐①♦s q✉❡ ♦ s❡t♦r OAM t❡♥❤❛ ár❡❛ θ

2✳ ❊♥tã♦ ♦ ❝❛rt❡s✐❛♥♦s ❞❡ ♠♦❞♦ q✉❡ ♦ s❡t♦r AOM â♥❣✉❧♦AOM\ t❡♠ ♠❡❞✐❞❛ θ✳

t❡♥❤❛ ár❡❛ θ

2✳ ❊♥tã♦ ♦ â♥❣✉❧♦ t❡♠ ♠❡❞✐✲ ❞❛ θ r❛❞✐❛♥♦s✱ ♣♦✐s

AOAM =

\ AOM

2 ×r

2

= θ

2 ❙❡❥❛AR ❛ r❡t❛ t❛♥❣❡♥t❡ ❛ ❤✐♣ér❜♦❧❡ ❡♠A✳ ❙❡❥❛ AR❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ ❡♠ A✳ ❆ss✐♠ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❞❡✜♥✐♠♦s ❛s ❢✉♥çõ❡s

❆ss✐♠✱ ❤✐♣❡r❜ó❧✐❝❛s ❝♦♠♦✿

ON =cosθ✱N M =senθ ❡ AR=tgθ✳ coshθ=ON✱ senhθ =N M ❡ tghθ=AR P♦rt❛♥t♦✱ ❆s ❞❡♠❛✐s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s sã♦✿ cotθ= 1

AR = 1 tgθ =

cosθ

senθ✱ cothθ= 1 AR = 1 tghθ = coshθ senhθ✳ secθ = 1

ON = 1

cosθ sechθ= 1 ON =

1 coshθ ❡ cossecθ = 1

N M = 1

senθ ❡cossechθ= 1 N M =

1 senhθ✳

❚❡♠♦s ♣❛r❛ ♦ ♣♦♥t♦ M✱ ❛s r❡❧❛çõ❡s✳ ❱❛♠♦s ❞❡❞✉③✐r ❛❧❣✉♠❛s r❡❧❛çõ❡s ❡♥tr❡ ❡st❛s ❢✉♥çõ❡s✳

(36)

❚❡♠♦s ♣❛r❛ ♦ ♣♦♥t♦M✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ P❛r❛ ♦ ♣♦♥t♦ M ❞❡ ❝♦♦r❞❡♥❛❞❛sX =ON

P✐tá❣♦r❛s Y =N M✱ ❛ss✐♠

X2

+Y2

= (ON)2

+ (N M)2

= 1 X2

−Y2

= (ON)2

−(N M)2

= 1

❡ ❧♦❣♦✱ ❡ ❧♦❣♦✱

cos2

θ+sen2

θ = 1✳ cosh2

θ₋senh2

θ = 1✳

❙❡♥❞♦ ♦ tr✐â♥❣✉❧♦ON M s❡♠❡❧❤❛♥t❡ ❛♦ ❙❡♥❞♦ ♦ tr✐â♥❣✉❧♦ON M s❡♠❡❧❤❛♥t❡ ❛♦ tr✐â♥❣✉❧♦ OAR✱ ❡♥tã♦ tr✐â♥❣✉❧♦ OAR✱ ❡♥tã♦✱

AR 1 = N M ON AR 1 = N M ON t❡♠✲s❡✱ t❡♠✲s❡✱

tgθ= senθ

cosθ tghθ=

senhθ coshθ✳

❈♦♠♦✱ ❈♦♠♦✱

cos2

θ+sen2

θ = 1 cosh2

θ₋senh2

θ = 1 ❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r cos2

θ ❞✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r cosh2

θ

1 + sen

2

θ cos2_θ =

1

cos2_θ✱ 1−

senh2

θ cosh2_θ =

1 cosh2_θ✱

♦✉ s❡❥❛✱ ♦✉ s❡❥❛✱

1 +tg2

θ=sec2_θ✳

1₋tgh2

θ =sech2_θ✳

❊ s❡ ❞✐✈✐❞✐r♠♦s ♣♦rsen2θ✱ ❡♥tã♦✿ ❊ s❡ ❞✐✈✐❞✐r♠♦s ♣♦r

senh2θ✱ ❡♥tã♦✿

cos2

θ

sen2_θ + 1 =

1 sen2_θ

cosh2

θ

senh2_θ −1 =

1 senh2_θ

❧♦❣♦✱ ❧♦❣♦✱

cot2

θ+ 1 =cossc2θ✳

coth2

θ₋1 = cossch2θ✳

❆♣❡s❛r ❞❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❡ ❛s ❢✉♥çõ❡s ❤✐♣❡r❜ó❧✐❝❛s s❡r❡♠ s❡♠❡❧❤❛♥t❡s✱ ♣♦ss✉❡♠ ❛❧❣✉♠❛s ❞✐❢❡r❡♥ç❛s✳

❱❡❥❛♠♦s ❛❧❣✉♠❛s✿

❚r✐❣♦♥♦♠étr✐❝❛s ❍✐♣❡r❜ó❧✐❝❛s

❛✮ ❖ senθ ❡ cosθ sã♦ ♣❡r✐ó❞✐❝♦s✱ ❝♦♠ ❛✮ ❖ senh ❡ ♦ cosh ♥ã♦ ♣♦ss✉❡♠

♣❡rí♦❞♦2π✳ ♣❡rí♦❞♦✳

(37)

❜✮ ❆ ❢✉♥çã♦senθ é ❧✐♠✐t❛❞❛✱ ❝♦♠ ❜✮ ❆ ❢✉♥çã♦ senhθ é ✐❧✐♠✐t❛❞❛✱

−1_≤senθ _≤1❡ ♦ cosθ t❛♠❜é♠ ✈❛r✐❛ ✈❛r✐❛♥❞♦ ❞❡ _−∞ ❛té +_∞ ❡ ♦ coshθ ❡♥tr❡−1 ❡+1✳ ✈❛r✐❛ ❞❡ +1 ❛ +_∞✳

❝✮ ❆tgθ ♣♦❞❡ ❛ss✉♠✐r q✉❛❧q✉❡r ✈❛❧♦r ❝✮ ❆ tghθ é ❧✐♠✐t❛❞❛✱₋1< tghθ < 1✳ ❡♥tr❡−∞ ❡+_∞✳

❉❡♠♦♥str❛çã♦✿ ❛✮ ◆♦ ❝❛s♦ ❞❛s ❢✉♥çõ❡s ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦✱ ♥ã♦ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ T ₆= 0 t❛❧ q✉❡ f(t+T) = f(t) _∈ R✳ ❖ s❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ é ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❡ ♦ ❝♦ss❡♥♦ ❤✐♣❡r❜ó❧✐❝♦ ♣❛r❛ q✉❛❧q✉❡r T _∈R✱f(t+T)< f(t) ♦✉ f(t+T)> f(t)✳ P♦rt❛♥t♦ ❡ss❛s ❢✉♥çõ❡s ♥ã♦ sã♦ ♣❡r✐ó❞✐❝❛s✳

❜✮ P❛r❛ θ _∈ R ❡ θ < 0✱ t❡♠♦s q✉❡ ₋θ > 0✱ s❡♥❞♦ ❛ss✐♠✱ eθ _{< e}₋θ_{✱ ♣♦rt❛♥t♦} eθ

−e−θ _<₀_{✱ ❡♥tã♦} e

θ₋_e−θ

2 < 0✱ ❧♦❣♦ senhθ < 0 ⋆✱ ❡ ♣❛r❛ θ ∈ R ❡ θ > 0✱ t❡♠♦s q✉❡ −θ <0✱ s❡♥❞♦ ❛ss✐♠✱e−θ _{< e}θ _{♣♦rt❛♥t♦} _eθ₋_e−θ _>₀_{✱ ❡♥tã♦} e

θ₋_e−θ

2 >0✱ ❧♦❣♦ senhθ > 0⋆⋆✳

❊♥tã♦ ❞❡∗ ❡ ⋆⋆✱ t❡♠♦s q✉❡ senhθ é ✐❧✐♠✐t❛❞♦✱ ✈❛r✐❛♥❞♦ ❞❡ _−∞ ❛+_∞✳ P❛r❛ θ = 0✱ t❡♠♦s q✉❡ cosh0 = e

0

+e−0

2 = 1 ∗✱ ❡ ♣❛r❛ θ ∈ R ❡ θ 6= 0✱ t❡♠♦s

(eθ ₋ ₁₎2

> 0✱ ♣♦rt❛♥t♦ e2θ _{+ 1} _> ₂_eθ_{✱ s❡} _eθ _> ₀ _{♣❛r❛ q✉❛❧q✉❡r} _θ _∈ _R_{✱ ❡♥tã♦} eθ₊_e−θ

2 >1∗ ∗✳

▲♦❣♦✱ ❞❡∗ ❡ ∗∗t❡♠♦s q✉❡ coshθ ✈❛r✐❛ ❞❡ 1❛ +_∞✳ ❝✮ P❛r❛ θ _∈ R✱ t❡♠♦s ₋e−θ _{< e}−θ_{✱ ❡♥tã♦} e

θ

−e−θ

2 <

eθ₊_e−θ

2 ✱ ♣♦rt❛♥t♦ senhθ < coshθ✱ ❝♦♠♦ tghθ = senhθ

coshθ ❧♦❣♦✱ | tghθ |< 1✱ ♦✉ s❡❥❛✱ −1 < tghθ < 1✳

❚❡♦r❡♠❛ ✷ ❆s ❢✉♥çõ❡s coshθ ❡ senhθ s❛t✐s❢❛③❡♠ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✱

coshθ= e

θ₊_e−θ

2

❡

(38)

senhθ = e

θ₋_e−θ

2 ✳

❉❡♠♦♥str❛çã♦✿

❋✐❣✉r❛ ✷✳✻✿ ❍✐♣ér❜♦❧❡ x2 −y2

= 1

❈♦♥❢♦r♠❡ ❋✐❣✉r❛ ✭✷✳✻✮✱ s❡❥❛ M ✉♠ ♣♦♥t♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ X2 −Y2

= 1✱ ❞❡

❢♦r♠❛ q✉❡ ❞❡t❡r♠✐♥❡ ✉♠ â♥❣✉❧♦ ❞❡ ♠❡❞✐❞❛ θ✱ ❡♥tã♦ AON M = θ

2✳ ◆♦s ❡✐①♦s X ❡ Y✱ ♦ ♣♦♥t♦ M t❡♠ ❝♦♦r❞❡♥❛❞❛s

X =OF =coshθ❡ Y =F M =senhθ ♥♦s ❡✐①♦s x ❡ y✱ ❛s ❝♦♦r❞❡♥❛❞❛s sã♦✿

x=OP ❡y=OQ ❙✉❜st✐t✉✐♥❞♦ ♥❛s ❡q✉❛çõ❡s ❛❜❛✐①♦✱ t❡♠♦s✿

OP =x=

√

2

2 (X−Y) =

√

2

2 (coshθ−senhθ)

❡

(39)

OQ=y=

√

2

2 (X+Y) =

√

2

2 (coshθ+senhθ)✳ ❆s ❝♦♦r❞❡♥❛❞❛s ❞♦ ♣♦♥t♦ N✱ ♥♦s ❡✐①♦sX ❡ Y

X = 1 ❡ Y = 0 ❡ ♥♦s ❡✐①♦s x ❡y✱

x=OR=

√

2

2 ❡ y=OS=

√

2

2 ✳

▲♦❣♦ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛s ár❡❛s ❞❛s r❡❣✐õ❡sP RN M ❡ QSN M ♣♦r✿

AP RN M =

1 2ln OR OP = 1 2ln √ 2 2 √ 2

2 (coshθ−senhθ)

❂−1₂ln(coshθ₋senhθ)

❡

AQSN M =

1 2ln OQ OS = 1 2ln √ 2

2 (coshθ+senhθ)

√

2 2

❂1

2ln(coshθ+senhθ)✳ ❈♦♠♦✱AON M =AP RN M✱ t❡♠♦s

θ

2 =−

1

2ln(coshθ−senhθ) ✭✷✳✻✮ ❡ ❝♦♠♦ AON M =AQSN M✱ t❡♠♦s

θ

2 =

1

2ln(coshθ+senhθ). ✭✷✳✼✮ ❆♣❧✐❝❛♥❞♦ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡♠ ✭✷✳✺✮ ❡ ✭✷✳✻✮✱ ✈❡♠

e−θ ₌_coshθ

−senhθ ✭✷✳✽✮

❡

eθ ₌_coshθ₊_senhθ. _{✭✷✳✾✮} P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦coshθ✱ ❜❛st❛ s♦♠❛r♠♦s ❛s ❡q✉❛çõ❡s ✭✷✳✼✮ ❡ ✭✷✳✽✮