❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡
❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s
†♣♦r
❏♦sé ■✈❡❧t♦♥ ❙✐q✉❡✐r❛ ▲✉st♦s❛
s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
▼❛✐♦✴✷✵✶✼ ❏♦ã♦ P❡ss♦❛ ✲ P❇
† ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❆P❊❙ ✲ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦✲
L972t Lustosa, José Ivelton Siqueira.
A transformada de Laplace e algumas aplicações / José Ivelton Siqueira Lustosa.- João Pessoa, 2017. 89 f. :
Orientador: Uberlandio Batista Severo.
Dissertação (Mestrado) – UFPB/CCEN/PROFMAT
1. Matemática. 2. Transformada de Laplace. 3. Equações Diferenciais Ordinárias. 4. Aplicações - Matemática. I. Título.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ s❡r ♠❛✐♦r ❞♦ ❯♥✐✈❡rs♦✱ ♣♦r ♠❡ ❞á ❢♦rç❛s ❡ ❝♦♥❝❡❞❡r ❛ ✈✐tór✐❛ ❡♠ ♠❛✐s ❡ss❛ ❡t❛♣❛ ❞❡ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠❡✉ ✐r♠ã♦ ❆♥tô♥✐♦ ❙✐q✉❡✐r❛ ▲✉st♦s❛ ♣♦r t❡r ❛❥✉✲ ❞❛❞♦ ♥❛ ❝♦♥❢❡❝çã♦ ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s✳
❆♦ Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦ ♣❡❧♦ ❝♦♥✈✐t❡✱ ❛♣♦✐♦ ❡ ♦r✐❡♥t❛çã♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s q✉❡ ♣r❡❝✐s❡✐ ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✳
➚ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ▼❛r✐❛ ❙♦❝♦rr♦ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❛♣♦✐♦ ❞❡♠♦♥str❛❞♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ❝✉rs♦✳
➚ ❝♦♦r❞❡♥❛çã♦ ❡ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞♦ P❘❖❋▼❆❚ ✲ ❯❋P❇✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ t✉r♠❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❊❞s♦♥ ❆r❛ú❥♦✱ ♣❡❧❛ ✉♥✐ã♦ ❡♠ ❧♦♥❣♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦s✱ q✉❡ ❢♦r❛♠ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❡ss❛ ✈✐tór✐❛✳
➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ q✉❡ ♥❛ ❜✉s❝❛ ❞❛ ♠❡❧❤♦r✐❛ ❞♦ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ✈✐❛❜✐❧✐③♦✉ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞♦ P❘❖❋▼❆❚✳
❉❡❞✐❝❛tór✐❛
❆ t♦❞♦s ♦s q✉❡ s❡ ❛❧❡❣r❛♠ ❝♦♠ ♦ ♥♦ss♦ s✉❝❡ss♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡ ❡①♣❧♦r❛♠♦s s✉❛ ❛♣❧✐✲ ❝❛çã♦ ♥❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ❧✐♥❡❛r❡s✱ ❛s q✉❛✐s ♠♦❞❡❧❛♠ ✈ár✐♦s ❢❡♥ô♠❡♥♦s ♥❛s ár❡❛s ❞❡ ❋ís✐❝❛✱ ❊♥❣❡♥❤❛r✐❛✱ ❆✉t♦♠❛çã♦ ■♥❞✉str✐❛❧ ❡ ♥❛ ♣ró♣r✐❛ ▼❛t❡♠át✐❝❛✳ ❚❛✐s ❝♦♥❤❡❝✐♠❡♥t♦s sã♦ ❞❡ s✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ❝✉r✲ s♦s s✉♣❡r✐♦r❡s q✉❡ ❛❜r❛♥❣❡♠ t❛✐s ár❡❛s✳ ❆♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦✱ ♣r♦♣r✐❡❞❛❞❡s ❡ ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡ ❛❜♦r❞❛♠♦s ✈ár✐♦s ♣r♦❜❧❡♠❛s ♥❛s ár❡❛s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✳ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s✳ ❆♣❧✐❝❛çõ❡s✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② t❤❡ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠ ❛♥❞ ❡①♣❧♦r❡ ✐ts ❛♣♣❧✐❝❛t✐♦♥ ✐♥ s♦❧✲ ✈✐♥❣ s♦♠❡ ❧✐♥❡❛r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✇❤✐❝❤ ♠♦❞❡❧ ✈❛r✐♦✉s ♣❤❡♥♦♠❡♥❛ ✐♥ t❤❡ ❛r❡❛s ♦❢ P❤②s✐❝s✱ ❊♥❣✐♥❡❡r✐♥❣✱ ■♥❞✉str✐❛❧ ❆✉t♦♠❛t✐♦♥ ❛♥❞ ▼❛t❤❡♠❛t✐❝s ✐ts❡❧❢✳ ❙✉❝❤ ❦♥♦✇❧❡❞❣❡ ✐s ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ ✐♥ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥ ❝♦✉rs❡s ❝♦✈❡r✐♥❣ s✉❝❤ ❛r❡❛s✳ ❲❡ ♣r❡s❡♥t t❤❡ ❞❡✜♥✐t✐♦♥✱ ♣r♦♣❡rt✐❡s ❛♥❞ ♠❛✐♥ r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ t❤❡ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠ ❛♥❞ ❛❞❞r❡ss s❡✈❡r❛❧ ♣r♦❜❧❡♠s ✐♥ t❤❡ ❛r❡❛s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳
❑❡②✇♦r❞s✿ ▲❛♣❧❛❝❡ ❚r❛♥s❢♦r♠✳ ▲✐♥❡❛r ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❆♣♣❧✐❝❛t✐♦♥s✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ①✐✐✐
✶ ❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✿ ❆❧❣✉♠❛s ❉❡✜♥✐çõ❡s✱ Pr♦♣r✐❡❞❛❞❡s ❡
Pr✐♥❝✐♣❛✐s ❘❡s✉❧t❛❞♦s ✶
✶✳✶ ❍✐stór✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❡ L[f(t)] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✹ ❆ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✺ ❚❡♦r❡♠❛s ❙♦❜r❡ ❉❡s❧♦❝❛♠❡♥t♦ ❡ ❉❡r✐✈❛❞❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✶✵ ✶✳✺✳✶ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ 1.4✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✶✳✺✳✷ ❋♦r♠❛ ■♥✈❡rs❛ ❞♦ Pr✐♠❡✐r♦ ❚❡♦r❡♠❛ ❞♦ ❉❡s❧♦❝❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺✳✸ ❋✉♥çã♦ ❉❡❣r❛✉ ❯♥✐tár✐♦ ♦✉ ❋✉♥çã♦ ❞❡ ❍❡❛✈✐s✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✺✳✹ ❋♦r♠❛ ■♥✈❡rs❛ ❞♦ ❙❡❣✉♥❞♦ ❚❡♦r❡♠❛ ❞❛ ❚r❛♥s❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✻ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ❉❡r✐✈❛❞❛s✱ ■♥t❡❣r❛✐s ❡ ❋✉♥çõ❡s P❡r✐ó❞✐❝❛s ✶✼ ✶✳✻✳✶ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ✉♠❛ ❋✉♥çã♦ P❡r✐ó❞✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✼ ❋✉♥çã♦ ❉❡❧t❛ ❞❡ ❉✐r❛❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✷✻
✷✳✶ ❆♣❧✐❝❛çã♦ ♥❛ ❘❡s♦❧✉çã♦ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✶✳✶ ❙♦❧✉çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠
❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✶✳✷ ❙♦❧✉çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠
♥ã♦ ❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✶✳✸ ❙♦❧✉çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠
❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✶✳✹ ❙♦❧✉çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ▲✐♥❡❛r❡s ❞❡ ❙❡❣✉♥❞❛ ❖r✲
❞❡♠ ♥ã♦ ❍♦♠♦❣ê♥❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✶✳✺ ❙♦❧✉çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ▲✐♥❡❛r❡s ❞❡ ❖r❞❡♠ ❙✉♣❡r✐♦r ✸✽ ✷✳✷ ❆♣❧✐❝❛çã♦ ♥❛ ❘❡s♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s ❡♠ ❊♥❣❡♥❤❛r✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✸ ❆♣❧✐❝❛çã♦ ♥❛ ❘❡s♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s ❡♠ ❆✉t♦♠❛çã♦ ■♥❞✉str✐❛❧ ✳ ✳ ✳ ✹✼ ✷✳✸✳✶ ▼♦❞❡❧❛❣❡♠ ❞❡ ❙✐st❡♠❛s ❞❡ ❈♦♥tr♦❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✷✳✸✳✷ ❈♦♥str✉çã♦ ❞❛ ❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛ ♣❛r❛ ✉♠ ❈✐r❝✉✐t♦ ❘▲❈ ✺✵ ✷✳✸✳✸ ❘❡s♣♦st❛ ❞❡ ✉♠ ❙✐st❡♠❛ ❛ P❛rt✐r ❞❡ s✉❛ ❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛ ✺✷ ✷✳✹ ❖❜t❡♥çã♦ ❞❛s ❙♦❧✉çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s q✉❡ ▼♦✲
❞❡❧❛♠ ❆❧❣✉♥s ❋❡♥ô♠❡♥♦s ❡♠ ❋ís✐❝❛ ❡ ▼❛t❡♠át✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✹✳✶ ❙♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❖r❞✐♥ár✐❛ q✉❡ ▼♦❞❡❧❛ ♦ ▼♦✲
✈✐♠❡♥t♦ ❞❡ ✉♠ ❈♦r♣♦ ❡♠ ◗✉❡❞❛ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✹✳✷ ❙♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❖r❞✐♥ár✐❛ q✉❡ ▼♦❞❡❧❛ ♦ ▼♦✲
✈✐♠❡♥t♦ ❞❡ ✉♠ ❙✐st❡♠❛ ▼❛ss❛ ▼♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✷✳✹✳✸ ❙♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❖r❞✐♥ár✐❛ ❞❡ ✉♠ ❈✐r❝✉✐t♦ ❘▲❈ ✺✽ ✷✳✹✳✹ ❙♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❖r❞✐♥ár✐❛ q✉❡ ▼♦❞❡❧❛ ❛ ❉❡✲
✢❡①ã♦ ❞❡ ✉♠❛ ❱✐❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✷✳✹✳✺ ❙♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ ❖r❞✐♥ár✐❛ q✉❡ ▼♦❞❡❧❛ ♦ Pr♦✲
❜❧❡♠❛ ❞❛ ❈❛♣✐t❛❧✐③❛çã♦ ❈♦♥tí♥✉❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✷✳✹✳✻ ❙♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧ q✉❡ ▼♦❞❡❧❛ ❛ ▲❡✐ ❞♦ ❘❡s❢r✐❛✲
♠❡♥t♦ ❞❡ ◆❡✇t♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✷✳✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
❆ ✻✺
❆♣ê♥❞✐❝❡ ❆ ✻✺
❆✳✶ ❚❛❜❡❧❛ ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ❞❛s Pr✐♥❝✐♣❛✐s ❋✉♥çõ❡s ❡ s✉❛s ❘❡s♣❡❝t✐✈❛s ■♥✈❡rs❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
❇ ✻✼
❆♣ê♥❞✐❝❡ ❇ ✻✼
❇✳✶ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛1.2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ❈♦r♣♦ ❡♠ ◗✉❡❞❛ ▲✐✈r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✈ ✷ ❈✐r❝✉✐t♦ ❘▲❈ ❡♠ ❙ér✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✈✐ ✸ ❱✐❣❛ ❝♦♠ ❉❡✢❡①ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✈✐✐ ✹ ❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ①✈✐✐✐ ✶✳✶ ❋✉♥çã♦ ❉❡❣r❛✉ ❯♥✐tár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷ ●rá✜❝♦ ❞❛ ❱♦❧t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ P❧❛♥♦ ✲ tz ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✶✳✹ ❋✉♥çã♦ ❉❡♥t❡ ❞❡ ❙❡rr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶ ❱✐❣❛ ❊♥❣❛st❛❞❛ ♥♦s ❊①tr❡♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷ ❱✐❣❛ ❊♥❣❛st❛❞❛ ♥♦s ❊①tr❡♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸ ❱✐❣❛ ❊♥❣❛st❛❞❛ á ❊sq✉❡r❞❛ ❡ ▲✐✈r❡ á ❉✐r❡✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✹ ❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✺ ❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡ ❈♦♠♣♦st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✻ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ ❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛ ❡♠ ✉♠ ❉✐❣r❛♠❛ ❞❡ ❇❧♦❝♦s ✹✾ ✷✳✼ ❈✐r❝✉✐t♦ ❘▲❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✽ ❉✐❛❣r❛♠❛ ❞❡ ❇❧♦❝♦s ❞❡ ✉♠ ❈✐r❝✉✐t♦ ❘▲❈ ❡♠ ❙ér✐❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✷✳✾ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❙✐st❡♠❛ ▼❛ss❛ ▼♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✷✳✶✵ ❈✐r❝✉✐t♦ ❘▲❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
❆✳✶ ❚❛❜❡❧❛ ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ❞❛s Pr✐♥❝✐♣❛✐s ❋✉♥çõ❡s ❡ s✉❛s ❘❡s♣❡❝t✐✈❛s ■♥✈❡rs❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
■♥tr♦❞✉çã♦
P❛r❛ ❞❡s❡♥✈♦❧✈❡r♠♦s tr❛❜❛❧❤♦s ❡♠ ár❡❛s ❝♦♠♦ ▼❛t❡♠át✐❝❛✱ ❊♥❣❡♥❤❛r✐❛✱ ■♥❞ús✲ tr✐❛✱ ❊❝♦♥♦♠✐❛ ❡♥tr❡ ♦✉tr❛s✱ ❢r❡q✉❡♥t❡♠❡♥t❡✱ s♦♠♦s ❝♦♥❢r♦♥t❛❞♦s ❝♦♠ ♣r♦❜❧❡♠❛s ♦✉ ❢❡♥ô♠❡♥♦s q✉❡✱ ♣❛r❛ s❡r❡♠ ❡st✉❞❛❞♦s ♣r❡❝✐s❛♠ s❡r ❞❡s❝r✐t♦s✱ ♦✉ ♠♦❞❡❧❛❞♦s ❛tr❛✈és ❞❡ ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s✳
❙❡❣✉♥❞♦ ❇✐❡♠❜❡♥❣✉t ❡♠ ❬✶❪✿
✧❯♠ ♠♦❞❡❧♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ sí♠❜♦❧♦s ♦s q✉❛✐s ✐♥t❡r❛❣❡♠ ❡♥tr❡ s✐ r❡✲ ♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♠❛ ❝♦✐s❛✳ ❊st❛ r❡♣r❡s❡♥t❛çã♦ ♣♦❞❡ s❡ ❞❛r ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❞❡s❡♥❤♦ ♦✉ ✐♠❛❣❡♠✱ ✉♠ ♣r♦❥❡t♦✱ ✉♠ ❡sq✉❡♠❛✱ ✉♠ ❣rá✜❝♦✱ ✉♠❛ ❧❡✐ ♠❛t❡♠á✲ t✐❝❛✱ ❞❡♥tr❡ ♦✉tr❛s ❢♦r♠❛s✧✳ ◆❛ ♠❛t❡♠át✐❝❛✱ ♣♦r ❡①❡♠♣❧♦✱ ✧✉♠ ♠♦❞❡❧♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ sí♠❜♦❧♦s ❡ r❡❧❛çõ❡s ♠❛t❡♠át✐❝❛s q✉❡ tr❛❞✉③❡♠✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ✉♠ ❢❡♥ô♠❡♥♦ ❡♠ q✉❡stã♦✧✳
P❡r❝❡❜❡♠♦s ❡♥tã♦✱ q✉❡ ♠♦❞❡❧❛r ✉♠ ♣r♦❜❧❡♠❛ ♥ã♦ é s✐♠♣❧❡s✳ ❉❡✈❡♠♦s ❝♦♥❤❡❝❡r ❜❡♠ ♦s ❢✉♥❞❛♠❡♥t♦s ❞♦ ♣r♦❜❧❡♠❛ ❡st✉❞❛❞♦ ♣❛r❛ ❝❤❡❣❛r♠♦s ❛ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛t❡♠át✐❝❛ ❝♦❡r❡♥t❡✳ ➱ ✐♠♣♦rt❛♥t❡ ❞❡st❛❝❛r♠♦s t❛♠❜é♠ q✉❡✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ♠♦❞❡❧♦✱ é ♣r❡❝✐s♦ ♣❛ss❛r ♣❡❧♦ ♣r♦❝❡ss♦ ❞❡ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ q✉❡✱ s❡❣✉♥❞♦ ❇✐❡♠❜❡♥❣✉t ❡♠ ❬✶❪✱ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ❡t❛♣❛s✿
✧❛✮ ■♥t❡r❛çã♦ ❝♦♠ ♦ ❛ss✉♥t♦
◆❡st❛ ❡t❛♣❛✱ ❛ s✐t✉❛çã♦ ❛ s❡r ❡st✉❞❛❞❛ s❡rá ❞❡❧✐♥❡❛❞❛ ❡ ♣❛r❛ t♦r♥á✲❧❛ ♠❛✐s ❝❧❛r❛ ❞❡✈❡rá s❡r ❢❡✐t❛ ✉♠❛ ♣❡sq✉✐s❛ s♦❜r❡ ♦ ❛ss✉♥t♦ ❡s❝♦❧❤✐❞♦ ❛tr❛✈és ❞❡ ❧✐✈r♦s ♦✉ r❡✈✐st❛s ❡s♣❡❝✐❛❧✐③❛❞❛s✳
❜✮ ▼❛t❡♠❛t✐③❛çã♦
❊st❛ é ❛ ❢❛s❡ ♠❛✐s ❝♦♠♣❧❡①❛ ❡ ❞❡s❛✜❛❞♦r❛✱ ♣♦✐s é ♥❡st❛ q✉❡ s❡ ❞❛rá ❛ tr❛✲ ❞✉çã♦ ❞❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ❛ ❧✐♥❣✉❛❣❡♠ ♠❛t❡♠át✐❝❛✳ ❆ss✐♠✱ ❛ ✐♥t✉✐çã♦ ❡ ❛ ❝r✐❛t✐✈✐❞❛❞❡ sã♦ ❡❧❡♠❡♥t♦s ✐♥❞✐s♣❡♥sá✈❡✐s✳
❝✮ ▼♦❞❡❧♦ ♠❛t❡♠át✐❝♦
❖ ♠♦❞❡❧♦ ❝♦♥❝❧✉í❞♦ ❞❡✈❡rá ❝♦rr❡s♣♦♥❞❡r à s✐t✉❛çã♦✲♣r♦❜❧❡♠❛ ❛♣r❡s❡♥✲ t❛❞❛✧✳
P♦rt❛♥t♦✱ ♣❛r❛ ❢❛③❡r♠♦s ❡ss❛ ♠♦❞❡❧❛❣❡♠✱ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ❛s ✈❛r✐á✈❡✐s q✉❡ ✐♥✢✉❡♥❝✐❛♠ ♦ ♣r♦❜❧❡♠❛ ❢❛③❡♥❞♦ ♦ s✐st❡♠❛ s♦❢r❡r ✈❛r✐❛çõ❡s✱ ❝♦♠♦ t❛♠❜é♠ ♦ ❝♦♥❥✉♥t♦
❞❡ ❤✐♣ót❡s❡s ❧❡✈❛♥t❛❞❛s s♦❜r❡ ❛s ❝♦♥❞✐çõ❡s ❛♣r❡s❡♥t❛❞❛s ✐♥✐❝✐❛❧♠❡♥t❡ ♣❡❧♦ s✐st❡♠❛ ❛♥❛❧✐s❛❞♦✳ ❆ ❡str✉t✉r❛ ♠❛t❡♠át✐❝❛ ❛♣r❡s❡♥t❛❞❛ ♥❛s ❤✐♣ót❡s❡s sã♦ ❢❡rr❛♠❡♥t❛s ❡s✲ s❡♥❝✐❛✐s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦✱ q✉❡ s❡rá ❛ ❝❤❛✈❡ ♣❛r❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✳
❊♠ ♠✉✐t♦s ❝❛s♦s✱ ♦s ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s sã♦ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ♦✉ ❡q✉❛✲ çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♦✉ ♠❡s♠♦ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ♦✉ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ❆s ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s sã♦ ♠♦❞❡❧♦s ❣❡r❛❞♦s q✉❛♥❞♦ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♣r♦❜❧❡♠❛s ❡♠ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛✱ ❡♥q✉❛♥t♦ q✉❡ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s sã♦ ♠♦❞❡❧♦s ❣❡r❛❞♦s q✉❛♥❞♦ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♣r♦❜❧❡♠❛s ❡♠ ▼❛t❡♠át✐❝❛ ❈♦♥tí♥✉❛✳
➱ ✐♠♣♦rt❛♥t❡ ❞❡st❛❝❛r♠♦s ❛q✉✐ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ❡ ❈♦♥✲ tí♥✉❛✳ ❆ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ❡st✉❞❛ ♣r♦❜❧❡♠❛s ❞❡s❡♥✈♦❧✈✐❞♦s ❡♠ ❝♦♥❥✉♥t♦s q✉❡ ❣❡r❛❧♠❡♥t❡ sã♦ ✜♥✐t♦s ❡ ❡♥✉♠❡rá✈❡✐s ❡ ❛ ▼❛t❡♠át✐❝❛ ❈♦♥tí♥✉❛ ❡st✉❞❛ ♣r♦❜❧❡♠❛s ❞❡s❡♥✈♦❧✈✐❞♦s ❡♠ s✉❜❝♦♥❥✉♥t♦s ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
P❛r❛ ❙❝❤❡✐♥❡r♠❛♥ ❡♠ ❬✷❪✱ ❡ss❛ ❞✐❢❡r❡♥ç❛ é ❞❛❞❛ ❝♦♠♦ s❡ s❡❣✉❡✿
✧❆ ▼❛t❡♠át✐❝❛ ❈♦♥tí♥✉❛ ❝♦rr❡s♣♦♥❞❡ ❛ r❡❧ó❣✐♦s ❛♥❛❧ó❣✐❝♦s ❬✳✳✳❪ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ✉♠ r❡❧ó❣✐♦ ❛♥❛❧ó❣✐❝♦✱ ❡♥tr❡ 12 : 02pm❡ 12 : 03pm❤á ✉♠ ♥ú♠❡r♦
✐♥✜♥✐t♦ ❞❡ ❞✐❢❡r❡♥t❡s t❡♠♣♦s ♣♦ssí✈❡✐s✱ ♥❛ ♠❡❞✐❞❛ q✉❡ ♦ r❡❧ó❣✐♦ ♣❡r❝♦rr❡ ♦ ♠♦str❛❞♦r ❬✳✳✳❪✳ ❆ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ é ❝♦♠♣❛r❛❞❛ ❛ ✉♠ r❡❧ó❣✐♦ ❞✐❣✐t❛❧✱ ❡♠ q✉❡ ❤á ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ♣♦ssí✈❡❧ ❞❡ t❡♠♣♦s ❞✐❢❡r❡♥t❡s ❡♥tr❡12 : 02pm
❡ 12 : 03pm✳ ❯♠ r❡❧ó❣✐♦ ❞✐❣✐t❛❧ ♥ã♦ r❡❝♦♥❤❡❝❡ ❢✉♥çã♦ ❞❡ s❡❣✉♥❞♦s ❬✳✳✳❪ ❛
tr❛♥s✐çã♦ ❞❡ ✉♠ t❡♠♣♦ ♣❛r❛ ♦ ♣ró①✐♠♦ é ❜❡♠ ❞❡✜♥✐❞❛ ❡ s❡♠ ❛♠❜✐❣✉✐❞❛❞❡✧✳
P❛r❛ tr❛❜❛❧❤❛r ❝♦♠ ♣r♦❜❧❡♠❛s ❡♠ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛✱ ♦✉ s❡❥❛✱ ♣r♦❜❧❡♠❛s ❝✉❥♦ ♠♦❞❡❧♦ é ✉♠❛ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s ♦✉ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çã♦ ❞❡ ❞✐❢❡r❡♥ç❛s✱ ✉s❛✲ s❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❩ ♣❛r❛ ❝♦♥✈❡rt❡r ❛s ❡q✉❛çõ❡s ❣❡r❛❞❛s ♥♦ ♠♦❞❡❧♦ ❡♠ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s✱ ♦ q✉❡ t♦r♥❛ ♠❛✐s s✐♠♣❧❡s s✉❛ ❛♥á❧✐s❡✳ ❏á ♥♦ ❝❛s♦ ❞♦s ♣r♦❜❧❡♠❛s ❡♠ ▼❛t❡♠át✐❝❛ ❈♦♥tí♥✉❛✱ ❝✉❥♦ ♠♦❞❡❧♦ ❣❡r❛❧♠❡♥t❡ é ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦✉ ✉♠ s✐st❡♠❛ ❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ✉s❛✲s❡✱ ❣❡r❛❧♠❡♥t❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ tr❛♥s❢♦r♠❛r ♦ ♠♦❞❡❧♦ ❡♠ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛✳
❙❡❣✉♥❞♦ ❱❡♥t✉r✐ ❡♠ ❬✹❪✱ ❍s✉ ❝♦♠♣❛r❛ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❩ ❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❆ ❚r❛♥s❢♦r♠❛❞❛ ❩ é ✐♥tr♦❞✉③✐❞❛ ♣❛r❛ r❡♣r❡s❡♥t❛r s✐♥❛✐s ❞❡ t❡♠♣♦ ❞✐s❝r❡t♦ ✭♦✉ s❡q✉ê♥❝✐❛s✮ ♥♦ ❞♦♠í♥✐♦ ③✱ ♦♥❞❡ ③ é ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛✳ ❆ ❚r❛♥s❢♦r✲ ♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❝♦♥✈❡rt❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡♠ ❡q✉❛çõ❡s ❛❧❣é❜r✐❝❛s ✭♠✉✐✲ t❛s ✈❡③❡s ♣r❡s❡♥t❡ ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ❢❡♥ô♠❡♥♦s ♠❡❝â♥✐❝♦s ❡ ❡❧étr✐❝♦s✮✳ ❉❡ ✉♠ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❩ ❝♦♥✈❡rt❡ ❡q✉❛çõ❡s ❞❡ ❞✐❢❡r❡♥ç❛s ❡♠ ❡q✉❛✲ çõ❡s ❛❧❣é❜r✐❝❛s✱ s✐♠♣❧✐✜❝❛♥❞♦ ❛ss✐♠ ❛ ❛♥á❧✐s❡ ❞❡ s✐st❡♠❛s ❞✐s❝r❡t♦s ♥♦ t❡♠♣♦ ✭♣r❡s❡♥t❡ ❡♠ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♥tr♦❧❡✱ ♣r♦❣r❡ssõ❡s ❡ s✉❝❡ssõ❡s✮✳
❱❛❧❡ s❛❧✐❡♥t❛r✱ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ✈✐s❛ ✉s❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❝♦♠♦ ♠ét♦❞♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❝✉❥♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❣❡r❛ ❡q✉❛çõ❡s ❞✐❢❡✲ r❡♥❝✐❛✐s✳
❱❡❥❛♠♦s✱ ❛ s❡❣✉✐r✱ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❝♦♥❤❡❝✐❞♦s✱ ❝✉❥♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ❣❡r❛ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✳ ❊st❡s ♣r♦❜❧❡♠❛s sã♦ ❛❞❛♣t❛❞♦s ❞❡ ❬✺✱ ✻❪✳
Pr♦❜❧❡♠❛ ✶✿
❈♦r♣♦ ❡♠ ◗✉❡❞❛ ▲✐✈r❡
❙✉♣♦♥❤❛ q✉❡ ✉♠ ♦❜❥❡t♦ s❡❥❛ ❧❛♥ç❛❞♦ ❞♦ ❛❧t♦ ❞❡ ✉♠ ♣ré❞✐♦ ❞❡ ❛❧t✉r❛h✱ ❡♠ q✉❡❞❛
❧✐✈r❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✐♥✐❝✐❛❧ v0✳
❋✐❣✉r❛ ✶✿ ❈♦r♣♦ ❡♠ ◗✉❡❞❛ ▲✐✈r❡
❙❛❜❡♠♦s q✉❡ ❞✉r❛♥t❡ ❛ q✉❡❞❛ ♦ ♦❜❥❡t♦ ♣♦ss✉✐ ❛❝❡❧❡r❛çã♦ ❝♦♥st❛♥t❡g✭❣r❛✈✐❞❛❞❡✮✳
▲❡♠❜r❛♥❞♦ q✉❡ ❛ ❛❝❡❧❡r❛çã♦ é ❛ ❞❡r✐✈❛❞❛ ❞❛ ✈❡❧♦❝✐❞❛❞❡✱ ❝♦♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦✱ q✉❡ ♣♦r s✉❛ ✈❡③ é ❛ ❞❡r✐✈❛❞❛ ❞❛ ♣♦s✐çã♦ ❞♦ ♦❜❥❡t♦ ❡♠ r❡❧❛çã♦ ❛♦ t❡♠♣♦ t✱ ❡♥tã♦ s✉❛
tr❛❥❡tór✐❛ é ❞❡s❝r✐t❛ ♣♦r ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❛❞❛ ♣♦r
d2s
dt2 =−g, 0< t0 ≤t,
♦♥❞❡ t0 é ♦ ✐♥st❛♥t❡ ❡♠ q✉❡ ♦ ♦❜❥❡t♦ ❢♦✐ ❧❛♥ç❛❞♦✱ s ❞❡s❝r❡✈❡ ❛s ♣♦s✐çõ❡s ❞♦ ♦❜❥❡t♦
❞✉r❛♥t❡ ❛ q✉❡❞❛✱ ❡ ♦ s✐♥❛❧ ♥❡❣❛t✐✈♦ ❞❡ g é ♣♦rq✉❡ ♦ ♣❡s♦ ❞❡ ✉♠ ❝♦r♣♦ é ✉♠❛ ❢♦rç❛
❞✐r❡❝✐♦♥❛❞❛ ♣❛r❛ ❜❛✐①♦✱ ♦♣♦st❛ ❛♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦ ❞❛ tr❛❥❡tór✐❛✳
Pr♦❜❧❡♠❛ ✷✿
❈✐r❝✉✐t♦ ❡♠ ❙ér✐❡
❙❡❥❛ ✉♠ ❝✐r❝✉✐t♦ ❡❧étr✐❝♦ s✐♠♣❧❡s ❝♦♥t❡♥❞♦ ✉♠ ✐♥❞✉t♦r L✱ ✉♠ ❝❛♣❛❝✐t♦r C ❡ ✉♠
r❡s✐st♦r R✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ ✷✱ ❛❞❛♣t❛❞❛ ❞❡ ❬✶✵❪✳ ❚❡♠♦s q✉❡✱ ❛ ❞✐❢❡r❡♥ç❛ ❞❡
♣♦t❡♥❝✐❛❧ U ❡♠ ❝❛❞❛ ✉♠ ❞♦s ❡❧❡♠❡♥t♦s ❞♦ ❝✐r❝✉✐t♦ sã♦ ❞❛❞❛s ♣♦r
Uindutor =
Ldi
dt , Uresistor=Ri e Ucapacitor = q C,
♦♥❞❡ i é ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ❝♦rr❡♥t❡ ❡❧étr✐❝❛ ❡ q é ❛ ❝❛r❣❛ ❡❧étr✐❝❛✳
❋✐❣✉r❛ ✷✿ ❈✐r❝✉✐t♦ ❘▲❈ ❡♠ ❙ér✐❡
❆ ❙❡❣✉♥❞❛ ▲❡✐ ❞❡ ❑✐r❝❝❤♦✛ ❛✜r♠❛ q✉❡ ❛ ❞✐❢❡r❡♥ç❛ ❞❡ ♣♦t❡♥❝✐❛❧ v(t) ❡♠ ✉♠
❝✐r❝✉✐t♦ ❢❡❝❤❛❞♦ é ❛ s♦♠❛ ❞❛s ✈♦❧t❛❣❡♥s ♥♦ ❝✐r❝✉✐t♦✳ P♦rt❛♥t♦✱
v(t) =Uindutor +Uresistor+Ucapacitor =Ldi
dt +Ri+ q C.
.
❈♦♠♦ i= dq
dt✱ ❡♥tã♦ di dt =
d2q
dt2✳ P♦rt❛♥t♦✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ú❧t✐♠❛ ❡q✉❛çã♦✱ ♦❜t❡♠♦s
v(t) =Ld
2q
dt2 +R
dq dt +
q C,
❛ q✉❛❧ é ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳
Pr♦❜❧❡♠❛ ✸✿
▲❡✐ ❞♦ ❘❡s❢r✐❛♠❡♥t♦ ❞❡ ◆❡✇t♦♥
❙❡❥❛T(t)❛ t❡♠♣❡r❛t✉r❛ ❞❡ ✉♠ ❝♦r♣♦ ❡♠ ✉♠ ✐♥st❛♥t❡t✳ ❙❡ ❝♦❧♦❝❛r♠♦s ♦ ♠❡s♠♦
❡♠ ✉♠ ❛♠❜✐❡♥t❡ ❡♠ q✉❡ ❛ t❡♠♣❡r❛t✉r❛ é TA ❡ ❝♦♥s✐❞❡r❛r♠♦s q✉❡ dTdt r❡♣r❡s❡♥t❛
❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞❡ s✉❛ t❡♠♣❡r❛t✉r❛ ❝♦♠ r❡s♣❡✐t♦ ❛♦ t❡♠♣♦✱ ❡♥tã♦ ❛ ▲❡✐ ❞♦ ❘❡s❢r✐❛♠❡♥t♦ ❞❡ ◆❡✇t♦♥ é ❞❡s❝r✐t❛ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ♣❡❧❛ ❡q✉❛çã♦
dT
dt =k(T −TA),
q✉❡ é ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❖ ♣❛râ♠❡tr♦ k q✉❡ ❛♣❛r❡❝❡ ♥❛
❡q✉❛çã♦ é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡✳ ❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡✱ ♦ ❝♦r♣♦ ❡stá
❡s❢r✐❛♥❞♦✱ ❡♥tã♦ T > TA✱ ♦ q✉❡ ✐♠♣❧✐❝❛ k <0✳
Pr♦❜❧❡♠❛ ✹✿
❉❡✢❡①ã♦ ❉❡ ❱✐❣❛s
❈♦♥s✐❞❡r❡ ✉♠❛ ✈✐❣❛ ✉♥✐❢♦r♠❡✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ L✱ ❝♦♠ ❡①tr❡♠✐❞❛❞❡ ❡sq✉❡r❞❛
✜①❛ ❡♠ ✉♠ s✉♣♦rt❡ ❡ s✉❛ ❡①tr❡♠✐❞❛❞❡ ❞✐r❡✐t❛ s♦❧t❛✱ ❝♦♠♦ ♠♦str❛ ❛ ❋✐❣✉r❛ 3✳ ❙✉❛
❋✐❣✉r❛ ✸✿ ❱✐❣❛ ❝♦♠ ❉❡✢❡①ã♦
❉❡✢❡①ã♦ ❊❧ást✐❝❛D(x)♣❛r❛ s✉♣♦rt❛r ✉♠❛ ❝❛r❣❛W(x),♣♦r ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱
é ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ q✉❛rt❛ ♦r❞❡♠
EId
4D
dx4 =W(x),
♦♥❞❡ E é ♦ ♠ó❞✉❧♦ ❞❛ ❡❧❛st✐❝✐❞❛❞❡ ❞❡ ❨♦✉♥❣ ❡ I é ♦ ♠♦♠❡♥t♦ ❞❡ ✐♥ér❝✐❛ ❞❡ ✉♠❛
s❡çã♦ tr❛♥s✈❡rs❛❧ ❞❛ ✈✐❣❛✳
Pr♦❜❧❡♠❛ ✺✿
❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡
❯♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ é r❡♣r❡s❡♥t❛❞♦ ♣♦r s✉❜s✐st❡♠❛s ♦✉ ♣❧❛♥t❛s q✉❡ sã♦ ❝♦♥s✲ tr✉í❞♦s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❝♦♥s❡❣✉✐r ✉♠❛ s❛í❞❛ ❞❡s❡❥❛❞❛ ❝♦♠ ✉♠ ❞❡s❡♠♣❡♥❤♦ ❞❡s❡✲ ❥❛❞♦✱ ♣❛r❛ ✉♠❛ ❡♥tr❛❞❛ ❡s♣❡❝í✜❝❛ ❞❡✜♥✐❞❛✳
◆❛ ❋✐❣✉r❛ 4✱ t❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ ❡♠ s✉❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✳
❋✐❣✉r❛ ✹✿ ❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡ ❙✐♠♣❧❡s
❆ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❧✐♥❡❛r ❞❡ ❡♥és✐♠❛ ♦r❞❡♠✱
an
dnc(t)
dtn +an−1
dn−1c(t)
dtn−1 +...+aoc(t) = bm
dmr(t)
dtm +bm−1
dm−1r(t)
dtm−1 +...+bor(t),
❞❡s❝r❡✈❡ ♠✉✐t♦s s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡✱ ✐♥✈❛r✐❛♥t❡ ♥♦ t❡♠♣♦✱† ❝♦♠♦ r❡♣r❡s❡♥t❛❞♦ ♥❛
❋✐❣✉r❛ 4.
◆❡st❛ ❡q✉❛çã♦ t❡♠♦sc(t)❛ s❛í❞❛✱r(t)❛ ❡♥tr❛❞❛✱ai ❝♦♠ i= 0,1,2, ..., n❡bi ❝♦♠
i= 0,1,2, ..., m✱ sã♦ ♣❛râ♠❡tr♦s ❞♦ s✐st❡♠❛✳
◆♦s ú❧t✐♠♦s ❛♥♦s✱ ❛♣❡s❛r ❞❡ ❡st❛r♠♦s ♣❛ss❛♥❞♦ ♣♦r ✉♠❛ ❝r✐s❡ ❡❝♦♥ô♠✐❝❛✱ ♦ ❇r❛s✐❧ ♣❛ss♦✉ ♣♦r ✉♥s ♠♦♠❡♥t♦s ❡♠ q✉❡ ❤♦✉✈❡ ✉♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♥♦tá✈❡❧ ♥❛s ár❡❛s ❞❛ ■♥❞ústr✐❛✱ ❈♦♥str✉çã♦ ❈✐✈✐❧ ❡ ❊❞✉❝❛çã♦✳ ❍♦✉✈❡ ✉♠❛ ❡①♣❛♥sã♦ ❞❛s ✉♥✐✈❡rs✐❞❛❞❡s Pú❜❧✐❝❛s ❡ ■♥st✐t✉t♦s ❋❡❞❡r❛✐s ❞❡ ❊♥s✐♥♦✱ ❞♦s s✐st❡♠❛s ❞❡ s❛ú❞❡ ❡ ❞✐✈❡rs♦s ♦✉tr♦s s❡t♦r❡s✳ ❈♦♠ ❡ss❡ ❝r❡s❝✐♠❡♥t♦✱ s✉r❣✐✉ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♠ã♦ ❞❡ ♦❜r❛ q✉❛❧✐✜❝❛❞❛ ♣❛r❛ tr❛❜❛❧❤❛r ♥❡ss❡s s❡t♦r❡s ❡ ❛s ❯♥✐✈❡rs✐❞❛❞❡s ❡ ■♥st✐t✉t♦s ❋❡❞❡r❛✐s ♣❛ss❛r❛♠ ❛ ❝r✐❛r ♥♦✈♦s ❝✉rs♦s ♥❛s ár❡❛s té❝♥✐❝❛s ❡ t❡❝♥♦❧ó❣✐❝❛s✱ ❝♦♠♦ t❛♠❜é♠ ♥❛ ár❡❛ ❞❡ ❢♦r♠❛çã♦ ❞❡ ♣r♦❢❡ss♦r❡s✱ ❝✉❥♦ ♦❜❥❡t✐✈♦ é ❛ q✉❛❧✐✜❝❛çã♦ ❞❡ ♣❡ss♦❛s ♣❛r❛ s✉♣r✐r ❛s ♥❡❝❡ss✐❞❛❞❡s ❡①✐❣✐❞❛s ♣❡❧♦ ♠❡r❝❛❞♦ ❞❡ tr❛❜❛❧❤♦✳
❆ ❡①❡♠♣❧♦ ❞✐ss♦✱ ♦ ■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ P❛r❛í❜❛✱ ❈❛♠✲ ♣✉s ❈❛❥❛③❡✐r❛s✱ ❝r✐♦✉ ♦s ❝✉rs♦s ❞❡ ❆✉t♦♠❛çã♦ ■♥❞✉str✐❛❧✱ ❆♥á❧✐s❡s ❞❡ ❙✐st❡♠❛s✱ ▼❛✲ t❡♠át✐❝❛ ❡ ❊♥❣❡♥❤❛r✐❛ ❈✐✈✐❧✱ ❛❧é♠ ❞♦s ❝✉rs♦s té❝♥✐❝♦s ❞❡ ❊❧❡tr♦♠❡❝â♥✐❝❛ ❡ ❊❞✐✜❝❛✲ çõ❡s✳ ❆ ♠❛✐♦r✐❛ ❞❡ss❡s ❝✉rs♦s tr❛❜❛❧❤❛ ❡♠ s✉❛s ❡♠❡♥t❛s ❝♦♠ ♣r♦❜❧❡♠❛s ❛❞✈✐♥❞♦s ❞❛ ▼❛t❡♠át✐❝❛ ❈♦♥tí♥✉❛✳ ❈♦♠♦ ❞✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❡st❡s ♣r♦❜❧❡♠❛s sã♦ ♠♦❞❡❧❛❞♦s ♣♦r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ♦ ♦❜❥❡t✐✈♦ ❣❡r❛❧ ❞♦ tr❛❜❛❧❤♦ é✿
• ❝♦♠♣r❡❡♥❞❡r ♦s ❝♦♥❝❡✐t♦s ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✉t✐❧✐③❛♥❞♦✲♦s ❝♦♠♦ ♠é✲
t♦❞♦ ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s q✉❡ s✉r❣❡♠ ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ár❡❛s ❝♦♠♦ ❛ ▼❛t❡♠át✐❝❛✱ ❋ís✐❝❛ ❊♥❣❡♥❤❛r✐❛ ❡ ■♥❞ústr✐❛✳ ❖s ♦❜❥❡t✐✈♦s ❡s♣❡❝í✜❝♦s sã♦✿
† ❯♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ é ✐♥✈❛r✐❛♥t❡ ♥♦ t❡♠♣♦✱ q✉❛♥❞♦ ❛ s❛í❞❛c(t)♥ã♦ ❞❡♣❡♥❞❡ ❞♦ ✐♥st❛♥t❡ ❡♠
q✉❡ ❛ ❡♥tr❛❞❛r(t)é ❛♣❧✐❝❛❞❛✳
• ❉❡✜♥✐r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ❞❡s❝r❡✈❡♥❞♦ s✉❛s ♣r✐♥❝✐♣❛✐s ♣r♦♣r✐❡❞❛❞❡s
❡ r❡s✉❧t❛❞♦s❀
• ❆♣❧✐❝❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♣❛r❛ s♦❧✉❝✐♦♥❛r ❛❧❣✉♥s
t✐♣♦s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s q✉❡ s✉r❣❡♠ ❡♠ ♠♦❞❡❧❛❣❡♥s ❞❡ ♣r♦✲ ❜❧❡♠❛s ♥❛ ❊♥❣❡♥❤❛r✐❛ ❡ ■♥❞ústr✐❛❀
• ❖❜t❡r s♦❧✉çõ❡s ❞❡ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ♥❛s ár❡❛s ❞❛ ▼❛t❡♠át✐❝❛ ❡ ❞❛ ❋ís✐❝❛✱ ❝✉❥♦s
♠♦❞❡❧♦s sã♦ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳
❖ tr❛❜❛❧❤♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❝♦♠ ❛ s❡❣✉✐♥t❡ ❡str✉t✉r❛✿ ✉♠❛ ✐♥tr♦❞✉çã♦✱ ❞♦✐s ❝❛♣ít✉❧♦s ❡ ❞♦✐s ❛♣ê♥❞✐❝❡s✳
◆❛ ■♥tr♦❞✉çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ❡ ❈♦♥tí♥✉❛✱ ❛❜♦r❞❛♠♦s ❛❧❣✉♥s ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s q✉❡ ❣❡r❛♠ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡ ❞❡st❛❝❛✲ ♠♦s ♦s ♦❜❥❡t✐✈♦s ❣❡r❛✐s ❡ ❡s♣❡❝í✜❝♦s ❞♦ tr❛❜❛❧❤♦✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ✜③❡♠♦s ✉♠ ❡st✉❞♦ ❡s♣❡❝í✜❝♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ❛❜♦r❞❛♥❞♦ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s✱ t❡♦r❡♠❛s ❡ r❡s♦❧✉çã♦ ❞❡ ❡①❡♠♣❧♦s✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ❛♣❧✐❝❛çõ❡s✱ ❛❜♦r❞❛♥❞♦ ♣r♦❜❧❡♠❛s ❞❛s ár❡❛s ❞❡ ▼❛t❡♠át✐❝❛✱ ❊♥❣❡♥❤❛r✐❛✱ ❋ís✐❝❛ ❡ ■♥❞ústr✐❛✱ ❡♠ q✉❡ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ s♦❧✉❝✐♦♥á✲❧♦s✳
◆♦ ❆♣ê♥❞✐❝❡ ❆✱ t❡♠♦s ✉♠❛ t❛❜❡❧❛ ❞❛s ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ❞❛s ♣r✐♥❝✐♣❛✐s ❢✉♥çõ❡s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❚r❛♥s❢♦r♠❛❞❛s ■♥✈❡rs❛s ❡ ♥♦ ❆♣ê♥❞✐❝❡ ❇✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ 1.2✳
❈❛♣ít✉❧♦ ✶
❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✿
❆❧❣✉♠❛s ❉❡✜♥✐çõ❡s✱ Pr♦♣r✐❡❞❛❞❡s ❡
Pr✐♥❝✐♣❛✐s ❘❡s✉❧t❛❞♦s
❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ ❝❛♣ít✉❧♦ t❡♠ ❝♦♠♦ ❜❛s❡ ❛s r❡❢❡rê♥❝✐❛s ❬✺✱ ✼✱ ✽❪✳
✶✳✶ ❍✐stór✐❝♦
❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡† é ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ♦♣❡r❛❞♦r ✐♥✲
t❡❣r❛❧ ❛ ✉♠❛ ❢✉♥çã♦ f q✉❡ ❣❡r❛❧♠❡♥t❡ t❡♠ s❡✉ ❞♦♠í♥✐♦ ✈❛r✐❛♥❞♦ ♥♦ t❡♠♣♦✳
❆♣❡s❛r ❞♦ ♦♣❡r❛❞♦r s❡r ❜❛t✐③❛❞♦ ❝♦♠ ❡ss❡ ♥♦♠❡✱ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡♠ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❡ ❣r❛♥❞❡s ♠❛t❡♠át✐❝♦s ❡ ❢ís✐❝♦s✳
❖ s✉r❣✐♠❡♥t♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ✐♥t❡❣r❛❧ ❛♣❛r❡❝❡✉✱ ♣r✐♠❡✐r❛♠❡♥t❡✱ ❡♠ tr❛❜❛❧❤♦s ❞❡ ▲❡♦♥❤❛r❞ ❊✉❧❡r✱ q✉❡ ❝♦♥s✐❞❡r❛✈❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ♣❛r❛ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✳ ❖ ♣ró♣r✐♦ ▲❛♣❧❛❝❡ ❡♠ ✉♠ ❞❡ s❡✉s tr❛❜❛❧❤♦s ✐♥t✐t✉❧❛❞♦ ❞❡ ❚❤é♦r✐❡ ❆♥❛❧②t✐q✉❡ ❞❡s Pr♦❜❛❜✐❧✐t❡s✱ ♣✉❜❧✐❝❛❞♦ ❡♠1812✱ ❝✐t♦✉ ❊✉❧❡r ❝♦♠♦ ♦ ♣r✐♠❡✐r♦ ❛ ✐♥tr♦❞✉③✐r ♦ ✉s♦ ❞❡ss❛ tr❛♥s❢♦r♠❛❞❛✳ ❆♣❡♥❛s
❡♠1878é q✉❡ ♦ ♠ét♦❞♦ ✐♥t❡❣r❛❧ é ❜❛t✐③❛❞♦ ❝♦♠ ♦ ♥♦♠❡ ✧❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✧✳
■♥✐❝✐❛❧♠❡♥t❡✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❢♦✐ ❞❛❞❛ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥❡❣❛t✐✈♦s✳ ▼❛s ♥♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳■❳✱ P♦✐♥❝❛ré ❡ P✐♥❝❤❡r❧❡ ❞❡✜♥✐r❛♠ ❛ tr❛♥s❢♦r♠❛❞❛ ♥❛ ❢♦r♠❛ ❝♦♠♣❧❡①❛ ❡ ❛✐♥❞❛ ♠❛✐s t❛r❞❡ ❢♦✐ ❡st❡♥❞✐❞❛ ♣❛r❛ ❢✉♥çõ❡s ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s ♣♦r P✐❝❛r❞✳
❙❡❣✉♥❞♦ P❛❝❤❡❝♦ ❡♠ ❬✼❪✱ ♥♦ tr❛❜❛❧❤♦ ❞❡ ❇❛t❡♠❛♥(1910)❡stá ♣r❡s❡♥t❡ ❛ ♣r✐♠❡✐r❛
❛♣❧✐❝❛çã♦ ♠♦❞❡r♥❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✱ ♦♥❞❡ s❡ tr❛♥s❢♦r♠❛♠ ❡q✉❛çõ❡s † P✐❡rr❡ ❙✐♠♦♥ ▼❛rq✉✐s ❞❡ ▲❛♣❧❛❝❡ (1749−1827) ❆♣❡s❛r t❡r ♥❛s❝✐❞♦ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❢r❛♥❝❡s❛
❞❡ ❝❧❛ss❡ ❜❛✐①❛✱ t♦r♥♦✉✲s❡ ✉♠ r❡♥♦♠❛❞♦ ♠❛t❡♠át✐❝♦✱ ❢ís✐❝♦ ❡ ❛strô♥♦♠♦ ♣♦r t❡r ❡s❝r✐t♦ ✐♠♣♦rt❛♥t❡s tr❛❜❛❧❤♦s ♥❡ss❛s ár❡❛s✳ ■♥tr♦❞✉③✐✉ ❛ tr❛♥s❢♦r♠❛❞❛ q✉❡ ❧❡✈♦✉ s❡✉ ♥♦♠❡ ❡♠ ✉♠ tr❛❜❛❧❤♦ s♦❜r❡ t❡♦r✐❛ ❞❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s✳
✶✳✷✳ ❉❡✜♥✐çã♦
♦r✐✉♥❞❛s ❞♦ tr❛❜❛❧❤♦ ❞❡ ❘✉t❤❡r❢♦r❞ s♦❜r❡ ❞❡❝❛✐♠❡♥t♦ r❛❞✐❛t✐✈♦✱ r❡❣✐❞♦ ♣❡❧❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛
dP
dt =−λiP,
♣♦r ♠❡✐♦ ❞❡
P(s) = Z ∞
0
e−stP(t)dt, ✭✶✳✶✮
❝❤❡❣❛♥❞♦✱ ❛ss✐♠✱ ❛ ❡q✉❛çã♦ ❞❡s❡❥❛❞❛✳ ❊♠ (1.1) P(t)✱ é ❛ ❢✉♥çã♦ q✉❡ ❞❡s❝r❡✈❡ ♦ ♥ú✲
♠❡r♦ ❞❡ ♥ú❝❧❡♦s r❛❞✐♦❛t✐✈♦s ♣r❡s❡♥t❡ ❡♠ ✉♠❛ ❛♠♦str❛ ❛♣ós ✉♠ ❞❡t❡r♠✐♥❛❞♦ t❡♠♣♦✳ ❊♠ ❬✼❪✱ P❛❝❤❡❝♦ ❝✐t❛ ♦ tr❛❜❛❧❤♦ ❞❡ ❖❧✐✈❡r ❍❡❛✈✐s✐❞❡✱ ❞❡s❡♥✈♦❧✈✐❞♦ ♥❛ ár❡❛ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛✱ ❝✉❥♦ tít✉❧♦ é ❊❧❡tr♦♠❛❣♥❡t✐❝ ❚❤❡♦r②✱ ♦♥❞❡ sã♦ ❛♣r❡s❡♥t❛❞❛s té❝♥✐❝❛s ✉s❛♥❞♦ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ ❛✉①✐❧✐❛r ❡♥❣❡♥❤❡✐r♦s ❡❧❡tr✐❝✐st❛s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞❡s❡♥✈♦❧✈✐❞♦s ❡♠ s✉❛s ♣❡sq✉✐s❛s✳ ❊st❡ tr❛❜❛❧❤♦ ❝♦♠♣❧❡t❛ ♦ ♠ét♦❞♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ três ✈♦❧✉♠❡s✱ q✉❡ ❢♦r❛♠ ♣✉❜❧✐❝❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥♦s ❛♥♦s1894✱1899❡1914✳ ❍♦❥❡ ❤á ✉♠❛ ✈❛st❛ ❛♣❧✐❝❛✲
çã♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡♠ tr❛❜❛❧❤♦s ❞❡s❡♥✈♦❧✈✐❞♦s ❡♠ ✈ár✐❛s ár❡❛s ❝♦♠♦ ❛ ❊♥❣❡♥❤❛r✐❛✱ ❛ ❊❝♦♥♦♠✐❛✱ ❛ ■♥❞ústr✐❛ ❡ ♥❛ ♣ró♣r✐❛ ▼❛t❡♠át✐❝❛✳
✶✳✷ ❉❡✜♥✐çã♦
❆ ♠❛✐♦r ♣❛rt❡ ❞♦s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡ ♦ ✉s♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ t❡♠ ❝♦♠♦ ✈❛r✐❛çã♦ ♣r✐♥❝✐♣❛❧ ♦ t❡♠♣♦✳ ❆ss✐♠✱ é ❝♦♥✈❡♥✐❡♥t❡ ❞❡✜♥✐✲❧❛ ❝♦♠ ✉♠ ❞♦♠í♥✐♦ t
t❛❧ q✉❡ t ∈[0,∞).
❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛ f : [0,∞)→R ✉♠❛ ❢✉♥çã♦✳ ❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡f✱
q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r L[f(t)] ♦✉ L(f), é ❛ ❢✉♥çã♦
F(s) = Z ∞
0
e−stf(t)dt, ✭✶✳✷✮
♣❛r❛ t♦❞♦ s≥0 t❛❧ q✉❡ q✉❡ ❛ ✐♥t❡❣r❛❧ ❝♦♥✈✐r❥❛✳
❆ss✐♠✱ q✉❛♥❞♦ ❛ ✐♥t❡❣r❛❧ ✐♠♣ró♣r✐❛(1.2)❝♦♥✈❡r❣❡✱ ♦ r❡s✉❧t❛❞♦ é ✉♠❛ ❢✉♥çã♦ ❞❡
s✳ P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r s ∈ R ♦✉ s ∈ C✱ ✐ss♦ ❞❡♣❡♥❞❡rá ❞❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛ ❡♠
q✉❡ ♦ r❡s✉❧t❛❞♦ ❢♦r ❛♣❧✐❝❛❞♦✳
❆q✉✐✱ ✉s❛♠♦s ❛s ❧❡tr❛s ♠✐♥ús❝✉❧❛s ♣❛r❛ ✐♥❞✐❝❛r ❛ ❢✉♥çã♦ ❛ s❡r tr❛♥s❢♦r♠❛❞❛ ❡ ❛s ❧❡tr❛s ♠❛✐ús❝✉❧❛s ♣❛r❛ ❞❡♥♦t❛r ❛ ❢✉♥çã♦ tr❛♥s❢♦r♠❛❞❛✱ ♦✉ s❡❥❛✱ L[f(t)] = F(s).
❈♦♠♦ ❛ ✐♥t❡❣r❛❧ (1.2) é ✐♠♣ró♣r✐❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈ê✲❧❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
F(s) = lim b→∞
Z b
0
e−stf(t)dt
❞❡s❞❡ q✉❡ ♦ ❧✐♠✐t❡ ❡①✐st❛ ❡ s❡❥❛ ✜♥✐t♦✳
✶✳✸✳ ❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❡ L[f(t)]
❊①❡♠♣❧♦ ✶✳✶ ❊♥❝♦♥tr❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ ❛ ❢✉♥çã♦ ❢✭t✮ ❂ ✶✳ ❙♦❧✉çã♦✿
L(1) = Z ∞
0
e−stdt = lim b→∞
Z b
0
e−stdt.
❋❛③❡♥❞♦ u=−st✱ t❡♠♦s du=−sdt✳ ❆ss✐♠✱
L(1) = lim b→∞
Z b
0
eu
−du s
= lim b→∞
−1seu
b
0 = limb→∞
−1
s e
−sb+ 1
s
= 1
s
❞❡s❞❡ q✉❡ s >0.
❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦✱ ♦❜s❡r✈❡ q✉❡
Z ∞
0
e−st[αf(t) +βg(t)]dt=α
Z ∞
0
e−stf(t)dt+β
Z ∞
0
e−stg(t)dt.
P♦rt❛♥t♦✱ q✉❛♥❞♦ ❛♠❜❛s ❛ ✐♥t❡❣r❛✐s ❝♦♥✈❡r❣❡♠
L[αf(t) +βg(t)] =αL[f(t)] +βL[g(t)] = αF(s) +βG(s).
❆ss✐♠✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ L é ❧✐♥❡❛r✳
✶✳✸ ❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❡
L
[
f
(
t
)]
❆ ✐♥t❡❣r❛❧ q✉❡ ❞❡✜♥❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♥✈❡r❣❡✳ P♦r ❡①❡♠♣❧♦✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ L(2
t)❡ L(e t2
) ♥ã♦ ❡①✐st❡♠✳ ❆s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s
❣❛r❛♥t❡♠ ❛ ❡①✐stê♥❝✐❛ ❞❛ tr❛♥s❢♦r♠❛❞❛ ✭❝♦♥❢♦r♠❡ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✮✿
1❛)f é ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s ❡♠[0,∞)✱ ♦✉ s❡❥❛✱ ❡♠ t♦❞♦ ✐♥t❡r✈❛❧♦0≤a≤t≤b,❤á
❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡s ❡ t♦❞❛ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ é ❞❡ ♣r✐♠❡✐r❛ ❡s♣é❝✐❡✱ ✐st♦ é✱ ❡①✐st❡♠ ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s✳
2❛)f é ❞❡ ♦r❞❡♠ ❡①♣♦♥❡♥❝✐❛❧✱ ❝♦♥❢♦r♠❡ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r✳
❉❡✜♥✐çã♦ ✶✳✷ ✭❖r❞❡♠ ❡①♣♦♥❡♥❝✐❛❧✮✳ ❉✐③❡♠♦s q✉❡ f : [0,∞) → R é ❞❡ ♦r❞❡♠
❡①♣♦♥❡♥❝✐❛❧ s❡ ❡①✐st❡♠ c, M >0 ❡ T > 0 t❛✐s q✉❡ |f(t)| ≤M ect ♣❛r❛ t♦❞♦ t >❚✳
❙❡ f é ❝r❡s❝❡♥t❡✱ ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ s✐♠♣❧❡s♠❡♥t❡ ❞✐③ q✉❡ ♦ ❣rá✜❝♦ ❞❡ f ♥♦
✐♥t❡r✈❛❧♦ [T,∞) ❡stá ❛❜❛✐①♦ ❞♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ M ect ❝♦♠ c > 0✳
P♦r ❡①❡♠♣❧♦✱ ❛s ❢✉♥çõ❡s f(t) = t✱ f(t) = e−t ❡ f(t) = 2 cost sã♦ t♦❞❛s ❞❡ ♦r❞❡♠
❡①♣♦♥❡♥❝✐❛❧ ♣❛r❛ t >0.P❛r❛ ❡st❡s ❝❛s♦s✱ t❡♠♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱|t| ≤et✱
|e−t
| ≤et
❡ |2cost| ≤2et✳
✶✳✸✳ ❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❡ L[f(t)]
❚❡♦r❡♠❛ ✶✳✶ ✭❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ❞❛❞❛ ❋✉♥çã♦✮✳ ❙❡❥❛ c ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧✳ ❙❡ f(t) é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛
♣♦r ♣❛rt❡s ❡♠ [0,∞) ❡ ❞❡ ♦r❞❡♠ ❡①♣♦♥❡♥❝✐❛❧ ♣❛r❛ t > T✱ ❡♥tã♦✱ s✉❛ ❚r❛♥s❢♦r♠❛❞❛
❞❡ ▲❛♣❧❛❝❡ ❡①✐st❡ ♣❛r❛ s > c✳
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s✱ ♣♦r ❞❡✜♥✐çã♦✱ q✉❡
F(s) = Z ∞
0
e−stf(t)dt.
❈♦♠♦ T >0✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
F(s) = Z T
0
e−stf(t)dt+ Z ∞
T
e−stf(t)dt.
❙❡❥❛♠ I1 =
RT
0 e−
stf(t)dt ❡ I
2 =
R∞ T e−
stf(t)dt. ◆♦t❡ q✉❡ I
1 ❡①✐st❡✱ ♣♦✐s ♣♦❞❡ s❡r
❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞❡ ✐♥t❡❣r❛✐s ❡♠ ✐♥t❡r✈❛❧♦s ❡♠ q✉❡ e−stf(t) s❡❥❛ ❝♦♥tí♥✉❛✳
P♦r ♦✉tr♦ ❧❛❞♦✱
|I2| ≤
Z ∞ T
e−stf(t)
dt ≤M
Z ∞ T
e−stectdt =M
Z ∞ T
e−(s−c)tdt
= lim b→∞M
Z b
T
e−(s−c)tdt.
❋❛③❡♥❞♦ u=−(s−c)t✱ t❡♠♦s du=−(s−c)dt ❡✱ ♣♦rt❛♥t♦✱
lim b→∞M
Z b
T
e−(s−c)tdt = lim b→∞M
Z b
T
eu
−du s−c
= −M
s−cblim→∞
e−(s−c)t b
T
= −M
s−cblim→∞
e−(s−c)b−e−(s−c)T
= M
s−ce
−(s−c)T
.
▲♦❣♦✱ I2 ❝♦♥✈❡r❣❡ ♣❛r❛ s > c✳ P♦rt❛♥t♦✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡①✐st❡ ♣❛r❛
s > c✳
❊①❡♠♣❧♦ ✶✳✷ ❉❡t❡r♠✐♥❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛s ❢✉♥çõ❡s✿ ❛✮ f(t) = e−3t
❜✮ f(t) =sen2t
✶✳✸✳ ❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❡ L[f(t)]
❙♦❧✉çã♦✿ ❛✮ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱
L(e−3t) = Z ∞
0
e−ste−3tdt = lim b→∞
Z b
0
e−(s+3)tdt.
❋❛③❡♥❞♦ u=−(s+ 3)t✱ t❡♠♦s du=−(s+ 3)dt ❡✱ ♣♦rt❛♥t♦✱
L(e−3t) = lim b→∞
Z b
0
e−(s+3)tdt
= lim b→∞
Z b
0
eu
−du s+ 3
= −1
s+ 3blim→∞ e
−(s+3)t b
0
= −1
s+ 3blim→∞ e −(s+3)b
−1
= 1
s+ 3, s >−3.
❜✮ P♦r ❞❡✜♥✐çã♦✱
L(sen(2t)) = Z ∞
0
e−stsen(2t)dt = lim b→∞
Z b
0
e−stsen(2t)dt.
❋❛③❡♥❞♦ u = sen(2t) ❡ dv = e−stdt✱ t❡♠♦s du = 2 cos(2t)dt ❡ v = R
e−stdt = −e−st s .
P♦rt❛♥t♦✱
L(sen(2t)) = 0 +2
s
Z ∞
0
e−stcos(2t)dt.
❋❛③❡♥❞♦ ♥♦✈❛♠❡♥t❡ u = cos(2t) ❡ dv = e−stdt✱ t❡♠♦s du = −2 sen(2t)dt ❡ v = R
e−stdt = −e−st s ✳ ❉❛í✱
L[sen(2t)] = 2
s2 −
4
s2
Z ∞
0
e−stsen(2t)dt
= 2
s2 −
4
s2L[sen(2t)]
= 2
s2+ 4, s >0.
❊①❡♠♣❧♦ ✶✳✸ ❈❛❧❝✉❧❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❛ s❡❣✉✐♥t❡ ❢✉♥çã♦
f(t) =
0, s❡ 0≤t <5 3, s❡ t≥5.
✶✳✸✳ ❈♦♥❞✐çõ❡s ❙✉✜❝✐❡♥t❡s ♣❛r❛ ❛ ❊①✐stê♥❝✐❛ ❞❡ L[f(t)]
❙♦❧✉çã♦✿ ❈♦♠♦ ❛ ❢✉♥çã♦ é ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s✱ s✉❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ é ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞❡ ❞✉❛s ✐♥t❡❣r❛✐s✱ ✐st♦ é✱
L[f(t)] = Z ∞
0
e−stf(t) =
Z 5
0
e−stf(t)dt+ Z ∞
5
e−stf(t)dt
= 0 + 3 Z ∞
5
e−stdt
= 3 lim b→∞
Z b
5
e−stf(t)dt
= 3 lim b→∞
−e−st
s
b
5
= 3 lim b→∞
−e−sb s +
e−5s s
= 3e−
5s
s , s >0.
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ ❛❧❣✉♠❛s ❢✉♥çõ❡s✿ ❚❡♦r❡♠❛ ✶✳✷ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❛❧❣✉♠❛s ❢✉♥çõ❡s ❜ás✐❝❛s
✶✳ L[1] = 1
s✱ s >0
✷✳ L[t] = 1
s2✱ s >0
✸✳ L[tn] = n!
sn+1✱ s >0
✹✳ L[sen(kt)] = k
s2+k2✱ s > k
✺✳ L[cos(kt)] = s
s2+k2✱ s > k
✻✳ L[eat] = 1
s−a✱ s > a
✼✳ L[cosh(kt)] = s
s2−k2✱ s > k
✽✳ L[senh(kt)] = k
s2−k2✱ s > k✳
❆ ❞❡♠♦♥str❛çã♦ ❞❡ 1.❥á ❢♦✐ ❢❡✐t❛✳ ❖s ♦✉tr♦s ✐t❡♥s ❡stã♦ ❞❡♠♦♥str❛❞♦s ♥♦ ❆♣ê♥❞✐❝❡
❇✳
✶✳✹✳ ❆ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛
❊①❡♠♣❧♦ ✶✳✹ ❉❡t❡r♠✐♥❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ f(t) = cos2t✳
❙♦❧✉çã♦✿ ❙❛❜❡♠♦s q✉❡ cos2t = 1
2[1 + cos(2t)]✳ ❊♥tã♦ ♣❡❧❛ ❛ ❧✐♥❡❛r✐❞❛❞❡
L[cos2t] =L
1 + cos(2t) 2
= 1
2L[1] + 1
2L[cos(2t)].
❆❣♦r❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ 1.2, t❡♠♦s q✉❡L[1] = 1
s ❡L[cos(2t)] =
s
s2+4✳ ▲♦❣♦✱
L[cos2t] = 1 2 ·
1
s +
1 2·
s s2+ 4 =
1 2
2s2+ 4
s(s2+ 4)
= s
2+ 2
s(s2+ 4),
s >2✳
✶✳✹ ❆ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛
❆té ♦ ♠♦♠❡♥t♦✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦ f(t)✱
♦ ♥♦ss♦ ♦❜❥❡t✐✈♦ ❡r❛ tr❛♥s❢♦r♠❛r ❡ss❛ ❢✉♥çã♦ ❡♠ ♦✉tr❛ ❢✉♥çã♦ F(s) ♣♦r ♠❡✐♦ ❞♦
♣r♦❝❡ss♦ ❞❡ ✐♥t❡❣r❛çã♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✳
❆❣♦r❛✱ ♦❜❥❡t✐✈❛♠♦s tr❛❜❛❧❤❛r ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦✱ ✐st♦ é✱ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦
F(s)✱ ♣r♦❝✉r❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ f(t) t❛❧ q✉❡ s✉❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
s❡❥❛ F(s)✳ ❆ss✐♠✱ ❞❡✜♥✐♠♦sf(t)❝♦♠♦ ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡F(s)✱
q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r f(t) =L−1[F(s)]✳
Pr♦♣♦s✐çã♦ ✶✳✶ ❆ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❧✐♥❡❛r✐✲ ❞❛❞❡✳
❉❡♠♦♥str❛çã♦✿ P❛r❛ ♣r♦✈❛r♠♦s ✐ss♦✱ ❝♦♥s✐❞❡r❡♠♦sa❡b❝♦♥st❛♥t❡s ❡F(s)❡G(s)
❢✉♥çõ❡s tr❛♥s❢♦r♠❛❞❛s ❞❡ f ❡ g✳ ❊♥tã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
L−1[aF(s) +bG(s)] = L−1(aL[f(t)] +bL[g(t)]) =L−1(L[af(t) +bg(t)]) =af(t) +bg(t)
=aL−1[F(s)] +bL−1[G(s)].
P♦❞❡✲s❡ ♠♦str❛r t❛♠❜é♠ q✉❡ s❡f ❡g sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ♣♦r ♣❛rt❡s ❡♠[0,∞)
❡ ❞❡ ♦r❞❡♠ ❡①♣♦♥❡♥❝✐❛❧✱ ❡♥tã♦ s❡ L[f(t)] =L[g(t)]✱f ❡g sã♦ ❡ss❡♥❝✐❛❧♠❡♥t❡ ✐❣✉❛✐s✱
♦✉ s❡❥❛✱ ❡❧❛s ♣♦❞❡♠ s❡r ❞✐❢❡r❡♥t❡s s♦♠❡♥t❡ ♥♦s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡s ❞❡ f ❡ g✳ ■ss♦ ❣❛r❛♥t❡ q✉❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ❢✉♥çã♦ F(s) ♥ã♦ é
♥❡❝❡ss❛r✐❛♠❡♥t❡ ú♥✐❝❛✳
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❞❡ ▲❛♣❧❛❝❡ ♣❛r❛ ❛❧❣✉♠❛s ❢✉♥çõ❡s✳
✶✳✹✳ ❆ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛
❚❡♦r❡♠❛ ✶✳✸ ❱❛❧❡ ♦ s❡❣✉✐♥t❡✿
✶✳ (1) =L−1
1
s
✷✳ tn =L−1
n!
sn+1
n= 1,2,3, ...
✸✳ eat =L−1
1
s−a
✹✳ sen(kt) = L−1
k s2+k2
✺✳ cos(kt) = L−1
s s2+k2
✻✳ senh(kt) =L−1
k s2−k2
✼✳ cosh(kt) = L−1
s s2−k2
.
❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ s❡rá ♦♠✐t✐❞❛✱ ♣♦✐s ❞❡♠❛♥❞❛ ♦ ✉s♦ ❞❡ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s✳ ❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡ ❜✉s❝❛r ❡♠ ❬✶✶❪✳
❱❡❥❛♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❡st❡ r❡s✉❧t❛❞♦✳
❊①❡♠♣❧♦ ✶✳✺ ❉❡t❡r♠✐♥❛r ❛s ❚r❛♥s❢♦r♠❛❞❛s ■♥✈❡rs❛s ♣❡❞✐❞❛s ❛❜❛✐①♦✿
❛✮L−11 s3
❜✮ L−1 5 s2+49
.
❙♦❧✉çã♦✿ ❛✮ ❯s❛♥❞♦ ♦ ✐t❡♠ ✷✳ ❞♦ ❚❡♦r❡♠❛ 1.3✱ ❝♦♠ n = 2✱ ❡ s❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❡
❞✐✈✐❞✐r♠♦s ✭❛✮ ♣♦r 2!✱ ♦❜t❡♠♦s
L−1
1 s3 = 1 2!L −1 2! s3 = 1 2!t
2 = t2
2.
❜✮ Pr✐♠❡✐r♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
L−1
5
s2+ 49
= 5L−1
1
s2+ 49
.
❆❣♦r❛✱ ♦❜s❡r✈❡ q✉❡ s❡ k2 = 49 ❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❡ ❞✐✈✐❞✐r♠♦s ✭❜✮ ♣♦r7✱ ❡ ❛♣❧✐❝❛r♠♦s
♦ ✐t❡♠ 4.❞♦ ❚❡♦r❡♠❛ 1.3✱ ♦❜t❡♠♦s
L−1
5
s2+ 49
= 51
7L −1
7
s2 + 49
= 5
7sen(7t).
✶✳✹✳ ❆ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛
❊①✐st❡♠ ❝❛s♦s q✉❡ ♣❛r❛ ♦❜t❡r♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛✱ ♣r❡❝✐s❛♠♦s r❡❝♦rr❡r ❛ té❝♥✐❝❛ ❞❛s ❢r❛çõ❡s ♣❛r❝✐❛✐s✳ ❆❜♦r❞❛r❡♠♦s ❛❧❣✉♥s ❝❛s♦s ♥♦s ❡①❡♠♣❧♦s ❛ s❡❣✉✐r✳ ❊①❡♠♣❧♦ ✶✳✻ ❙❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ❢✉♥çã♦ ❢✭t✮ é
F(s) = s+ 3
s2−3s+ 2
❞❡t❡r♠✐♥❡ f(t) = L−1[F(s)].
❙♦❧✉çã♦✿ ❯s❛♥❞♦ ❢r❛çõ❡s ♣❛r❝✐❛✐s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
F(s) = s+ 3
s2−3s+ 2 =
A s−1 +
B s−2 =
A(s−2) +B(s−1) (s−1)(s−2) .
❉❛ ✐❣✉❛❧❞❛❞❡
s+ 3
(s−1)(s−2) =
A(s−2) +B(s−1) (s−1)(s−2) ,
t❡♠♦s
s+ 3 =A(s−2) +B(s−1),
❞❡ ♦♥❞❡ ♦❜t❡♠♦s A=−4❡ B = 5✳ P♦rt❛♥t♦✱
F(s) = −4
s−1 + 5
s−2
⇒L−1[F(s)] =−4L−1
1
s−1
+ 5L−1
1
s−2
=−4et+ 5e2t,
♦♥❞❡ ✉s❛♠♦s ♦ ✐t❡♠ 3. ❞♦ ❚❡♦r❡♠❛ 1.3✳
❊①❡♠♣❧♦ ✶✳✼ ❊♥❝♦♥tr❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ■♥✈❡rs❛ ❢✭t✮ ♣❛r❛
F(s) = 1
s2(s+ 4).
❙♦❧✉çã♦✿ ❯s❛♥❞♦ ❢r❛çõ❡s ♣❛r❝✐❛✐s✱ t❡♠♦s q✉❡
F(s) = 1
s2(s+ 4) =
A s +
B s2 +
C s+ 4 =
As(s+ 4) +B(s+ 4) +Cs2
s2(s+ 4) .
❉❛ ✐❣✉❛❧❞❛❞❡
1
s2(s+ 4) =
As(s+ 4) +B(s+ 4) +Cs2 s2(s+ 4) ,
✶✳✺✳ ❚❡♦r❡♠❛s ❙♦❜r❡ ❉❡s❧♦❝❛♠❡♥t♦ ❡ ❉❡r✐✈❛❞❛ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
s❡❣✉❡ q✉❡
1 = As(s+ 4) +B(s+ 4) +Cs2,
❞❡ ♦♥❞❡ ♦❜t❡♠♦s A=−1 16✱ B =
1
4 ❡ C =
1
16. ❆ss✐♠✱
F(s) = 1
s2(s+ 4) = −161
s +
1 4
s2 + 1 16
s+ 4
❡✱ ♣♦rt❛♥t♦✱
f(t) = L−1[F(s)] = − 1 16L
−1
1
s
+ 1
4L −1
1
s2
+ 1
16L −1
1
s+ 4
=− 1
16+ 1 4t+
1 16e
−4t
.
✶✳✺ ❚❡♦r❡♠❛s ❙♦❜r❡ ❉❡s❧♦❝❛♠❡♥t♦ ❡ ❉❡r✐✈❛❞❛ ❞❛
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡
❊♥❝♦♥tr❛r ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ❢✉♥çã♦f(t) ✉s❛♥❞♦ ❛♣❡♥❛s ❛ ❉❡✲
✜♥✐çã♦ 1.1 ♥❡♠ s❡♠♣r❡ é ❝♦♥✈❡♥✐❡♥t❡✱ ♣♦✐s ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ♦s ❝á❧❝✉❧♦s sã♦
❜❛st❛♥t❡s tr❛❜❛❧❤♦s♦s✳ P❛r❛ ❢❛❝✐❧✐t❛r ♦ ♣r♦❝❡ss♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ❛❧❣✉♥s t❡♦✲ r❡♠❛s q✉❡ ♣♦ss✐❜✐❧✐t❛rã♦ ❝♦♥str✉✐r ✉♠❛ ❧✐st❛ ❜❡♠ ♠❛✐s ❝♦♠♣❧❡t❛ ❞❡ tr❛♥s❢♦r♠❛❞❛s s❡♠ ♥❡❝❡ss✐t❛r ✉s❛r ❞✐r❡t❛♠❡♥t❡ ❛ ❞❡✜♥✐çã♦✳
▼❡s♠♦ s❡♥❞♦ ♣♦ssí✈❡❧ ❛ ❝♦♥str✉çã♦ ❞❡ t❛❜❡❧❛s ❡①t❡♥s✐✈❛s ❞❡ ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ é ✐♠♣♦rt❛♥t❡ s❛❜❡r♠♦s ❛s ❚r❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ❞❡ ❛❧❣✉♠❛s ❢✉♥çõ❡s ❜ás✐❝❛s✱ t❛✐s ❝♦♠♦ f(t) =t, f(t) =tn✱ f(t) =eat✱ f(t) = sen(kt)❡ f(t) = cos(kt)✳
❚❡♦r❡♠❛ ✶✳✹ ✭Pr✐♠❡✐r♦ ❚❡♦r❡♠❛ ❞♦ ❉❡s❧♦❝❛♠❡♥t♦✮✳ ❙❡❥❛ ❛ ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧✳ ❙❡ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ ❢✉♥çã♦ f : [0,∞)→R é F(s) ♣❛r❛s > c, ❡♥tã♦ ❛
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❛ ❢✉♥çã♦ g(t) = eatf(t)éG(s) = F(s
−a)♣❛r❛s−a > c.
❉❡♠♦♥str❛çã♦✿ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦✱ ♦❜t❡♠♦s
L[g(t)] = Z ∞
0
e−stg(t)dt = Z ∞
0
e−steatf(t)dt= Z ∞
0
e−(s−a)tf(t)dt =F(s−a)
♣❛r❛ s−a > c.
✶✳✺✳✶ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛
1.4
❊①❡♠♣❧♦ ✶✳✽ ❙❡❥❛♠ ❛ ❡ ❜ ❝♦♥st❛♥t❡s✳ ❈♦♥s✐❞❡r❡ ❛ ❢✉♥çã♦ f : [0,∞)−→ R ❞❛❞❛
♣♦r g(t) = ebtcos(at)✳ ❉❡t❡r♠✐♥❡ s✉❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡✳
