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❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ❢♦r P♦✇❡r ❯t✐❧✐t② ▼❛①✐♠✐③❛t✐♦♥ ▼❛r❝❡❧ ◆✉t③ ❊❚❍ ❩✉r✐❝❤ ❆♥❙t❆♣✶✵ ❈♦♥❢❡r❡♥❝❡ ❱✐❡♥♥❛✱ ✶✷✳✵✼✳✷✵✶✵ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✶ ✴ ✶✺

❇❛s✐❝ Pr♦❜❧❡♠ P♦✇❡r ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ❯ ( ① ) = ✶ ♣ ① ♣ ✱ ♣ ∈ ( −∞ , ✵ ) ∪ ( ✵ , ✶ ) ❯t✐❧✐t② ♠❛①✐♠✐③❛t✐♦♥✿ [∫ ❚ ] ♠❛① ❊ ❯ ( ❝ t ) ❞t + ❯ ( ❳ ❚ ( 휋, ❝ )) ✵ ♦✈❡r tr❛❞✐♥❣ ❛♥❞ ❝♦♥s✉♠♣t✐♦♥ str❛t❡❣✐❡s ( 휋, ❝ ) ❆✐♠✿ ❉❡s❝r✐❜❡ ♦♣t✐♠❛❧ str❛t❡❣✐❡s ✐♥ t❤❡ ❧✐♠✐ts ♣ → ✵✱ ✐✳❡✳ r❡❧❛t✐✈❡ r✐s❦ ❛✈❡rs✐♦♥ → ✶✳ ♣ → −∞ ✱ ✐✳❡✳ r❡❧❛t✐✈❡ r✐s❦ ❛✈❡rs✐♦♥ → + ∞ ✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✷ ✴ ✶✺

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆s②♠♣t♦t✐❝s ♣ → ✵ ✷ ❆s②♠♣t♦t✐❝s ♣ → −∞ ✭✵t❤ ♦r❞❡r✮ ✸ ❆s②♠♣t♦t✐❝s ♣ → −∞ ✭✶st ♦r❞❡r✮ ✹ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✸ ✴ ✶✺

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆s②♠♣t♦t✐❝s ♣ → ✵ ✷ ❆s②♠♣t♦t✐❝s ♣ → −∞ ✭✵t❤ ♦r❞❡r✮ ✸ ❆s②♠♣t♦t✐❝s ♣ → −∞ ✭✶st ♦r❞❡r✮ ✹ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✸ ✴ ✶✺

❯t✐❧✐t② ▼❛①✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠ ❞ r✐s❦② ❛ss❡ts✿ s❡♠✐♠❛rt✐♥❣❛❧❡ ❘ ♦❢ st♦❝❦ r❡t✉r♥s✱ ❘ ✵ = ✵ ( ) s♣♦t ♣r✐❝❡s ❙ = ℰ ( ❘ ✶ ) , . . . , ℰ ( ❘ ❞ ) s❛t✐s❢②✐♥❣ ◆❋▲❱❘ ✭❘❛♥❞♦♠✮ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ❯ t ( ① ) = ❯ ( ♣ ) ( ① ) = ❉ t ✶ ♣ ① ♣ , ♣ ∈ ( −∞ , ✵ ) ∪ ( ✵ , ✶ ) , t ✇✐t❤ ❉ > ✵ ❝à❞❧à❣ ❛❞❛♣t❡❞ ❛♥❞ ❉ , ❉ − ✶ ❜♦✉♥❞❡❞ ❋♦r ✜①❡❞ ♣ ❛♥❞ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❝❛♣✐t❛❧ ① ✵ > ✵✿ [ ∫ ❚ ] s✉♣ ✉ ♣ ( ① ✵ ) := ❊ ❯ t ( ❝ t ) ❞t + ❯ ❚ ( ❝ ❚ ) ( 휋, ❝ ) ∈풜 � �� � ✵ ❯ t ( ❝ t ) 휇 ∘ ( ❞t ) , ✇✐t❤ 휇 ∘ := ❞t + 훿 { ❚ } 휋 ∈ ▲ ( ❘ ) tr❛❞✐♥❣ str❛t❡❣②✱ ❝ ≥ ✵ ❝♦♥s✉♠♣t✐♦♥ ∫ t ∫ t ❲❡❛❧t❤✿ ❳ t ( 휋, ❝ ) = ① ✵ + ✵ ❳ s − ( 휋, ❝ ) 휋 s ❞❘ s − ✵ ❝ s ❞s ❆❞♠✐ss✐❜✐❧✐t②✿ ( 휋, ❝ ) ∈ 풜 ✐❢ ❳ ( 휋, ❝ ) > ✵ ❛♥❞ ❝ ❚ = ❳ ❚ ( 휋, ❝ ) ❊①✐st❡♥❝❡ ❬❑❛r❛t③❛s ❛♥❞ ➎✐t❦♦✈✐➣ ❆♦P✬✵✸❪✿ ■❢ ✉ ♣ ( ① ✵ ) < ∞ ✱ t❤❡r❡ ❡①✐sts ❛♥ [ ∫ ❚ ] ♦♣t✐♠❛❧ str❛t❡❣② (ˆ ❝ ) ∈ 풜 ✱ ✐✳❡✳✱ ✉ ♣ ( ① ✵ ) = ❊ ❝ t ) 휇 ∘ ( ❞t ) ✳ 휋, ˆ ✵ ❯ t (ˆ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✹ ✴ ✶✺

❯t✐❧✐t② ▼❛①✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠ ❞ r✐s❦② ❛ss❡ts✿ s❡♠✐♠❛rt✐♥❣❛❧❡ ❘ ♦❢ st♦❝❦ r❡t✉r♥s✱ ❘ ✵ = ✵ ( ) s♣♦t ♣r✐❝❡s ❙ = ℰ ( ❘ ✶ ) , . . . , ℰ ( ❘ ❞ ) s❛t✐s❢②✐♥❣ ◆❋▲❱❘ ✭❘❛♥❞♦♠✮ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ❯ t ( ① ) = ❯ ( ♣ ) ( ① ) = ❉ t ✶ ♣ ① ♣ , ♣ ∈ ( −∞ , ✵ ) ∪ ( ✵ , ✶ ) , t ✇✐t❤ ❉ > ✵ ❝à❞❧à❣ ❛❞❛♣t❡❞ ❛♥❞ ❉ , ❉ − ✶ ❜♦✉♥❞❡❞ ❋♦r ✜①❡❞ ♣ ❛♥❞ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❝❛♣✐t❛❧ ① ✵ > ✵✿ [ ∫ ❚ ] s✉♣ ✉ ♣ ( ① ✵ ) := ❊ ❯ t ( ❝ t ) ❞t + ❯ ❚ ( ❝ ❚ ) ( 휋, ❝ ) ∈풜 � �� � ✵ ❯ t ( ❝ t ) 휇 ∘ ( ❞t ) , ✇✐t❤ 휇 ∘ := ❞t + 훿 { ❚ } 휋 ∈ ▲ ( ❘ ) tr❛❞✐♥❣ str❛t❡❣②✱ ❝ ≥ ✵ ❝♦♥s✉♠♣t✐♦♥ ∫ t ∫ t ❲❡❛❧t❤✿ ❳ t ( 휋, ❝ ) = ① ✵ + ✵ ❳ s − ( 휋, ❝ ) 휋 s ❞❘ s − ✵ ❝ s ❞s ❆❞♠✐ss✐❜✐❧✐t②✿ ( 휋, ❝ ) ∈ 풜 ✐❢ ❳ ( 휋, ❝ ) > ✵ ❛♥❞ ❝ ❚ = ❳ ❚ ( 휋, ❝ ) ❊①✐st❡♥❝❡ ❬❑❛r❛t③❛s ❛♥❞ ➎✐t❦♦✈✐➣ ❆♦P✬✵✸❪✿ ■❢ ✉ ♣ ( ① ✵ ) < ∞ ✱ t❤❡r❡ ❡①✐sts ❛♥ [ ∫ ❚ ] ♦♣t✐♠❛❧ str❛t❡❣② (ˆ ❝ ) ∈ 풜 ✱ ✐✳❡✳✱ ✉ ♣ ( ① ✵ ) = ❊ ❝ t ) 휇 ∘ ( ❞t ) ✳ 휋, ˆ ✵ ❯ t (ˆ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✹ ✴ ✶✺

❯t✐❧✐t② ▼❛①✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠ ❞ r✐s❦② ❛ss❡ts✿ s❡♠✐♠❛rt✐♥❣❛❧❡ ❘ ♦❢ st♦❝❦ r❡t✉r♥s✱ ❘ ✵ = ✵ ( ) s♣♦t ♣r✐❝❡s ❙ = ℰ ( ❘ ✶ ) , . . . , ℰ ( ❘ ❞ ) s❛t✐s❢②✐♥❣ ◆❋▲❱❘ ✭❘❛♥❞♦♠✮ ✉t✐❧✐t② ❢✉♥❝t✐♦♥ ❯ t ( ① ) = ❯ ( ♣ ) ( ① ) = ❉ t ✶ ♣ ① ♣ , ♣ ∈ ( −∞ , ✵ ) ∪ ( ✵ , ✶ ) , t ✇✐t❤ ❉ > ✵ ❝à❞❧à❣ ❛❞❛♣t❡❞ ❛♥❞ ❉ , ❉ − ✶ ❜♦✉♥❞❡❞ ❋♦r ✜①❡❞ ♣ ❛♥❞ ❣✐✈❡♥ ✐♥✐t✐❛❧ ❝❛♣✐t❛❧ ① ✵ > ✵✿ [ ∫ ❚ ] s✉♣ ✉ ♣ ( ① ✵ ) := ❊ ❯ t ( ❝ t ) ❞t + ❯ ❚ ( ❝ ❚ ) ( 휋, ❝ ) ∈풜 � �� � ✵ ❯ t ( ❝ t ) 휇 ∘ ( ❞t ) , ✇✐t❤ 휇 ∘ := ❞t + 훿 { ❚ } 휋 ∈ ▲ ( ❘ ) tr❛❞✐♥❣ str❛t❡❣②✱ ❝ ≥ ✵ ❝♦♥s✉♠♣t✐♦♥ ∫ t ∫ t ❲❡❛❧t❤✿ ❳ t ( 휋, ❝ ) = ① ✵ + ✵ ❳ s − ( 휋, ❝ ) 휋 s ❞❘ s − ✵ ❝ s ❞s ❆❞♠✐ss✐❜✐❧✐t②✿ ( 휋, ❝ ) ∈ 풜 ✐❢ ❳ ( 휋, ❝ ) > ✵ ❛♥❞ ❝ ❚ = ❳ ❚ ( 휋, ❝ ) ❊①✐st❡♥❝❡ ❬❑❛r❛t③❛s ❛♥❞ ➎✐t❦♦✈✐➣ ❆♦P✬✵✸❪✿ ■❢ ✉ ♣ ( ① ✵ ) < ∞ ✱ t❤❡r❡ ❡①✐sts ❛♥ [ ∫ ❚ ] ♦♣t✐♠❛❧ str❛t❡❣② (ˆ ❝ ) ∈ 풜 ✱ ✐✳❡✳✱ ✉ ♣ ( ① ✵ ) = ❊ ❝ t ) 휇 ∘ ( ❞t ) ✳ 휋, ˆ ✵ ❯ t (ˆ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✹ ✴ ✶✺

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆s②♠♣t♦t✐❝s ♣ → ✵ ✷ ❆s②♠♣t♦t✐❝s ♣ → −∞ ✭✵t❤ ♦r❞❡r✮ ✸ ❆s②♠♣t♦t✐❝s ♣ → −∞ ✭✶st ♦r❞❡r✮ ✹ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✹ ✴ ✶✺

❆s②♠♣t♦t✐❝s ♣ → ✵ ❢♦r ❉ ≡ ✶ ❝ Pr♦♣❡♥s✐t② t♦ ❝♦♥s✉♠❡ 휅 := ❳ ( 휋, ❝ ) ✳ ❚❤❡♦r❡♠ ✭ ♣ → ✵✮ ∙ ❆ss✉♠❡ ✉ ♣ ✵ ( ① ✵ ) < ∞ ❢♦r s♦♠❡ ♣ ✵ > ✵ ✳ ❚❤❡♥ ✶ ˆ 휅 t ( ♣ ) → ✉♥✐❢♦r♠❧② ✐♥ t✱ P✲❛✳s✳ ✶ + ❚ − t ∙ ■❢ ❙ ✐s ❝♦♥t✐♥✉♦✉s✱ ✐♥ ▲ ✷ 휋 ( ♣ ) → 휆 ˆ ❧♦❝ ( ▼ ) . ❙ ❝♦♥t✐♥✉♦✉s ✰ ◆❋▲❱❘ ⇒ str✉❝t✉r❡ ❝♦♥❞✐t✐♦♥✿ ∫ s♦♠❡ 휆 ∈ ▲ ✷ ❘ = ▼ + ❞ ⟨ ▼ ⟩ 휆, ❧♦❝ ( ▼ ) . ▲✐♠✐t ✐s t❤❡ ❧♦❣✲♦♣t✐♠❛❧ str❛t❡❣②✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✺ ✴ ✶✺

❆s②♠♣t♦t✐❝s ♣ → ✵ ❢♦r ❉ ≡ ✶ ❝ Pr♦♣❡♥s✐t② t♦ ❝♦♥s✉♠❡ 휅 := ❳ ( 휋, ❝ ) ✳ ❚❤❡♦r❡♠ ✭ ♣ → ✵✮ ∙ ❆ss✉♠❡ ✉ ♣ ✵ ( ① ✵ ) < ∞ ❢♦r s♦♠❡ ♣ ✵ > ✵ ✳ ❚❤❡♥ ✶ ˆ 휅 t ( ♣ ) → ✉♥✐❢♦r♠❧② ✐♥ t✱ P✲❛✳s✳ ✶ + ❚ − t ∙ ■❢ ❙ ✐s ❝♦♥t✐♥✉♦✉s✱ ✐♥ ▲ ✷ 휋 ( ♣ ) → 휆 ˆ ❧♦❝ ( ▼ ) . ❙ ❝♦♥t✐♥✉♦✉s ✰ ◆❋▲❱❘ ⇒ str✉❝t✉r❡ ❝♦♥❞✐t✐♦♥✿ ∫ s♦♠❡ 휆 ∈ ▲ ✷ ❘ = ▼ + ❞ ⟨ ▼ ⟩ 휆, ❧♦❝ ( ▼ ) . ▲✐♠✐t ✐s t❤❡ ❧♦❣✲♦♣t✐♠❛❧ str❛t❡❣②✳ ▼❛r❝❡❧ ◆✉t③ ✭❊❚❍✮ ❘✐s❦ ❆✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝s ✺ ✴ ✶✺