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❆ ❝♦♠♣❧❡①✲❛♥❛❧②s✐s ❢r✐❡♥❞❧② ❢♦r♠ ♦❢ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✇✐t❤ ❛ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♣♦t❡♥t✐❛❧

❉♠✐tr② P♦♥♦♠❛r❡✈✶,✷

✶■♥st✐t✉t❡ ♦❢ ❆♥❛❧②s✐s ✫ ❙❝✐❡♥t✐✜❝ ❈♦♠♣✉t✐♥❣✱ ❚❯ ❲✐❡♥✱ ❆✉str✐❛ ✷❙t✳ P❡t❡rs❜✉r❣ ❉❡♣❛rt♠❡♥t ♦❢ ❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡ ❘❆❙✱ ❘✉ss✐❛

❈♦♥t❛❝t✿ ❞♠✐tr②✳♣♦♥♦♠❛r❡✈❅❛s❝✳t✉✇✐❡♥✳❛❝✳❛t

❙❡♣t❡♠❜❡r ✹✱ ✷✵✶✾

■❱P ❢♦r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥

• ✐✈❡♥ ❛ (①) s✳t✳ ❛ (①) = ✵ ❢♦r ① > ✵✱ ❝♦♥s✐❞❡r ❛♥ ✐♥✐t✐❛❧✲✈❛❧✉❡

♣r♦❜❧❡♠ ❢♦r t❤❡ st❛t✐♦♥❛r② ✇❛✈❡ ❡q✉❛t✐♦♥ ✉′′ (①) + ❛ (①)✉ (①) = ✵, ① > ✵, ✉ (✵) = ✉✵, ✉′ (✵) = ✉✶. ❈❧❛ss✐❝s✿ ✉ ① ❛ ① ✉ ① ✵

✈ ✉ ✉

✈ ① ✈✷ ① ❛ ① ✵✳ ✭ ❘✐❝❝❛t✐ ❡q✉❛t✐♦♥ ✮ ❆ ♠♦r❡ ✐♥t❡r❡st✐♥❣ t♦♦❧✿ ✉ ① ❛ ① ✉ ① ✵ ❩ ① ❢ ① ❩ ① ✭ ❄❄❄ ✮

■❱P ❢♦r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥

• ✐✈❡♥ ❛ (①) s✳t✳ ❛ (①) = ✵ ❢♦r ① > ✵✱ ❝♦♥s✐❞❡r ❛♥ ✐♥✐t✐❛❧✲✈❛❧✉❡

♣r♦❜❧❡♠ ❢♦r t❤❡ st❛t✐♦♥❛r② ✇❛✈❡ ❡q✉❛t✐♦♥ ✉′′ (①) + ❛ (①)✉ (①) = ✵, ① > ✵, ✉ (✵) = ✉✵, ✉′ (✵) = ✉✶. ❈❧❛ss✐❝s✿ ✉′′ (①) + ❛ (①)✉ (①) = ✵

✈:=✉′/✉

⇐ ⇒ ✈′ (①) + ✈✷ (①) + ❛ (①) = ✵✳ ✭ ❘✐❝❝❛t✐ ❡q✉❛t✐♦♥ ✮ ❆ ♠♦r❡ ✐♥t❡r❡st✐♥❣ t♦♦❧✿ ✉ ① ❛ ① ✉ ① ✵ ❩ ① ❢ ① ❩ ① ✭ ❄❄❄ ✮

■❱P ❢♦r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥

• ✐✈❡♥ ❛ (①) s✳t✳ ❛ (①) = ✵ ❢♦r ① > ✵✱ ❝♦♥s✐❞❡r ❛♥ ✐♥✐t✐❛❧✲✈❛❧✉❡

♣r♦❜❧❡♠ ❢♦r t❤❡ st❛t✐♦♥❛r② ✇❛✈❡ ❡q✉❛t✐♦♥ ✉′′ (①) + ❛ (①)✉ (①) = ✵, ① > ✵, ✉ (✵) = ✉✵, ✉′ (✵) = ✉✶. ❈❧❛ss✐❝s✿ ✉′′ (①) + ❛ (①)✉ (①) = ✵

✈:=✉′/✉

⇐ ⇒ ✈′ (①) + ✈✷ (①) + ❛ (①) = ✵✳ ✭ ❘✐❝❝❛t✐ ❡q✉❛t✐♦♥ ✮ ❆ ♠♦r❡ ✐♥t❡r❡st✐♥❣ t♦♦❧✿ ✉′′ (①) + ❛ (①)✉ (①) = ✵

...

⇐ ⇒ ❩ ′ (①) = ❢ (①)¯ ❩ (①). ✭ ❄❄❄ ✮

❙♦♠❡ ❡❧❡♠❡♥t❛r② tr❛♥s❢♦r♠❛t✐♦♥s✿ ✈❡❝t♦r✐s❛t✐♦♥

✉′′ (①) + ❛ (①)✉ (①) = ✵ ⇐ ⇒ ❯′ (①) = ❆ (①)❯ (①), ❯ (①) :=

• α (①) ✉ (①)

β (①) ✉′ (①) + γ (①) ✉ (①)

• ✱

α (①)✱ β (①) = ✵✱ ❆ (①) :=    α′ α − γ β α β γ′ − β❛ α − β′ + γ αβ γ β′ + γ β    (①)✳

❙♦♠❡ ❡❧❡♠❡♥t❛r② tr❛♥s❢♦r♠❛t✐♦♥s✿ ♠✐①✐♥❣

❯′ (①) = ❆ (①)❯ (①)

❱ (①)=P❯(①)

⇐ ⇒ ❱ ′ (①) = ❇ (①)❱ (①), ❆ (①) =    α′ α − γ β α β γ′ − β❛ α − β′ + γ αβ γ β′ + γ β    (①)✱ ❇ (①) := P❆ (①)P−✶ =: ♣ r s q

• (①)✱

P ✐s ❛ ❝♦♥st❛♥t ✐♥✈❡rt✐❜❧❡ ♠❛tr✐①✳

▲❡ss ❡❧❡♠❡♥t❛r② ❜✉t ✐♠♣♦rt❛♥t tr❛♥s❢♦r♠❛t✐♦♥

❱ ′ (①) = ❇ (①)❱ (①)

❲ (①)=❙(①)❱ (①)

⇐ ⇒ ❲ ′ (①) = ◆ (①)❲ (①), ❇ (①) = ♣ r s q

• (①)✱

❙ (①) := exp

• −

①

✵ ♣ (t) ❞t

• ✵

✵ exp

• −

①

✵ q (t) ❞t

• ✱

◆ (①) :=

• ✵

r (①) exp

• −

①

✵ (♣ (t) − q (t)) ❞t

• s (①) exp

①

✵ (♣ (t) − q (t)) ❞t

• ✵
• ✳

❚❤✐s ✐s ❞✉❡ t♦✿

❱ ♣ ✵ ✵ q ❱ ❡

① ✵ ♣ t ❞t

✵ ✵ ❡

① ✵ q t ❞t

❡

① ✵ ♣ t ❞t

✵ ✵ ❡

① ✵ q t ❞t

❱ ✳

▲❡ss ❡❧❡♠❡♥t❛r② ❜✉t ✐♠♣♦rt❛♥t tr❛♥s❢♦r♠❛t✐♦♥

❱ ′ (①) = ❇ (①)❱ (①)

❲ (①)=❙(①)❱ (①)

⇐ ⇒ ❲ ′ (①) = ◆ (①)❲ (①), ❇ (①) = ♣ r s q

• (①)✱

❙ (①) := exp

• −

①

✵ ♣ (t) ❞t

• ✵

✵ exp

• −

①

✵ q (t) ❞t

• ✱

◆ (①) :=

• ✵

r (①) exp

• −

①

✵ (♣ (t) − q (t)) ❞t

• s (①) exp

①

✵ (♣ (t) − q (t)) ❞t

• ✵
• ✳

❚❤✐s ✐s ❞✉❡ t♦✿

❱ ′ − ♣ ✵ ✵ q

• ❱ =
• ❡

①

✵ ♣(t)❞t

✵ ✵ ❡

①

✵ q(t)❞t

❡−

①

✵ ♣(t)❞t

✵ ✵ ❡−

①

✵ q(t)❞t

• ❱

′ ✳

❆♥ ♦❜s❡r✈❛t✐♦♥

❲ ′ (①) = ◆ (①)❲ (①), ① > ✵, ❲ (✵) = ❲✵,

◆ (①) =

• ✵

r (①) ❡−

①

✵ (♣(t)−q(t))❞t

s (①) ❡

①

✵ (♣(t)−q(t))❞t

✵

• ✳

◆♦t❡ t❤❛t ✐❢

r (①) ❡−

①

✵ (♣(t)−q(t))❞t = s (①) ❡

①

✵ (♣(t)−q(t))❞t =: ❢ (①)✱

t❤❡♥ ❲ (①) = (cosh ❢ (①)■ + sinh ❢ (①)❙) ❲✵✱ ✇❤❡r❡ ■ := ✶ ✵ ✵ ✶

• , ❙ :=

✵ ✶ ✶ ✵

• ✳

❍✐♥t✿ ❲r✐t❡ P✐❝❛r❞ ✐t❡r❛t✐♦♥ ❛♥❞ ✉s❡ ❙✷♠ = ■✱ ❙✷♠+✶ = ❙✱ ♠ ∈ N✳

❆♥♦t❤❡r ♦❜s❡r✈❛t✐♦♥

❲ ′ (①) = ◆ (①)❲ (①),

◆ (①) =

• ✵

r (①) ❡−

①

✵ (♣(t)−q(t))❞t

s (①) ❡

①

✵ (♣(t)−q(t))❞t

✵

• ✳

❙❡❝♦♥❞ ❜❡st s✐t✉❛t✐♦♥✿ r (①) ❡−

①

✵ (♣(t)−q(t))❞t = ¯

s (①) ❡

①

✵ (¯

♣(t)−¯ q(t))❞t

❢ ① ✱ ✐✳❡✳ ✐❢ ♣ ① q ① ✐s ♣✉r❡❧② ✐♠❛❣✐♥❛r②✱ ✇❡ ❥✉st ♥❡❡❞ r ① s ① ✦ ▼✉❝❤ ❡❛s② t♦ ✐♠♣♦s❡ ✦

❆♥♦t❤❡r ♦❜s❡r✈❛t✐♦♥

❲ ′ (①) = ◆ (①)❲ (①),

◆ (①) =

• ✵

r (①) ❡−

①

✵ (♣(t)−q(t))❞t

s (①) ❡

①

✵ (♣(t)−q(t))❞t

✵

• ✳

❙❡❝♦♥❞ ❜❡st s✐t✉❛t✐♦♥✿ r (①) ❡−

①

✵ (♣(t)−q(t))❞t = ¯

s (①) ❡

①

✵ (¯

♣(t)−¯ q(t))❞t=: ❢ (①)✱

✐✳❡✳ ✐❢ ♣ (①) − q (①) ✐s ♣✉r❡❧② ✐♠❛❣✐♥❛r②✱ ✇❡ ❥✉st ♥❡❡❞ r (①) = ¯ s (①) ✦ ▼✉❝❤ ❡❛s② t♦ ✐♠♣♦s❡ ✦

❉❡❛❧✐♥❣ ✇✐t❤ ❛♥ ✐♥t❡r❡st✐♥❣ ♠❛tr✐①

❲ ′ (①) = ◆ (①)❲ (①), ❲ (✵) = ❲✵,

◆ (①) =

• ✵

❢ (①) ¯ ❢ (①) ✵

• ✳

P✐❝❛r❞ ✐t❡r❛t✐♦♥ ♣r♦❝❡❞✉r❡ ❣✐✈❡s ❛ s♦❧✉t✐♦♥ r❡♣r❡s❡♥t❛t✐♦♥✿ ❲ (①) = ❈❢ (①) ❙❢ (①) ❙¯

❢ (①)

❈¯

❢ (①)

• ❲✵,

✇❤❡r❡

• ❈ ′

❢ (①) = ❢ (①) ❙¯ ❢ (①),

❙′

❢ (①) = ❢ (①) ❈¯ ❢ (①),

❈❢ (✵) = ✶, ❙❢ (✵) = ✵.

❋✐♥❛❧ r❡❞✉❝t✐♦♥

• ❈ ′

❢ (①) = ❢ (①) ❙¯ ❢ (①),

❙′

❢ (①) = ❢ (①) ❈¯ ❢ (①),

❈❢ (✵) = ✶, ❙❢ (✵) = ✵. ❚❤❡s❡ ❡q✉❛t✐♦♥s ❝❛♥ ❜❡ ❞❡❝♦✉♣❧❡❞✿ ❩± (①) := ❈❢ (①) ± ❙❢ (①) ⇒

• ❩ ′

± (①) = ±❢ (①) ¯

❩± (①), ❩± (✵) = ✶. ❍❡♥❝❡ ✐t ❛❧❧ ❜♦✐❧s ❞♦✇♥ t♦ s♦❧✈✐♥❣ ♦♥❧② ♦♥❡ ❖❉❊✿

• ❩ ′ (①) = ❢ (①) ¯

❩ (①) , ❩ (✵) = ✶.

❊①❛♠♣❧❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s

✉′′ (①) + ❛ (①)✉ (①) = ✵ ⇐ ⇒ ❯′ (①) = ❆ (①)❯ (①)

❱ (①)=P❯(①)

⇐ ⇒ ❱ ′ (①) = ❇ (①)❱ (①)

❬✶❪ ❑✳ ▲♦r❡♥③✱ ❚✳ ❏❛❤♥❦❡✱ ❈✳ ▲✉❜✐❝❤✱ ❆❞✐❛❜❛t✐❝ ✐♥t❡❣r❛t♦rs ❢♦r ❤✐❣❤❧② ♦s❝✐❧❧❛t♦r② s❡❝♦♥❞✲♦r❞❡r ❧✐♥❡❛r ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❡✐❣❡♥❞❡❝♦♠♣♦s✐t✐♦♥ ✭✷✵✵✺✮✳

❯ (①) :=

• α (①) ✉ (①)

β (①) ✉′ (①) + γ (①) ✉ (①)

• =
• ✉ (①)

❛−✶/✷ (①)✉′ (①)

• ✱

❆ (①) :=

• α′

α − γ β α β γ′−β❛ α

− β′+γ

αβ γ β′+γ β

• (①) =
• ✵
• ❛ (①)

−

• ❛ (①)

✵

• =√

❛(①)P−✶

• −✐

✵ ✵ ✐

• P

+

• ✵

✵ ✵ − ❛′(①)

✷❛(①)

• ✱

❇ (①) := P❆ (①)P−✶ = ✐√❛ − ❛′

✹❛ ✐❛′ ✹❛

− ✐❛′

✹❛

−✐√❛ − ❛′

✹❛

• (①)✱

P =

✶ √ ✷

✐ ✶ ✶ ✐

• ✳

❊①❛♠♣❧❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s

✉′′ (①) + ❛ (①)✉ (①) = ✵ ⇐ ⇒ ❯′ (①) = ❆ (①)❯ (①)

❱ (①)=P❯(①)

⇐ ⇒ ❱ ′ (①) = ❇ (①)❱ (①)

❬✷❪ ❆✳ ❆r♥♦❧❞✱ ◆✳ ❇❡♥ ❆❜❞❛❧❧❛❤✱ ❈✳ ◆❡❣✉❧❡s❝✉✱ ❲❑❇✲❜❛s❡❞ s❝❤❡♠❡s ❢♦r t❤❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✐♥ t❤❡ s❡♠✐✲❝❧❛ss✐❝❛❧ ❧✐♠✐t ✭✷✵✶✶✮✳

❯ (①) :=

• α (①) ✉ (①)

β (①) ✉′ (①) + γ (①) ✉ (①)

• =
• ❛✶/✹ (①)✉ (①)

✶ ❛✶/✹(①)✉′ (①) + ❛′(①) ✹❛✺/✹(①)✉ (①)

• ✱

❆ (①) :=

• α′

α − γ β α β γ′−β❛ α

− β′+γ

αβ γ β′+γ β

• (①) =
• ✵
• ❛ (①)

−

• ❛ (①)

✵

• =√

❛(①)P−✶

• −✐

✵ ✵ ✐

• P

+

• ✵

✵

❛′′(①) ✽❛✸/✷(①) − ✺(❛′(①))✷ ✸✷❛✺/✷(①)

✵

• ✱
• =:❜(①)

❇ (①) := P❆ (①)P−✶ = ✐ √❛ − ❜

• ❜

❜ −✐ √❛ − ❜

• (①)✱

P =

✶ √ ✷

✐ ✶ ✶ ✐

• ✳

❊①❛♠♣❧❡ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s

❱ ′ (①) = ❇ (①)❱ (①)

❲ (①)=❙(①)❱ (①)

⇐ ⇒ ❲ ′ (①) = ◆ (①)❲ (①),

■♥ ❬✶❪ ✿

❇ (①) = ✐√❛ − ❛′

✹❛ ✐❛′ ✹❛

− ✐❛′

✹❛

−✐√❛ − ❛′

✹❛

• (①) ,

◆ (①) =   ✵

✐❛′(①) ✹❛(①) ❡−✷✐ ①

✵

√

❛(t)❞t

− ✐❛′(①)

✹❛(①) ❡✷✐ ①

✵

√

❛(t)❞t

✵   .

■♥ ❬✷❪ ✿

❇ (①) = ✐ √❛ − ❜

• ❜

❜ −✐ √❛ − ❜

• (①) ,

◆ (①) =   ✵ ❜ (①)❡−✷✐

①

✵

√ ❛(t)−❜(t)

• ❞t

❜ (①)❡✷✐

①

✵

√ ❛(t)−❜(t)

• ❞t

✵   ,

❜ (①) :=

❛′′(①) ✽❛✸/✷(①) − ✺(❛′(①))✷ ✸✷❛✺/✷(①).

◆♦t❡✿ ✐♥ ❜♦t❤ ❬✶❪✲❬✷❪✱ ✇❡ ❤❛✈❡ ◆ (①) =

✵ ❢ ¯ ❢ ✵

• (①)

❢♦r s♦♠❡ ❢ (①)✱ ❛♥❞ ❤❡♥❝❡ ❛❝❤✐❡✈❡ ❛ r❡❞✉❝t✐♦♥ t♦ ❩ ′ (①) = ❢ (①)¯

❩ (①)✳

❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❩ ′ (①) = ❢ (①) ¯ ❩ (①)

❊✣❝✐❡♥t ❤②❜r✐❞ ❛s②♠♣t♦t✐❝❛❧✲♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❢♦r ✐♥✐t✐❛❧✲✈❛❧✉❡ ♣r♦❜❧❡♠s ✴ ❜♦✉♥❞❛r②✲✈❛❧✉❡ ♣r♦❜❧❡♠s ✭❜② ❛ ✏s❤♦♦t✐♥❣ ♠❡t❤♦❞✑✮ ✐♥ ❛ s❡♠✐✲❝❧❛ss✐❝❛❧ r❡❣✐♠❡ ❢♦r t❤❡ ✇❛✈❡ ❡q✉❛t✐♦♥ ✭❈ (①) ≡ ❛✵ (①) > ✵✮ ✴ ❝♦✉♣❧❡❞ s②st❡♠ ♦❢ ♦s❝✐❧❧❛t♦rs ✭❈ (①) ✐s ❛ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐①✮✱ ❛s ✐♥ ❬✶❪✲❬✷❪✱ ✉′′ (①) + ✶ ǫ✷ ❈ (①)✉ (①) = ✵, ① > ✵, ✵ < ǫ ≪ ✶. ❆❢t❡r t❤❡ r❡❞✉❝t✐♦♥✱ ♥♦ ♥❡❡❞ t♦ tr❛❝❡ ♠❛tr✐① str✉❝t✉r❡ ♦❢ P✐❝❛r❞ ✐t❡r❛t✐♦♥s✿ t❤❡ s❝❛❧❛r ♣r♦❜❧❡♠ ②✐❡❧❞s ❡①❛❝t❧② t❤❡ s❛♠❡ r❡s✉❧t ✦   P✐❝❛r❞ ✐t❡r❛t✐♦♥s ❢♦r ❩ ′ (①) = ❢ (①)¯ ❩ (①)✱ ❩ (✵) = ✶ ✿ ❩ (①) = ✶ + ①

✵ ❢ (t)❞t +

①

✵ ❢ (t✷)

t✷

✵ ¯

❢ (t✶)❞t✶❞t✷ + . . .  

❙♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❩ ′ (①) = ❢ (①) ¯ ❩ (①)

❲ ′ (①) = ◆ (①)❲ (①), ① > ✵, ❲ (✵) = ❲✵✳ ❋♦r ❬✷❪✿

◆ (①) =   ✵ ǫ❜✵ (①)❡− ✷✐

ǫ

①

✵

√ ❛✵(t)−ǫ✷❜✵(t)

• ❞t

ǫ❜✵ (①)❡

✷✐ ǫ

①

✵

√ ❛✵(t)−ǫ✷❜✵(t)

• ❞t

✵   ,

• =:¯

❢✵ (①)

❜✵ (①) :=

❛′′

✵ (①)

✽❛✸/✷

✵

(①) − ✺(❛′

✵(①)) ✷

✸✷❛✺/✷

✵

(①)✳

• ❩ ′

± (①) = ±ǫ❢✵ (①) ¯

❩± (①), ❩± (✵) = ✶.

⇒

• ❈❢✵ (①) = ✶

✷ (❩+ (①) + ❩− (①)) ,

❙❢✵ (①) = ✶

✷ (❩+ (①) − ❩− (①)) .

⇒ ❲ (①) = ❈❢✵ (①) ❙❢✵ (①) ❙¯

❢✵ (①)

❈¯

❢✵ (①)

• ❲✵.

❙♦♠❡ ❛❧t❡r♥❛t✐✈❡ r❡❢♦r♠✉❧❛t✐♦♥s ♦❢ ❩ ′ (①) = ❢ (①) ¯ ❩ (①)

Prü❢❡r t②♣❡ ❡q✉❛t✐♦♥ ❘❡♣r❡s❡♥t ❢ = |❢ | ❡✐Φ✱ ❳ = ❘❡✐Θ ❢♦r s♦♠❡ ❘ = ❘ (①) ≥ ✵✱ Φ = Φ (①)✱ Θ = Θ (①) ∈ R✳ ❚❤❡♥✿ Θ′ = − |❢ | sin (✷Θ − Φ), (log ❘)′ = ❘❡

• ❢ ❡−✷✐Θ

✳ ❇❛❝❦ t♦ ❛ ❧✐♥❡❛r ✷♥❞ ♦r❞❡r ❖❉❊✿ ✶ ¯ ❢ (t) ❞ ❞t ✶ ❢ (t) ❞ ❞t ❳ = ❳

❚♦✇❛r❞s ♥❡✇ ❝❧❛ss❡s ♦❢ ✏✐♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s

✏■♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s ✭≡ ❡①❛❝t❧② s♦❧✈❛❜❧❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✮ ❩ ′ (①) = ❢ (①)¯ ❩ (①), ① > ✵, ❩ (✵) = ✶. ❈♦♥s✐❞❡r ❡✳❣✳ ❢ (①) = ❡−α✵①✱ ❘❡ α✵ > ✵✳ ❚❛❦❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ✭ˆ ❋ (❦) := ∞

✵ ❡✐❦①❋ (①) ❞①✮✿

−✐❦ ˆ ❩ (❦) = ˆ ❩ (− (❦ + ✐ ¯ α✵)) + ✶, ❦ ∈ R✱ ⇒ −✐❦ ˆ ❩ (−❦) = ˆ ❩ (❦ − ✐ ¯ α✵) + ✶, ❦ ∈ R✱ ❙✐♥❝❡

ˆ ❩ (❦)✱ ˆ ❩ (−❦) ∈ ❍+ ✭❜♦✉♥❞❡❞ ❛♥❛❧②t✐❝ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡✮✱

−✐ (❦ + ✐ ¯ α✵) ˆ ❩ (− (❦ + ✐ ¯ α✵)) = ˆ ❩ (❦) + ✶, ❦ ∈ R,

✐❦❩ ❦ ✶

❩ ❦

✶

✵

✐❦ ✶ ❦ ❦ ✐

✵ ✳

❚♦✇❛r❞s ♥❡✇ ❝❧❛ss❡s ♦❢ ✏✐♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s

✏■♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s ✭≡ ❡①❛❝t❧② s♦❧✈❛❜❧❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✮ ❩ ′ (①) = ❢ (①)¯ ❩ (①), ① > ✵, ❩ (✵) = ✶. ❈♦♥s✐❞❡r ❡✳❣✳ ❢ (①) = ❡−α✵①✱ ❘❡ α✵ > ✵✳ ❚❛❦❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ✭ˆ ❋ (❦) := ∞

✵ ❡✐❦①❋ (①) ❞①✮✿

−✐❦ ˆ ❩ (❦) = ˆ ❩ (− (❦ + ✐ ¯ α✵)) + ✶, ❦ ∈ R✱ ⇒ −✐❦ ˆ ❩ (−❦) = ˆ ❩ (❦ − ✐ ¯ α✵) + ✶, ❦ ∈ R✱ ❙✐♥❝❡

ˆ ❩ (❦)✱ ˆ ❩ (−❦) ∈ ❍+ ✭❜♦✉♥❞❡❞ ❛♥❛❧②t✐❝ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡✮✱

−✐ (❦ + ✐ ¯ α✵) ˆ ❩ (− (❦ + ✐ ¯ α✵)) = ˆ ❩ (❦) + ✶, ❦ ∈ R,

• =−✐❦ ˆ

❩(❦)−✶

❩ ❦

✶

✵

✐❦ ✶ ❦ ❦ ✐

✵ ✳

❚♦✇❛r❞s ♥❡✇ ❝❧❛ss❡s ♦❢ ✏✐♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s

✏■♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s ✭≡ ❡①❛❝t❧② s♦❧✈❛❜❧❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✮ ❩ ′ (①) = ❢ (①)¯ ❩ (①), ① > ✵, ❩ (✵) = ✶. ❈♦♥s✐❞❡r ❡✳❣✳ ❢ (①) = ❡−α✵①✱ ❘❡ α✵ > ✵✳ ❚❛❦❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ✭ˆ ❋ (❦) := ∞

✵ ❡✐❦①❋ (①) ❞①✮✿

−✐❦ ˆ ❩ (❦) = ˆ ❩ (− (❦ + ✐ ¯ α✵)) + ✶, ❦ ∈ R✱ ⇒ −✐❦ ˆ ❩ (−❦) = ˆ ❩ (❦ − ✐ ¯ α✵) + ✶, ❦ ∈ R✱ ❙✐♥❝❡

ˆ ❩ (❦)✱ ˆ ❩ (−❦) ∈ ❍+ ✭❜♦✉♥❞❡❞ ❛♥❛❧②t✐❝ ✐♥ t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡✮✱

−✐ (❦ + ✐ ¯ α✵) ˆ ❩ (− (❦ + ✐ ¯ α✵)) = ˆ ❩ (❦) + ✶, ❦ ∈ R,

• =−✐❦ ˆ

❩(❦)−✶

= ⇒ ˆ ❩ (❦) = −

✶+¯ α✵−✐❦ ✶+❦(❦+✐ ¯ α✵)✳

❚♦✇❛r❞s ♥❡✇ ❝❧❛ss❡s ♦❢ ✏✐♥t❡❣r❛❜❧❡✑ ♣♦t❡♥t✐❛❧s

❩ ′ (①) = ❢ (①)¯ ❩ (①), ① > ✵, ❩ (✵) = ✶. ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ♦♥❡ ❝❛♥ ❛tt❡♠♣t✳✳✳ ❢ (①) = ▼

❥=✶ ❝❥❡−α❥①✱ ❝❥ ∈ C,

❘❡ α❥ > ✵❀

❢ (①) = ∞

✵ ❡−♣①ψ (♣)❞♣

❢♦r s♦♠❡ ψ (♣)✳ ✳✳✳

❙✉♠♠❛r② ✫ ❖✉t❧♦♦❦

❲❡ ❤❛✈❡ ❞✐s❝✉ss❡❞ ❤♦✇ ■❱P ❢♦r ❛ ❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥ ✇✐t❤ ❛ ♥♦♥✲✈❛♥✐s❤✐♥❣ ♣♦t❡♥t✐❛❧ ❝❛♥ ❜❡ r❡❞✉❝❡❞ ✐♥ ♠❛♥② ✇❛②s t♦ ❛♥ ❡q✉❛t✐♦♥ ♦❢ ❛♥ ❡①tr❡♠❡❧② s✐♠♣❧❡ ❢♦r♠ ❩ ′ (①) = ❢ (①)¯ ❩ (①)✳ ❇❡s✐❞❡s t❤❡ ♥❡❛t ❧♦♦❦✱ t❤❡ ♦❜t❛✐♥❡❞ ❡q✉❛t✐♦♥ ❤❛s ❛❞✈❛♥t❛❣❡s ❢♦r ❜♦t❤ ♥✉♠❡r✐❝❛❧ ❛♥❞ t❤❡♦r❡t✐❝❛❧ ✐♥✈❡st✐❣❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ✐t ❢❡❡❧s ❧✐❦❡ t❤❡ ♠❛✐♥ ❛❞✈❛♥t❛❣❡ ♦❢ s✉❝❤ ❛ r❡❢♦r♠✉❧❛t✐♦♥ ✐s ②❡t t♦ ❜❡ ✉♥❞❡rst♦♦❞✳✳✳ ✳✳✳ ❛s ✇❡❧❧ ❛s ✐ts ❣❡♥❡r❛❧✐③❛t✐♦♥s✳

❚❤❛♥❦ ❨♦✉✦