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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 ✹✱ ✷✵✶✾

❈❧❛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 > ✵ , ✉ ′ ( ✵ ) = ✉ ✶ . ✉ ( ✵ ) = ✉ ✵ ,

❆ ♠♦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✐♦♥ ✮

■❱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 ) , ⇐ ⇒ � � α ( ① ) ✉ ( ① ) ❯ ( ① ) := ✱ α ( ① ) ✱ β ( ① ) � = ✵✱ β ( ① ) ✉ ′ ( ① ) + γ ( ① ) ✉ ( ① )  α ′  α − γ α β β   ❆ ( ① ) :=  ( ① ) ✳ γ ′ − β ❛ − β ′ + γ β ′ + γ  γ α αβ β

❙♦♠❡ ❡❧❡♠❡♥t❛r② tr❛♥s❢♦r♠❛t✐♦♥s✿ ♠✐①✐♥❣ ❱ ( ① )= P❯ ( ① ) ❯ ′ ( ① ) = ❆ ( ① ) ❯ ( ① ) ❱ ′ ( ① ) = ❇ ( ① ) ❱ ( ① ) , ⇐ ⇒  α ′  α − γ α β β   ❆ ( ① ) =  ( ① ) ✱ γ ′ − β ❛ − β ′ + γ β ′ + γ  γ α αβ β � ♣ � r ❇ ( ① ) := P❆ ( ① ) P − ✶ =: ( ① ) ✱ s q P ✐s ❛ ❝♦♥st❛♥t ✐♥✈❡rt✐❜❧❡ ♠❛tr✐①✳

❚❤✐s ✐s ❞✉❡ t♦✿ ① ① ✵ ♣ t ❞t ✵ ♣ t ❞t ♣ ✵ ❡ ✵ ❡ ✵ ✳ ❱ ❱ ❱ ① ① ✵ q ✵ q t ❞t ✵ q t ❞t ✵ ❡ ✵ ❡ ▲❡ss ❡❧❡♠❡♥t❛r② ❜✉t ✐♠♣♦rt❛♥t tr❛♥s❢♦r♠❛t✐♦♥ ❲ ( ① )= ❙ ( ① ) ❱ ( ① ) ❱ ′ ( ① ) = ❇ ( ① ) ❱ ( ① ) ❲ ′ ( ① ) = ◆ ( ① ) ❲ ( ① ) , ⇐ ⇒ � ♣ � r ❇ ( ① ) = ( ① ) ✱ s q � exp � ① � � � − ✵ ♣ ( t ) ❞t ✵ ❙ ( ① ) := ✱ � ① � � ✵ exp − ✵ q ( t ) ❞t � ① � � � � ✵ r ( ① ) exp − ✵ ( ♣ ( t ) − q ( t )) ❞t ✳ ◆ ( ① ) := �� ① � s ( ① ) exp ✵ ( ♣ ( t ) − q ( t )) ❞t ✵

▲❡ss ❡❧❡♠❡♥t❛r② ❜✉t ✐♠♣♦rt❛♥t tr❛♥s❢♦r♠❛t✐♦♥ ❲ ( ① )= ❙ ( ① ) ❱ ( ① ) ❱ ′ ( ① ) = ❇ ( ① ) ❱ ( ① ) ❲ ′ ( ① ) = ◆ ( ① ) ❲ ( ① ) , ⇐ ⇒ � ♣ � r ❇ ( ① ) = ( ① ) ✱ s q � exp � ① � � � − ✵ ♣ ( t ) ❞t ✵ ❙ ( ① ) := ✱ � ① � � ✵ exp − ✵ q ( t ) ❞t � ① � � � � ✵ r ( ① ) exp − ✵ ( ♣ ( t ) − q ( t )) ❞t ✳ ◆ ( ① ) := �� ① � s ( ① ) exp ✵ ( ♣ ( t ) − q ( t )) ❞t ✵ ❚❤✐s ✐s ❞✉❡ t♦✿ � ♣ � � �� � � ′ � ① � ① � ✵ ♣ ( t ) ❞t ❡ − ✵ ♣ ( t ) ❞t ✵ ❡ ✵ ✵ ❱ ′ − ❱ = ❱ ✳ � ① � ① ✵ q ✵ q ( t ) ❞t ❡ − ✵ q ( t ) ❞t ✵ ❡ ✵

❆♥ ♦❜s❡r✈❛t✐♦♥ ❲ ′ ( ① ) = ◆ ( ① ) ❲ ( ① ) , ① > ✵ , ❲ ( ✵ ) = ❲ ✵ , � � ① � r ( ① ) ❡ − ✵ ( ♣ ( t ) − q ( t )) ❞t ✵ ◆ ( ① ) = ✳ � ① ✵ ( ♣ ( t ) − q ( t )) ❞t s ( ① ) ❡ ✵ � ① � ① ✵ ( ♣ ( t ) − q ( t )) ❞t = s ( ① ) ❡ ✵ ( ♣ ( t ) − q ( t )) ❞t =: ❢ ( ① ) ✱ r ( ① ) ❡ − ◆♦t❡ t❤❛t ✐❢ t❤❡♥ ❲ ( ① ) = (cosh ❢ ( ① ) ■ + sinh ❢ ( ① ) ❙ ) ❲ ✵ ✱ � ✶ � ✵ � � ✵ ✶ ✇❤❡r❡ ■ := , ❙ := ✳ ✵ ✶ ✶ ✵ ❍✐♥t✿ ❲r✐t❡ P✐❝❛r❞ ✐t❡r❛t✐♦♥ ❛♥❞ ✉s❡ ❙ ✷ ♠ = ■ ✱ ❙ ✷ ♠ + ✶ = ❙ ✱ ♠ ∈ N ✳

❢ 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 ) ❡ − ✵ ( ♣ ( t ) − q ( t )) ❞t ✵ ◆ ( ① ) = ✳ � ① ✵ ( ♣ ( t ) − q ( t )) ❞t s ( ① ) ❡ ✵ ❙❡❝♦♥❞ ❜❡st s✐t✉❛t✐♦♥✿ � ① � ① ✵ ( ♣ ( t ) − q ( t )) ❞t = ¯ ✵ (¯ ♣ ( t ) − ¯ q ( t )) ❞t r ( ① ) ❡ − s ( ① ) ❡

❆♥♦t❤❡r ♦❜s❡r✈❛t✐♦♥ ❲ ′ ( ① ) = ◆ ( ① ) ❲ ( ① ) , � � ① � r ( ① ) ❡ − ✵ ( ♣ ( t ) − q ( t )) ❞t ✵ ◆ ( ① ) = ✳ � ① ✵ ( ♣ ( t ) − q ( t )) ❞t s ( ① ) ❡ ✵ ❙❡❝♦♥❞ ❜❡st s✐t✉❛t✐♦♥✿ � ① � ① ✵ ( ♣ ( t ) − q ( t )) ❞t = ¯ ✵ (¯ ♣ ( t ) − ¯ q ( t )) ❞t =: ❢ ( ① ) ✱ r ( ① ) ❡ − s ( 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 ) = ❲ ✵ , ✇❤❡r❡ ❙ ¯ ❢ ( ① ) ❈ ¯ ❢ ( ① ) � ❈ ′ ❙ ′ ❢ ( 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 ) , ❩ ( ✵ ) = ✶ .