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■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❖♥ ❛ ▼✉❧t✐✲P❤②s✐❝s ❈♦✉♣❧✐♥❣ ▼❡❝❤❛♥✐s♠✳ ❚❤❡ ✸ r❞ ◆❛❥♠❛♥ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙♣❡❝tr❛❧ Pr♦❜❧❡♠s ❢♦r ❖♣❡r❛t♦rs ❛♥❞ ▼❛tr✐❝❡s ❘❛✐♥❡r P✐❝❛r❞ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s ❚❯ ❉r❡s❞❡♥✱ ●❡r♠❛♥② ❇✐♦❣r❛❞ ✷✵✶✸

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚❤❡ ❙❤❛♣❡ ♦❢ ❊✈♦❧✉t✐♦♥❛r② ❊q✉❛t✐♦♥s✳ ●❡♥❡r❛❧ ❋♦r♠ ♦❢ ❊✈♦❧✉t✐♦♥❛r② Pr♦❜❧❡♠s✿ ∂ ✵ ❱ + ❆❯ = ❢ ♦♥ R , ❱ = M ❯ . ❊✈♦❧✉t✐♦♥❛r② ❊q✉❛t✐♦♥✿ ( ∂ ✵ M + ❆ ) ❯ = ❢ . ❙♦❧✉t✐♦♥ ❚❤❡♦r②✿ ❉♦❡s t❤❡ ♦♣❡r❛t♦r ( ∂ ✵ M + ❆ ) − ✶ ❡①✐st ❛s ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r ♠❛♣♣✐♥❣ ♦♥ ❛ s✉✐t❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡❄ ❲❤✐❝❤ ✏s✉✐t❛❜❧❡✑ ❍✐❧❜❡rt s♣❛❝❡❄

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚❤❡ ❙❤❛♣❡ ♦❢ ❊✈♦❧✉t✐♦♥❛r② ❊q✉❛t✐♦♥s✳ ●❡♥❡r❛❧ ❋♦r♠ ♦❢ ❊✈♦❧✉t✐♦♥❛r② Pr♦❜❧❡♠s✿ ∂ ✵ ❱ + ❆❯ = ❢ ♦♥ R , ❱ = M ❯ . ❊✈♦❧✉t✐♦♥❛r② ❊q✉❛t✐♦♥✿ ( ∂ ✵ M + ❆ ) ❯ = ❢ . ❙♦❧✉t✐♦♥ ❚❤❡♦r②✿ ❉♦❡s t❤❡ ♦♣❡r❛t♦r ( ∂ ✵ M + ❆ ) − ✶ ❡①✐st ❛s ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r ♠❛♣♣✐♥❣ ♦♥ ❛ s✉✐t❛❜❧❡ ❍✐❧❜❡rt s♣❛❝❡❄ ❲❤✐❝❤ ✏s✉✐t❛❜❧❡✑ ❍✐❧❜❡rt s♣❛❝❡❄

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❊①♣♦♥❡♥t✐❛❧ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ t �→ ❡①♣ ( − ρ t ) ✱ ρ ∈ R ✱ ❣❡♥❡r❛t❡s ❛ ✇❡✐❣❤t❡❞ ▲ ✷ ✲s♣❛❝❡ ❍ ρ , ✵ ( R , C ) ❜② ❝♦♠♣❧❡t✐♦♥ ♦❢ t❤❡ s♣❛❝❡ ˚ ❈ ∞ ( R , C ) ♦❢ s♠♦♦t❤ ❝♦♠♣❧❡①✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s ✇✐t❤ ❝♦♠♣❛❝t s✉♣♣♦rt ✇✳r✳t✳ �·|·� ρ , ✵ ✭♥♦r♠✿ |·| ρ , ✵ ✮ � ( ϕ , ψ ) �→ R ϕ ( t ) ψ ( t ) ❡①♣ ( − ✷ ρ t ) ❞t . ❚✐♠❡✲❞✐✛❡r❡♥t✐❛t✐♦♥ ∂ ✵ ❛s ❛ ❝❧♦s❡❞ ♦♣❡r❛t♦r ✐♥ ❍ ρ , ✵ ( R , C ) ✐♥❞✉❝❡❞ ❜② ˚ ❈ ∞ ( R , C ) ⊆ ❍ ρ , ✵ ( R , C ) → ❍ ρ , ✵ ( R , C ) , ϕ �→ ϕ ′ .

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❚✐♠❡✲❞✐✛❡r❡♥t✐❛t✐♦♥ ∂ ✵ ✐s ❛ ♥♦r♠❛❧ ♦♣❡r❛t♦r ✐♥ ❍ ρ , ✵ ( R , C ) ∂ ✵ = Re ∂ ✵ + i Im ∂ ✵ = ✶ ✵ )+ i ✶ ✷ ( ∂ ✵ + ∂ ∗ ✷ i ( ∂ ✵ − ∂ ∗ ✵ ) ✇✐t❤ Re ∂ ✵ ✱ Im ∂ ✵ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♠✉t✐♥❣ r❡s♦❧✈❡♥ts✿ Re ∂ ✵ = ρ . ❋♦r ρ ∈ R \{ ✵ } ✿ ❝♦♥t✐♥✉♦✉s ✐♥✈❡rt✐❜✐❧✐t② ♦❢ ∂ ✵ ✱ ✐✳❡✳ ✵ ∈ ρ ( ∂ ✵ ) ✭r❡s♦❧✈❡♥t s❡t✮✿ σ ( ∂ ✵ ) = i R + ρ ✭s♣❡❝tr✉♠✮✳

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❚✐♠❡✲❞✐✛❡r❡♥t✐❛t✐♦♥ ∂ ✵ ✐s ❛ ♥♦r♠❛❧ ♦♣❡r❛t♦r ✐♥ ❍ ρ , ✵ ( R , C ) ∂ ✵ = Re ∂ ✵ + i Im ∂ ✵ = ✶ ✵ )+ i ✶ ✷ ( ∂ ✵ + ∂ ∗ ✷ i ( ∂ ✵ − ∂ ∗ ✵ ) ✇✐t❤ Re ∂ ✵ ✱ Im ∂ ✵ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦rs ✇✐t❤ ❝♦♠♠✉t✐♥❣ r❡s♦❧✈❡♥ts✿ Re ∂ ✵ = ρ . ❋♦r ρ ∈ R \{ ✵ } ✿ ❝♦♥t✐♥✉♦✉s ✐♥✈❡rt✐❜✐❧✐t② ♦❢ ∂ ✵ ✱ ✐✳❡✳ ✵ ∈ ρ ( ∂ ✵ ) ✭r❡s♦❧✈❡♥t s❡t✮✿ σ ( ∂ ✵ ) = i R + ρ ✭s♣❡❝tr✉♠✮✳

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠✿ ✉♥✐t❛r② ❡①t❡♥s✐♦♥ ♦❢ ❈ ∞ ( R , C ) ⊆ ❍ ρ , ✵ ( R , C ) → ❍ ✵ , ✵ ( R , C ) = ▲ ✷ ( R , C ) ˚ ϕ �→ L ρ ϕ ✶ � √ ✇✐t❤ L ρ ϕ ( ① ) = R ❡①♣ ( − i ① t ) ❡①♣ ( − ρ t ) ϕ ( t ) ❞t , ① ∈ R . ✷ π ✐s s♣❡❝tr❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r Im ∂ ✵ ✿ Im ∂ ✵ = L − ✶ ∂ ✵ = L − ✶ ρ ♠ ✵ L ρ , ( i ♠ ✵ + ρ ) L ρ . ρ ❍❡r❡ ♠ ✵ ✐s t❤❡ s❡❧❢❛❞❥♦✐♥t ♠✉❧t✐♣❧✐❝❛t✐♦♥✲❜②✲❛r❣✉♠❡♥t ♦♣❡r❛t♦r ✐♥ ▲ ✷ ( R , C ) ✿ ( ♠ ✵ ϕ )( ① ) = ① ϕ ( ① ) ❢♦r ① ∈ R ❛♥❞ ϕ ∈ ˚ ❈ ∞ ( R , C ) ✳

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❚❤❡ ❝❛♥♦♥✐❝❛❧ ❡①t❡♥s✐♦♥ ♦❢ ∂ ✵ t♦ t❤❡ ❳ ✲✈❛❧✉❡❞ ❝❛s❡✱ ❳ ❛ ❍✐❧❜❡rt s♣❛❝❡✱ ✐♥❤❡r✐ts t❤❡ ♥♦r♠❛❧✐t②✿ ∂ ✵ ✐s st✐❧❧ ❛ ♥♦r♠❛❧ ♦♣❡r❛t♦r ✐♥ ❍ ρ , ✵ ( R , ❳ ) ρ = Re ∂ ✵ . ❲✐t❤ t❤❡ ❡①t❡♥❞❡❞ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ L ρ : ❍ ρ , ✵ ( R , ❳ ) → ▲ ✷ ( R , ❳ ) ✇❡ st✐❧❧ ❣❡t ∂ ✵ = L − ✶ ( i ♠ ✵ + ρ ) L ρ . ρ

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❚❤❡ ❝❛♥♦♥✐❝❛❧ ❡①t❡♥s✐♦♥ ♦❢ ∂ ✵ t♦ t❤❡ ❳ ✲✈❛❧✉❡❞ ❝❛s❡✱ ❳ ❛ ❍✐❧❜❡rt s♣❛❝❡✱ ✐♥❤❡r✐ts t❤❡ ♥♦r♠❛❧✐t②✿ ∂ ✵ ✐s st✐❧❧ ❛ ♥♦r♠❛❧ ♦♣❡r❛t♦r ✐♥ ❍ ρ , ✵ ( R , ❳ ) ρ = Re ∂ ✵ . ❲✐t❤ t❤❡ ❡①t❡♥❞❡❞ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ L ρ : ❍ ρ , ✵ ( R , ❳ ) → ▲ ✷ ( R , ❳ ) ✇❡ st✐❧❧ ❣❡t ∂ ✵ = L − ✶ ( i ♠ ✵ + ρ ) L ρ . ρ

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❚❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❛s ❛ ◆♦r♠❛❧ ❖♣❡r❛t♦r ❚❤❡ ❝❛♥♦♥✐❝❛❧ ❡①t❡♥s✐♦♥ ♦❢ ∂ ✵ t♦ t❤❡ ❳ ✲✈❛❧✉❡❞ ❝❛s❡✱ ❳ ❛ ❍✐❧❜❡rt s♣❛❝❡✱ ✐♥❤❡r✐ts t❤❡ ♥♦r♠❛❧✐t②✿ ∂ ✵ ✐s st✐❧❧ ❛ ♥♦r♠❛❧ ♦♣❡r❛t♦r ✐♥ ❍ ρ , ✵ ( R , ❳ ) ρ = Re ∂ ✵ . ❲✐t❤ t❤❡ ❡①t❡♥❞❡❞ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ L ρ : ❍ ρ , ✵ ( R , ❳ ) → ▲ ✷ ( R , ❳ ) ✇❡ st✐❧❧ ❣❡t ∂ ✵ = L − ✶ ( i ♠ ✵ + ρ ) L ρ . ρ

■♥tr♦❞✉❝t✐♦♥ ❙♦❧✉t✐♦♥ ❚❤❡♦r② ❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s ❙✉♠♠❛r② ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ▼❛t❡r✐❛❧ ▲❛✇ ❖♣❡r❛t♦rs ❛s ❋✉♥❝t✐♦♥s ♦❢ t❤❡ ❚✐♠❡ ❉❡r✐✈❛t✐✈❡ ❲❡ ❛❧s♦ ❤❛✈❡ t❤❛t ✶ ∂ − ✶ = L − ✶ i ♠ ✵ + ρ L ρ , ρ ✵ ❛♥❞ s♦ ◆ ◆ ✶ ▼ ❦ ∂ − ❦ = L − ✶ ∑ ∑ ▼ ❦ ( i ♠ ✵ + ρ ) ❦ L ρ ρ ✵ ❦ = ✵ ❦ = ✵ ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r ♦♣❡r❛t♦rs ▼ ❦ ♦♥ ❳ ❛s ❝♦❡✣❝✐❡♥ts✱ ❦ = ✵ ,..., ◆ ✳ ◆♦t❡ t❤❛t ❢♦r ρ ∈ ] ✵ , ∞ [ � ① � = ✶ � � ∂ − ✶ � ∂ − ✶ � � ρ ❛♥❞ ✵ ϕ ( ① ) = − ∞ ϕ ( t ) ❞t ✵ ❢♦r ❛❧❧ ϕ ∈ ˚ ❈ ∞ ( R ) ❛♥❞ ① ∈ R ✳