Singularity-based Approach in a Pade-Chebyshev Resolution of the Gibbs Phenomenon

(1)

❙✐♥❣✉❧❛r✐t②✲❜❛s❡❞ ❆♣♣r♦❛❝❤

✐♥ ❛ P❛❞é✲❈❤❡❜②s❤❡✈ ❘❡s♦❧✉t✐♦♥

♦❢ t❤❡ ●✐❜❜s P❤❡♥♦♠❡♥♦♥

❆r♥❡❧ ▲✳ ❚❛♠♣♦s

∗

_{❏♦s❡ ❊r♥✐❡ ❈✳ ▲♦♣❡}

†

_{❏❛♥ ❙✳ ❍❡st❤❛✈❡♥}

‡

❆❜str❛❝t✖ ❲❡ ♣r❡s❡♥t ❛ s✐♥❣✉❧❛r✐t②✲❜❛s❡❞ ❛♣✲ ♣r♦❛❝❤ t♦ r❡s♦❧✈❡ t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥ t❤❛t ❛♣✲ ♣❡❛rs ✐♥ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t✐❡s✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ ❝♦♥✲ s✐❞❡r t❤❡ ♠♦r❡ ❞✐✣❝✉❧t ❝❛s❡ ✇❤❡r❡ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❛r❡ ♥♦t ❦♥♦✇♥✳ ❚❤❡ ✐❞❡♥t✐✲ ✜❝❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ s✐♥❣✉❧❛r✐t✐❡s ✐s ❝❛rr✐❡❞ ♦✉t ✉s✐♥❣ ❛ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❲❡ ♣r♦✈✐❞❡ ♥✉✲ ♠❡r✐❝❛❧ ❡①❛♠♣❧❡s t♦ ✐❧❧✉str❛t❡ t❤❡ ♠❡t❤♦❞✱ ✐♥❝❧✉❞✐♥❣ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦♥ ♣♦st♣r♦❝❡ss✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ ❝♦rr✉♣t❡❞ ❜② t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✳

❑❡②✇♦r❞s✿ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✱ ❢✉♥❝t✐♦♥ r❡❝♦♥✲ str✉❝t✐♦♥✱ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥

✶ ■♥tr♦❞✉❝t✐♦♥

❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ s♠♦♦t❤ ❢✉♥❝t✐♦♥s ❜② ❋♦✉r✐❡r s❡r✐❡s ♦r ❜② tr✉♥❝❛t❡❞ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥s ✐♥ ❣❡♥✲ ❡r❛❧ ✐s ❦♥♦✇♥ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧❧② ❝♦♥✈❡r❣❡♥t ❛♥❞ ❤✐❣❤❧② ❛❝❝✉r❛t❡ ❬✷✱✹❪✳ ❋♦r ❢✉♥❝t✐♦♥s ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s✱ ❤♦✇❡✈❡r✱ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛ ♣❛rt✐❛❧ s✉♠ ♦❢ ♦rt❤♦❣♦♥❛❧ s❡r✐❡s ✐s ❛❞✲ ✈❡rs❡❧② ❛✛❡❝t❡❞ ✐♥ t❤❡ ❛r❡❛ ♦✈❡r ✇❤✐❝❤ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❝❝✉r✱ ❛ ♣r♦❜❧❡♠ ✇❤✐❝❤ ❤❛s ❝♦♠❡ t♦ ❜❡ ❦♥♦✇♥ ❛s t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✳ ❚❤✐s ♣❤❡♥♦♠❡♥♦♥ ♠❛♥✐❢❡sts ✐♥ ❛♥ ♦s❝✐❧❧❛t♦r② ❜❡❤❛✈✐♦r ❛t t❤❡ ✈✐❝✐♥✐t② ♦❢ t❤❡ ❥✉♠♣s✳

❆ ❝❧❛ss ♦❢ t❡❝❤♥✐q✉❡s ❛✐♠❡❞ ❛t r❡s♦❧✈✐♥❣ ●✐❜❜s ♣❤❡✲ ♥♦♠❡♥♦♥ ❝♦♠♣r✐s❡s P❛❞é✲t②♣❡ ❛♣♣r♦①✐♠❛t✐♦♥s✳ ❆♥ ❛♣✲ ♣r♦①✐♠❛♥t ♦❢ t❤✐s t②♣❡ ❡♥❥♦②s t❤❡ ❛❞✈❛♥t❛❣❡ ♦❢ ✉t✐❧✐③✐♥❣ ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ❛s t❤✐s ❦✐♥❞ ♦❢ ❢✉♥❝t✐♦♥ ✐s ❜r♦❛❞❡r ❛♥❞ r✐❝❤❡r ✐♥ ❢♦r♠ t❤❛♥ ❛ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ✐s ❝♦♥s✐❞❡r❡❞ t❤❡ s✐♠♣❧❡st ❢✉♥❝t✐♦♥ t❤❛t ❝❛♥ ❤❛✈❡ s✐♥❣✉❧❛r✐t✐❡s✱ ❛♥❞ ❤❡♥❝❡ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ t❤❡ ♣♦❧❡s ♦❢ ❛ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛♥t ❜❡✲ ✐♥❣ ❝❧♦s❡ ❡♥♦✉❣❤ t♦ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜❡✐♥❣ ❛♣♣r♦①✐♠❛t❡❞ ❬✷✱✺❪✳

❙♦♠❡ P❛❞é✲❜❛s❡❞ ♠❡t❤♦❞s ✇♦r❦ ✇✐t❤♦✉t r❡q✉✐r✐♥❣ ✐♥✲ ❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❥✉♠♣ ❧♦❝❛t✐♦♥s✳ ❍♦✇❡✈❡r✱ ❧♦❝❛t✐♥❣ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t✐❡s ❝❛♥ ❜❡❝♦♠❡ ❛ r❡❧❡✈❛♥t ✐ss✉❡ ✇❤❡♥ t❤❡ ❛❝t✉❛❧ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❡①♣❧✐❝✐t❧② ❦♥♦✇♥✳ ■♥ ♠❛♥② ∗_{❉✐✈✐s✐♦♥ ♦❢ P❤②s✐❝❛❧ ❙❝✐❡♥❝❡s ❛♥❞ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢}

t❤❡ P❤✐❧✐♣♣✐♥❡s ❱✐s❛②❛s✱ ▼✐❛❣✲❛♦✱ ■❧♦✐❧♦✱ P❤✐❧✐♣♣✐♥❡s✳ ❊♠❛✐❧✿ ❛❧✲ t❛♠♣♦s❅②❛❤♦♦✳❝♦♠

†_{■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ t❤❡ P❤✐❧✐♣♣✐♥❡s ❉✐❧✐✲}

♠❛♥✱ ◗✉❡③♦♥ ❈✐t②✱ P❤✐❧✐♣♣✐♥❡s✳ ❊♠❛✐❧✿ ❡r♥✐❡❅♠❛t❤✳✉♣❞✳❡❞✉✳♣❤

‡_{❉✐✈✐s✐♦♥ ♦❢ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❇r♦✇♥ ❯♥✐✈❡rs✐t②✱ Pr♦✈✐✲}

❞❡♥❝❡✱ ❘✳■✳✱ ❯❙❆✳ ❊♠❛✐❧✿ ❏❛♥✳❍❡st❤❛✈❡♥❅❜r♦✇♥✳❡❞✉

❝❛s❡s✱ ❢♦r ✐♥st❛♥❝❡✱ ✐♥✈♦❧✈✐♥❣ s♣❡❝tr❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s ♦❢ ♥♦♥s♠♦♦t❤ s♦❧✉t✐♦♥s t♦ s♦♠❡ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛✲ t✐♦♥s✱ t❤❡ s♦❧✉t✐♦♥ ❝♦♠❡s ✐♥ t❤❡ ❢♦r♠ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ t❤❛t ❛r❡ ❝♦♥t❛♠✐♥❛t❡❞ ❜② ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✳ ❆s t❤❡s❡ ❞❛t❛ ❛r❡ ♥♦✐s②✱ t❤❡ st❛♥❞❛r❞ ♣r♦❝❡❞✉r❡ ✐s t♦ ♣♦st✲ ♣r♦❝❡ss t❤❡♠ t♦ ❝♦rr❡❝t t❤❡ ♣❤❡♥♦♠❡♥♦♥✳ ❖♥❡ ✇❛② t❤✐s ❝❛♥ ❜❡ ❞♦♥❡✱ ❛s ❞❡♠♦♥str❛t❡❞ ✐♥ ❬✶✱✺❪✱ ✐s t♦ ✉s❡ P❛❞é✲t②♣❡ ❛♣♣r♦①✐♠❛t✐♦♥✳ ❚❤✐s P❛❞é ♣♦st♣r♦❝❡ss✐♥❣ ❛♣✲ ♣r♦❛❝❤✱ ❤♦✇❡✈❡r✱ ♠❛② t✉r♥ ♦✉t t♦ ❜❡ ❧❡ss s✉❝❝❡ss❢✉❧ ✉♥✲ ❧❡ss ❢❡❞ ✇✐t❤ s♦♠❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ♣♦ss✐❜❧❡ ❥✉♠♣ ♣♦s✐t✐♦♥s ✇❤✐❝❤✱ ❛s ♥♦t❡❞ ✐♥ ❬✺❪✱ ❝❛♥ ❜❡ ❛❞✈❛♥t❛❣❡♦✉s ❢♦r ✐ts ❡✛❡❝t✐✈❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ❆s ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ ♠❛② ♥♦t s❤♦✇ ❡①♣❧✐❝✐t❧② t❤❡ ❡①✐st❡♥❝❡ ❛♥❞ ✇❤❡r❡❛❜♦✉ts ♦❢ ♣♦ss✐❜❧❡ ❥✉♠♣s✱ t♦ s♦♠❡❤♦✇ ❧♦❝❛t❡ t❤❡♠ ❝❛♥ ❜❡❝♦♠❡ ✐♠♣❡r❛t✐✈❡✳

❆ st✉❞② ❜② ❉r✐s❝♦❧❧ ❛♥❞ ❋♦r♥❜❡r❣ ❬✷❪ r❡✈❡❛❧s ❥✉st ❤♦✇ s✐❣♥✐✜❝❛♥t t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❥✉♠♣ ❧♦❝❛t✐♦♥s ❝❛♥ ❜❡ ✐♥ ❝♦rr❡❝t✐♥❣ t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✳ ❘❡❛❧✐③✐♥❣ t❤❛t t❤❡ ♣♦❧❡s ❛✈❛✐❧❛❜❧❡ ✐♥ ❛ r❛t✐♦♥❛❧ ❛♣♣r♦①✐♠❛♥t ❞♦ ♥♦t ✐♥tr✐♥s✐✲ ❝❛❧❧② ❛♥❞ ❛❞❡q✉❛t❡❧② r❡♣r♦❞✉❝❡ t❤❡ ❥✉♠♣ ❜❡❤❛✈✐♦rs ♦❢ ❛ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥f✱ t❤❡② ❞❡✈✐s❡❞ ❛♥ ❛♣♣r♦❛❝❤ t❤❛t ✐♥❝♦r♣♦r❛t❡s t❤❡ ❥✉♠♣ ❧♦❝❛t✐♦♥s ✐♥t♦ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦❝❡ss✳ ❆ s✐♠✐❧❛r ❛♣♣r♦❛❝❤ t❤❛t ✐♠❜✐❜❡s t❤✐s ❝♦♥❝❡♣t ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❞✐s✲ ❝✉ss❡❞ ✐♥ ❬✾❪✳

❚❤✐s ♣❛♣❡r ✐s ❛♥❝❤♦r❡❞ ♦♥ t❤❡ ❙✐♥❣✉❧❛r P❛❞é ✲ ❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥ ❞✐s❝✉ss❡❞ ✐♥ ❬✾❪✱ ❛ ❜r✐❡❢ r❡✈✐❡✇ ♦❢ ✇❤✐❝❤ ✐s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳ ❙❡❝t✐♦♥ ✸ ❞✐s✲ ❝✉ss❡s ❛ P❛❞é✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ✐♥ ✐❞❡♥t✐❢②✐♥❣ s✐♥❣✉❧❛r✐t✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❙❡❝t✐♦♥ ✹ ❢♦❝✉s❡s ♦♥ t❤❡ ♥✉♠❡r✐❝❛❧ r❡✲ s✉❧ts ♦❢ t❤❡ ❙P❈ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ r❡❝♦♥str✉❝t✐♥❣ ❛ t❡st ❢✉♥❝t✐♦♥ ❛♥❞ ♣♦st♣r♦❝❡ss✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛✳

✷ ❆

❙✐♥❣✉❧❛r✐t②✲❜❛s❡❞

P❛❞é✲

❈❤❡❜②s❤❡✈ ❘❡s♦❧✉t✐♦♥

❚❤❡ ❈❤❡❜②s❤❡✈ ❡①♣❛♥s✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥f ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

f(x) =c0 2 +

∞

X

n=1

(2)

✇❤❡r❡ Tn(x)❛r❡ t❤❡ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧s ❞❡✜♥❡❞ ❛s

Tn(x) = cos (nθ), θ = cos−1₍_x₎_{✱ ❛♥❞} _x_∈ _[₋₁_,_1]_{✳ ❚❤❡}

❝♦❡✣❝✐❡♥tscn ❛r❡ ❣✐✈❡♥ ❜②

cn = 2

π

ˆ 1

−1

f(x)Tn(x)

√

1−x2 dx

❛♥❞ ♠❛② ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ✉s✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ●❛✉ss✲ ❈❤❡❜②s❤❡✈ q✉❛❞r❛t✉r❡ r✉❧❡

ˆ 1

−1

h(x)ω(x)dx∼=

m

X

k=1

Akh(xk), ✭✷✮

✇❤❡r❡ {xk} ❛r❡ t❤❡ ③❡r♦s ♦❢ t❤❡ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧s

Tm(x) = cos (mθ)✱ h(x) = f(x)Tn(x)✱ ω(x) = 1

√

1−x2✱

❛♥❞Ak = π

m ❢♦r ❛❧❧k✳

❇② t❤❡ s✉❜st✐t✉t✐♦♥ z = eiθ_{✱ ❡①♣❛♥s✐♦♥ ✭✶✮ ✐s tr❛♥s✲} ❢♦r♠❡❞ ✐♥t♦

f(z) = 1 2

∞ X′

n=0

cnzn₊

∞

X′

n=0

cnz−n

,

✇❤❡r❡ t❤❡ ♣r✐♠❡❞ s✉♠ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ ✜rst t❡r♠ ✐s ❤❛❧✈❡❞✳ ▲❡t

g(z) =

∞

X′

k=0

cnzn. ✭✸✮

❲❡ r❡❢❡r t♦g(z)❛s t❤❡ tr❛♥s❢♦r♠❡❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s ❛s✲

s♦❝✐❛t❡❞ ✇✐t❤ f(z)✱ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② ✇✐t❤f(x)✳

▲❡t f(x) ❜❡ ❛ ♣✐❡❝❡✇✐s❡ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ [−1,1] ✇✐t❤ s ❥✉♠♣ ❧♦❝❛t✐♦♥s ❛t x= ξk ∈ [−1,1]✱ k = 1, . . . , s✱ ❛♥❞ ❝♦♥s✐❞❡r ✐ts ❛ss♦❝✐❛t❡❞ tr❛♥s❢♦r♠❡❞ ❈❤❡❜②✲ s❤❡✈ s❡r✐❡s ✭✸✮✳ ❚❤❡ ❙✐♥❣✉❧❛r P❛❞é✲❈❤❡❜②s❤❡✈ ✭❙P❈✮ ❛♣✲ ♣r♦①✐♠❛♥t t♦f(x) ♦❢ ♦r❞❡r(N, M, V1, . . . , Vs) ✐s ❞❡✜♥❡❞

❜② t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥

R(z) =

PN(z) +

s

X

k=1

RVk(z) log

1−_eiθzk

QM(z) , ✭✹✮

✇❤❡r❡z=eicos−1

(x) _❛♥❞

PN(z) =

N

X

j=0

pjzj_, _QM₍_z_{) =} M

X

j=0

qjzj₆_{= 0}_,

RVk(z) = Vk

X

j=0

r_j(k)zj_, _k_{= 1}_{, . . . , s,}

s✉❝❤ t❤❛t

QM(z)g(z)−[PN(z) +U(z)] =O zη+1_,

✇✐t❤

U(z) =

s

X

k=1

RVk(z) log

1− z

eiθk

❛♥❞

η=N+M +s+

s

X

k=1

Vk.

❚❤❡ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ♦❢ ♣♦❧②♥♦♠✐❛❧sPN✱QM✱ ❛♥❞ RVk ❛r❡ t❤❡♥ ❝♦♠♣✉t❡❞ t❤r♦✉❣❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r s②s✲ t❡♠ ♦❢η+ 1❡q✉❛t✐♦♥s ✐♥η+ 2✈❛r✐❛❜❧❡s✿

M

X∗

j=0

cN₋j+tqj−

V1

X

j=0

a(1)_N₋_j₊_tr(1)_j −· · ·−

Vs

X

j=0

a(_Ns)₋_j₊_tr_j(s)= 0,

M

X∗

j=0

cl₋jqj− V1

X

j=0

a(1)_l₋_jr(1)_j − · · · −

Vs

X

j=0

a(_l₋s)_jr(_js)=pl,

✇❤❡r❡ t = 1, . . . , η−N✱ l = 0, . . . , N✱ ❛♥❞ t❤❡ ❛st❡r✐s❦✲ ♠❛r❦❡❞ s✉♠♠❛t✐♦♥ ✐♥❞✐❝❛t❡s t❤❛t t❤❡ t❡r♠ ✇✐t❤ c0 ✐s

❤❛❧✈❡❞✳ ❲❡ ♥♦t❡ t❤❛t ✐♥ t❤✐s s②st❡♠✱ cn = 0✱ ❢♦rn <0✳

■t s❤♦✉❧❞ ❜❡ ♥♦t❡❞ t♦♦ t❤❛t t❤❡a(nk)❛r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ log 1− z

eiθk

❛♥❞ a(nk) = 0✱ ❢♦r n≤0✳ ❆❝❝♦r❞✐♥❣❧②✱R(N,M)(z)❛♣♣r♦①✐♠❛t❡sg(z)✇❤✐❝❤

✐♠♣❧✐❡s t❤❛t t❤❡ r❡❛❧ ♣❛rt ♦❢ R❛♣♣r♦①✐♠❛t❡s f(x)✳

✸ ❆♣♣r♦①✐♠❛t❡ ❏✉♠♣ ▲♦❝❛t✐♦♥s ♦❢

❛ ❉✐s❝♦♥t✐♥✉♦✉s ❋✉♥❝t✐♦♥

❚❤❡r❡ ❤❛✈❡ ❜❡❡♥ st✉❞✐❡s ♦♥ ❧♦❝❛t✐♥❣ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t✐❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ❬✸✱✼❪ ❛♥❞ s♦♠❡ ♦❢ t❤❡s❡ ❡①♣❧♦r❡ t❤❡ ❝♦♥♥❡❝✲ t✐♦♥ ❜❡t✇❡❡♥ ❥✉♠♣ ❧♦❝❛t✐♦♥s ❛♥❞ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ s❡r✐❡s ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳ ❊st✐♠❛t✐♥❣ ❥✉♠♣ ❧♦❝❛t✐♦♥s ✉s✲ ✐♥❣ P❛❞é ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ✐♥tr♦❞✉❝❡❞ ✐♥ ❬✷❪ ❛♥❞ ✐ts ❛♣♣❧✐✲ ❝❛❜✐❧✐t② ✐s ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ t❤❛t ❛ P❛❞é ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ s❡r✐❡s ❡①♣❛♥s✐♦♥ ♦❢ ❛ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥f ❧✐❦❡❧② ❧❡❛❞s t♦ ❛♥ ♦r❞✐♥❛r② ♣♦❧❡ ❛t ❛ ❥✉♠♣ ❧♦✲ ❝❛t✐♦♥✳ ❆s ♦✉r ❛♣♣r♦❛❝❤ ✐s ❢♦✉♥❞❡❞ ♦♥ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❡ ❢✉rt❤❡r ♣✉rs✉❡ t❤✐s ✐❞❡❛ t♦ ❣❡♥❡r❛t❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❥✉♠♣ ❧♦❝❛t✐♦♥s ♦❢ ❞✐s❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s✳

❋♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f✱ ❛ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐✲ ♠❛♥t ♦❢ ♦r❞❡r(N, M)♠❛② ❜❡ ❞❡✜♥❡❞ ❛s

Rf′(z) =

(Pf′)_N(z)

(Qf′)

M(z)

, ✭✺✮

✇❤❡r❡ z=eicos−1

(x)_❛♥❞

(Pf′)_N(z) =

N

X

j=0

(pf′)_jzj,

(Qf′)_M(z) =

M

X

j=0

(qf′)_jzj 6= 0,

s✉❝❤ t❤❛t

(Qf′)

M(z)g′(z)−(Pf′)N(z) =O z

N+M+1_.

❋✐♥❞✐♥❣ t❤❡ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts ♦❢ ♣♦❧②♥♦♠✐❛❧s (Pf′)

N ❛♥❞(Qf′)

(3)

s②st❡♠✿                      M X j=0

i(N+λ−j+ 1)cN+λ−j+1(qf′)_j = 0,

λ= 0,1,2, . . . , M−1, M

X

j=0

i(λ−j)cλ−j(qf′)_j = (pf′)_λ,

λ= 1,2. . . , N,

✇❤❡r❡ i = √−1 ❛♥❞ t❤❡ ❡①♣❛♥s✐♦♥ ❝♦❡✣❝✐❡♥ts ct = 0

❢♦r ❡❛❝❤t <0✳ ❲❡ r❡♠❛r❦ t❤❛tRf′ ❛♣♣r♦①✐♠❛t❡sg′(z)

✇❤✐❝❤ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❡❞ ❈❤❡❜②s❤❡✈ s❡✲ r✐❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ f(x) ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❡ r❡❛❧ ♣❛rt ♦❢

Rf′ ❛♣♣r♦①✐♠❛t❡s f′(x)✳

❘❡❝❛❧❧✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❦♥♦✇ t❤❛t θ = cos−1₍_x₎ _✇✐t❤ _x _∈ _[₋₁_,_1] _❛♥❞ _θ _∈

[0, π]✳ ❚❤✐s ❞❡✜♥❡s ❛ ♠❛♣♣✐♥❣ ❢r♦♠ [−1,1] ♦♥t♦ [0, π]✳

❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ z = eiθ _{❝♦♥s❡q✉❡♥t❧② ♠❛♣s} _[₋₁_,_1] t♦ t❤❡ ✉♣♣❡r ❤❛❧❢ ♦❢ t❤❡ ✉♥✐t ❝✐r❝❧❡ ✐♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ❛t ✇❤✐❝❤

eiθ

= 1✳ ◆♦✇ ❝♦♥s✐❞❡r t❤❡ P❛❞é✲❈❤❡❜②s❤❡✈

❛♣♣r♦①✐♠❛♥tRf′ t♦ g′✳ ▲❡tz₀ ❜❡ ❛ ③❡r♦ ♦❢ (Qf′)

M ♦r ❛ ♣♦❧❡ ♦❢ Rf′✳ ❲❡ ❤❛✈❡z₀ =eiθ0 ❢♦r s♦♠❡θ₀ ∈[0, π]✳ ❇②

t❤❡ ✐♥✈❡rs❡ ♠❛♣♣✐♥❣✱|z0|= 1✐♠♣❧✐❡s t❤❛tz0❝♦rr❡s♣♦♥❞s

t♦ ❛ ♣♦✐♥tx0✐♥[−1,1]✳ ❆sz0✐s ❛ s✐♥❣✉❧❛r✐t②✱x0♠✉st ❜❡

❛ ❥✉♠♣ ♦❢f(x)✐♥[−1,1].❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡z0= cosθ0+

isinθ0✱ t❤❡ ❥✉♠♣ ✐s ❧♦❝❛t❡❞ ❛tx0= cosθ0=ℜe(z0)✳

❚❤❡ ✐♠♠❡❞✐❛t❡❧② ♣r❡❝❡❞✐♥❣ ❞✐s❝✉ss✐♦♥ ♠❛② ❜❡ s✉♠♠❛✲ r✐③❡❞ ❜② st❛t✐♥❣ t❤❛t ❛ ♣♦❧❡ z0♦❢ Rf′ ❢♦r ✇❤✐❝❤|z0|= 1

❝♦rr❡s♣♦♥❞s t♦ ❛ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ f(x) ✐♥ [−1,1]

✇❤✐❝❤ ♦❝❝✉rs ❛t x = ℜe(z0)✳ ❚❤✐s ♣r♦✈✐❞❡s ❛ s✐♠♣❧❡

❝r✐t❡r✐♦♥ ❜② ✇❤✐❝❤ ✇❡ ♠❛② ❜❡ ❛❜❧❡ t♦ ❧♦❝❛t❡ ❛ ❥✉♠♣ ❞✐s✲ ❝♦♥t✐♥✉✐t② ♦❢ ❛ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ✉s✐♥❣ t❤❡ P❛❞é✲❈❤❡❜②s❤❡✈ ❛♣♣r♦①✐♠❛♥t ♦❢ ✐ts ❞✐✛❡r❡♥t✐❛t❡❞ s❡r✐❡s ❡①♣❛♥s✐♦♥✳ ❆s st❛t❡❞✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤♦s❡ ③❡r♦s ♦❢ (Qf′)

M ❢♦r ✇❤✐❝❤ t❤❡ ♠♦❞✉❧✉s ✐s ❡q✉❛❧ t♦ ✭♦r ❛♣♣r♦①✐♠❛t❡❧②✮ ✶ ✐♥ ♦r❞❡r t♦ ✐❞❡♥t✐❢② t❤❡ ③❡r♦t❤✲♦r❞❡r ❥✉♠♣s ♦❢ t❤❡ ❢✉♥❝t✐♦♥✳

✹ ◆✉♠❡r✐❝❛❧ ❘❡s✉❧ts

❲❡ ✜rst ✐♠♣❧❡♠❡♥t t❤❡ ❙P❈ ♠❡t❤♦❞ t♦ r❡❝♦♥str✉❝t t❤❡ ❢♦❧❧♦✇✐♥❣ t❡st ❢✉♥❝t✐♦♥

f(x) =

     √

1−x2_, ₀_≤_x_≤₁

0, −1/2≤x <0

−x−1, −1≤x <−1/2.

✭✻✮

❆s ❛ s❡❝♦♥❞ ❡①❛♠♣❧❡✱ ✇❡ s❤♦✇ ❤♦✇ t❤❡ ♠❡t❤♦❞ r❡✲ ❝♦✈❡rs ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ s❡t t❤❛t ✐s ❝♦♥t❛♠✐♥❛t❡❞ ❜② t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✳ ❋♦r t❤✐s ❝❛s❡✱ ✇❡ ❝♦♥s✐❞❡r ❛ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ ❢r♦♠ t❤❡ ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈✐s❝♦✉s ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥✿

∂u ∂t +u

∂u ∂x =ǫ

∂2_u

∂x2, x∈[−1,1], ǫ= 0.001 ✭✼✮

✇✐t❤ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

u(−1, t) =u(1, t) = 0 ✭✽✮

❛♥❞ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥

u(x,0) =−tanh

_x_{+ 0}_.₅

2ǫ

+ 1. ✭✾✮

✹✳✶ ❘❡❝♦♥str✉❝t✐♥❣

f

❚❤❡ ❡①❛❝t ❈❤❡❜②s❤❡✈ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❞❡✲ ✜♥❡❞ ❜② ✭✻✮ ❛r❡ ❣✐✈❡♥ ❜②

cn=     

−23+ 2+√3

π , n= 0

1+√3

π −

√

3 4π −

1

3, n= 1

k, n≥2,

✇❤❡r❡

k = 2nsin

nπ

2 −nsin 2nπ3 −

√

3 cos 2nπ₃ −2

n2−1

π

+ 2

nπsin

2nπ

3 .

❙✐♥❝❡ f ❤❛s ❦♥♦✇♥ ❞✐s❝♦♥t✐♥✉✐t② ❛tx= 0 ❛♥❞x=−12✱

✐ts ❙P❈ ❛♣♣r♦①✐♠❛♥t ✐s ❞❡t❡r♠✐♥❡❞ ❜②

P(z) +R1(z)L1(z) +R2(z)L2(z)

Q(z) ,

✇❤❡r❡

L1(z) = log

1−z_i

❛♥❞

L2(z) = log "

1− z

exp 2πi

3

#

.

(4)

❋✐❣✉r❡ ✷✿ ❈♦♥tr❛st ❜❡t✇❡❡♥ t❤❡ ❡①❛❝t f ❛♥❞ ✐ts ❙P❈ ✭✶✺✱✶✷✱✶✵✱✶✵✮ ❛♣♣r♦①✐♠❛♥t

❋✐❣✉r❡ ✸✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ♣♦✐♥t✇✐s❡ ❡rr♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✭❛✮ P❈✭✶✺✱✶✷✮ ❛♥❞ ✭❜✮ ❙P❈ ✭✶✺✱✶✷✱✶✵✱✶✵✮ ❛♣♣r♦①✲ ✐♠❛♥ts t♦f

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐s❝✉ss✐♦♥✱ ✇❡ ❞❡♥♦t❡ ❜② ❙P❈

(N, M, V1, . . . , Vs) ❛♥ ❙P❈ ❛♣♣r♦①✐♠❛♥t ♦❢ ♦r❞❡r

(N, M, V1, . . . , Vs) ✇❤✐❧❡ ✐ts ❝♦rr❡s♣♦♥❞✐♥❣ P❛❞é✲

❈❤❡❜②s❤❡✈ ✭P❈✮ ❛♣♣r♦①✐♠❛♥t ♦❢ ♦r❞❡r (N, M) ✐s

❞❡♥♦t❡❞ ❜② P❈(N, M)✳

❋✐❣✉r❡ ✶ s❤♦✇s t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥ ✐♥ ❛ P❈ ❛♣✲ ♣r♦①✐♠❛t✐♦♥ ♦❢ f✳ ❚❤❡ ♦s❝✐❧❧❛t✐♦♥ ❝❛✉s❡❞ ❜② t❤❡ ♣❤❡✲ ♥♦♠❡♥♦♥ ✐s ♣r❛❝t✐❝❛❧❧② ❡❧✐♠✐♥❛t❡❞ ✉♣♦♥ t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✬s s✐♥❣✉❧❛r✐t✐❡s ✐♥t♦ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦✲ ❝❡ss ❛s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✳ ❆♥ ❙P❈ ❛♣♣r♦①✐♠❛♥t ♦❢f ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷ ❛❣❛✐♥st t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❡①❛❝t ❢✉♥❝t✐♦♥✳ ❚❤❡ r❡❝♦♥str✉❝t✐♦♥ ✐s r❡♠❛r❦❛❜❧② ❣♦♦❞ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❡①❛❝t ❢✉♥❝t✐♦♥ ✐s ❤❛r❞❧② ♥♦t✐❝❡❛❜❧❡✳ ❆s s❤♦✇♥ ✐♥ ❋✐❣✲ ✉r❡ ✸✱ t❤✐s ✐♠♣r❡ss✐✈❡ r❡s✉❧t ❜② t❤❡ ❙P❈ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❝❧❡❛r❧② ♠❛r❦❡❞ ❜② ❛♥ ✐♠♣r♦✈❡❞ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♣♦✐♥t✲ ✇✐s❡ ❡rr♦r ❞r❛✇♥ ✐♥ ❧♦❣❛r✐t❤♠✐❝ s❝❛❧❡✳

✹✳✷ ❘❡❝♦✈❡r✐♥❣ ❙♦❧✉t✐♦♥ t♦ ❇✉r❣❡r✬s

❊q✉❛t✐♦♥

◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❜② s♣❡❝tr❛❧ ♠❡t❤♦❞ ❣❡♥❡r❛t❡s ❛ s❡t ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛ t❤❛t ✐s ❝♦rr✉♣t❡❞ ❜② t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥ ✐♥ t❤❡ s❡♥s❡ t❤❛t s♦❧✉t✐♦♥s t♦ s✉❝❤ ❡q✉❛t✐♦♥ ❛r❡ ❦♥♦✇♥ t♦ ❞❡✈❡❧♦♣ s❤❛r♣ ❣r❛❞✐❡♥t ✐♥ t✐♠❡ ❬✶❪✳ ❍❡r❡ ✇❡ ♣r❡s❡♥t s♦♠❡ r❡s✉❧ts ♦♥ t❤❡ ✉s❡ ♦❢ t❤❡ ❙P❈ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ ♣♦st♣r♦❝❡ss ♦r ✏❝❧❡❛♥ ✉♣✑ t❤❡ ❞❛t❛ ✐♥♦r❞❡r t♦ r❡❝♦✈❡r t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ✈✐s❝♦✉s ❇✉r❣❡r✬s ❡q✉❛t✐♦♥ ❞❡✜♥❡❞ ✐♥ ✭✼✮✲✭✾✮✳ ❚❤✐s ❡q✉❛t✐♦♥ ✐s ❛ s✉✐t❛❜❧❡ ♠♦❞❡❧ ❢♦r t❡st✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s ❢♦r ✢♦✇s ✇❤❡r❡ st❡❡♣ ❣r❛❞✐❡♥ts ♦r s❤♦❝❦s ❛r❡ ❛♥t✐❝✐♣❛t❡❞ ❜❡❝❛✉s❡ ✐t ❛❧❧♦✇s ❡①❛❝t s♦❧✉t✐♦♥s ❢♦r ♠❛♥② ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐♥✐t✐❛❧ ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❬✶❪✳ ■t s❤♦✉❧❞ ❜❡ ♥♦t❡❞ t❤❛t t❤❡ ♣♦st♣r♦❝❡ss✐♥❣ ♥❡❡❞s ♦♥❧② t♦ ❜❡ ❛♣♣❧✐❡❞ ❛t t✐♠❡ ❧❡✈❡❧s ❛t ✇❤✐❝❤ ❛ ✏❝❧❡❛♥✑ s♦❧✉t✐♦♥ ✐s ❞❡s✐r❡❞✱ ❛♥❞ ♥♦t ❛t ❡✈❡r② t✐♠❡ st❡♣ ❬✽❪✳

❋✐❣✉r❡ ✹✿ ❆♣♣r♦①✐♠❛t❡ s❤♦❝❦ ❧♦❝❛t✐♦♥ ♦❢ u ❛t x =

−0.4932143✇❤❡♥t= 0✱ ✉s✐♥❣ P❈✕■❈ ✭✸✱✸✮ ✇✐t❤m= 100

❋✐❣✉r❡ ✺✿ ❆♣♣r♦①✐♠❛t❡ s❤♦❝❦ ❧♦❝❛t✐♦♥ ♦❢ u ❛t x =

(5)

❋✐❣✉r❡ ✻✿ ❈♦♥str❛st ❜❡t✇❡❡♥ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ✐ts P❈✭✸✱✸✮ ❛♣♣r♦①✐♠❛♥t ❛tt= 0

❋✐❣✉r❡ ✼✿ ❈♦♥str❛st ❜❡t✇❡❡♥ t❤❡ ❡①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ✐ts P❈✭✸✱✸✮ ❛♣♣r♦①✐♠❛♥t ❛tt= 0.1

■♥ t❤✐s ❝❛s❡✱ t❤❡ tr❛♥s❢♦r♠❡❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s ❢♦r t❤❡ s♦❧✉t✐♦♥ ❛ss✉♠❡s ❡①♣❛♥s✐♦♥ ❝♦❡✣❝✐❡♥ts t❤❛t ❛r❡ ❛♣♣r♦①✲ ✐♠❛t❡❞ ✉s✐♥❣ ✭✷✮✳ ❚❤❡ ✐♥♣✉t ❞❛t❛ ❛r❡ ❣✐✈❡♥ ❛t t❤❡ m ●❛✉ss✲❈❤❡❜②s❤❡✈ q✉❛❞r❛t✉r❡ ♣♦✐♥ts✳ ❲♦r❦✐♥❣ ♦♥ t❤❡ ❛s✲ s✉♠♣t✐♦♥ t❤❛t t❤❡r❡ ♠❛② ❜❡ s♦♠❡ ✐♥❤❡r❡♥t ❥✉♠♣ ❞✐s❝♦♥t✐✲ ♥✉✐t✐❡s ♦r s❤❛r♣ ❣r❛❞✐❡♥t ♥♦t ❦♥♦✇♥ ♦r r❡❛❞✐❧② ♦❜s❡r✈❛❜❧❡ ❢r♦♠ t❤❡ ❞❛t❛✱ ✇❡ ✜rst s❡❡❦ t❤❡ ❧♦❝❛t✐♦♥s ♦❢ t❤❡s❡ ♣♦ss✐❜❧❡ ❥✉♠♣s ♦r s❤♦❝❦s ✐♥ t❤❡ ❞❛t❛ ❜② ✇❛② ♦❢ t❤❡ P❛❞é ❛♣♣r♦①✐✲ ♠❛t✐♦♥ ❛♣♣❧✐❡❞ t♦ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ ❡①♣❛♥s✐♦♥ t❤❛t r❡♣✲ r❡s❡♥ts t❤❡ s♦❧✉t✐♦♥u✳ ■♥❝♦r♣♦r❛t✐♥❣ t❤❡ r❡s✉❧t✐♥❣ s❤♦❝❦ ✐♥❢♦r♠❛t✐♦♥ ✐♥t♦ t❤❡ ❙P❈ ❛♣♣r♦①✐♠❛t✐♦♥ ❣❡♥❡r❛t❡s ❛ r❡✲ ❝♦♥str✉❝t❡❞u✳ ❋♦r ✐❧❧✉str❛t✐♦♥✱ ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ t✐♠❡ t= 0❛♥❞ t= 0.1✳ ❯♥❞❡r ❡❛❝❤ ❝❛s❡✱ ✇❡ t❛❦❡

❛s ✐♥♣✉ts s♦♠❡ ❝♦♠♣✉t❡❞ ❞❛t❛ t❤❛t s❡r✈❡ ❛s ✈❛❧✉❡s ♦❢u ❛t t❤❡ ❣✐✈❡♥m●❛✉ss✲❈❤❡❜②s❤❡✈ ♣♦✐♥ts✳

❋✐❣✉r❡ ✹ ♣r♦❞✉❝❡❞ ❜② t❤❡ P❈✭✸✱✸✮ ❛♣♣r♦①✐♠❛♥t t♦ t❤❡ ❞✐✛❡r❡♥t✐❛t❡❞ tr❛♥s❢♦r♠❡❞ ❈❤❡❜②s❤❡✈ ❡①♣❛♥s✐♦♥ ❛ss♦❝✐✲ ❛t❡❞ ✇✐t❤ u s❤♦✇s t❤❛t ❛t t = 0 ❛ ♣♦ss✐❜❧❡ ❥✉♠♣ ♦r

❋✐❣✉r❡ ✽✿ ❈♦♥tr❛st ❜❡t✇❡❡♥ ❡①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ✐ts ❙P❈✭✸✱✸✱✸✮ ❛♣♣r♦①✐♠❛♥t ❛t t= 0

❋✐❣✉r❡ ✾✿ ❈♦♥tr❛st ❜❡t✇❡❡♥ ❡①❛❝t s♦❧✉t✐♦♥ ❛♥❞ ✐ts ❙P❈✭✸✱✸✱✸✮ ❛♣♣r♦①✐♠❛♥t ❛t t= 0.1

s❤♦❝❦ ♦❝❝✉rs s♦♠❡✇❤❡r❡ ✈❡r② ❝❧♦s❡ t♦x=−0.5✳ ❚❤❡ ③❡✲

r♦s ♦❢ t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ t❤❡ P❈✭✸✱✸✮ ❛♣♣r♦①✐♠❛♥t ❛r❡

−0.0076362 ❛♥❞ −0.4932143±0.8696562i ✳ ❚❤❡ ❝♦♠✲ ♣❧❡① ③❡r♦ ❣✐✈❡s ❛ ♠♦❞✉❧✉s ♦❢ ✵✳✾✾✾✼✽✶✶ ✇❤✐❝❤ str♦♥❣❧② ✐♥❞✐❝❛t❡s t❤❛t ❛ s❤♦❝❦ ♦❝❝✉rs ❛t x=−0.4932143✳ ❚❤✐s

❝♦♥✜r♠s ✇❤❛t t❤❡ ♣❧♦t s❤♦✇s✳ ❋♦r t❤❡ ❝❛s❡ ✇❤❡♥t= 0.1✱

t❤❡ P❈✭✸✱✸✮ ❛♣♣r♦①✐♠❛t✐♦♥ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺ ✐♥❞✐❝❛t❡s t❤❛t t❤❡r❡ ✐s ❛ s❤♦❝❦ ✈❡r② ♥❡❛r x=−0.4✳ ❚❤❡ ③❡r♦s ♦❢

t❤❡ ❞❡♥♦♠✐♥❛t♦r ♦❢ t❤❡ P❈✭✸✱✸✮ ❛♣♣r♦①✐♠❛♥t ✐♥ t❤✐s ❝❛s❡ ❛r❡−0.4066582±0.9140550i❛♥❞−1.6168483✳ ❚❤❡ ❝♦♠✲

♣❧❡① ③❡r♦ ❣✐✈❡s ❛ ♠♦❞✉❧✉s ♦❢ ✶✳✵✵✵✹✸✸✼ ✐♠♣❧②✐♥❣ t❤❛t ❛ s❤♦❝❦ ❧♦❝❛t✐♦♥ ✐s ❛t x=−0.4066582✱ ✇❤✐❝❤ ✐s ✇❤❛t t❤❡

♣❧♦t s❡❡♠s t♦ s✉❣❣❡st✳ ■♥ ❝♦♥s✐❞❡r❛t✐♦♥ ♦❢ t❤❡ t✇♦ ❞✐❢✲ ❢❡r❡♥t s❤♦❝❦ ♣♦s✐t✐♦♥s ❛t t✇♦ ❞✐✛❡r❡♥t ♣♦✐♥ts ✐♥ t✐♠❡✱ ✇❡ ♥♦t❡ t❤❛t t❤❡ ❇✉r❣❡rs✬ s♦❧✉t✐♦♥ ✐♥✈♦❧✈❡s t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛ s❤♦❝❦ ♦r ❛ s❤❛r♣ ❣r❛❞✐❡♥t✳

(6)

❋✐❣✉r❡ ✶✵✿ ❈♦♠♣❛r✐s♦♥ ♦❢ ♣♦✐♥t✇✐s❡ ❡rr♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✭❛✮ P❈✭✸✱✸✮ ❛♥❞ ✭❜✮ ❙P❈ ✭✸✱✸✱✸✮ ❛♣♣r♦①✐♠❛♥ts ❛t t= 0

❋✐❣✉r❡ ✶✶✿ ❈♦♠♣❛r✐s♦♥ ♦❢ ♣♦✐♥t✇✐s❡ ❡rr♦r ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✭❛✮ P❈✭✸✱✸✮ ❛♥❞ ✭❜✮ ❙P❈ ✭✸✱✸✱✸✮ ❛♣♣r♦①✐♠❛♥ts ❛t t= 0.1

❋✐❣✉r❡s ✼ ❛♥❞ ✾ ❢♦r t = 0.1✳ ❚❤❡② ❛r❡ ♣❧♦tt❡❞ ❛❣❛✐♥st

t❤❡ ❡①❛❝t s♦❧✉t✐♦♥✳ ❇♦t❤ ❛♣♣r♦①✐♠❛♥ts ✐♥ t❤❡ t✇♦ ❝❛s❡s t❛❦❡ t❤❡ P❈ ❛♣♣r♦①✐♠❛t❡❞ ❥✉♠♣ ❧♦❝❛t✐♦♥s✱ t❤❛t ✐s✱ t❤❡ ❥✉♠♣ ❛t x = −0.4932143 ❢♦r t = 0 ❛♥❞ t❤❡ ❥✉♠♣ ❛t

x=−0.4066582 ❢♦r t = 0.1. ❚❤❡ ❙P❈ r❡s✉❧ts ❛r❡ q✉✐t❡ ✐♠♣r❡ss✐✈❡ ♥♦t✇✐t❤st❛♥❞✐♥❣ t❤❡ ❢❛❝t t❤❡ ✇❡ ♦♥❧② ✉s❡ ❧♦✇ ♦r❞❡r ❛♣♣r♦①✐♠❛♥ts t♦ ❣❡♥❡r❛t❡ t❤❡♠✳ ❈♦♠♣❛r✐s♦♥s ♦❢ t❤❡✐r r❡s♣❡❝t✐✈❡ ♣♦✐♥t✇✐s❡ ❡rr♦r ❝♦♥✈❡r❣❡♥❝❡ ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡s ✶✵ ❛♥❞ ✶✶✳

✺ ❈♦♥❝❧✉s✐♦♥

❚❤❡ ❙✐♥❣✉❧❛r P❛❞é✲❈❤❡❜②s❤❡✈ ✭❙P❈✮ ❛♣♣r♦①✐♠❛t✐♦♥ ❞❡♠♦♥str❛t❡s ❤♦✇ ❛ P❛❞é✲❈❤❡❜②s❤❡✈ ✭P❈✮ r❡❝♦♥str✉❝✲ t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ s✐♥❣✉❧❛r✐t✐❡s ✐s ❣r❡❛t❧② ❡♥❤❛♥❝❡❞ ❜② ✉t✐❧✐③✐♥❣ ✐ts s✐♥❣✉❧❛r✐t✐❡s ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦✲

❝❡ss✳ ■❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ❛r❡ ❦♥♦✇♥✱ t❤❡ ❙✐♥❣✉❧❛r P❛❞é✲❈❤❡❜②s❤❡✈ ✭❙P❈✮ ❛♣♣r♦①✐♠❛t✐♦♥ r❡♠❛r❦❛❜❧② r❡✲ ❝♦♥str✉❝ts s✉❝❤ ❢✉♥❝t✐♦♥✳ ❯♥❞❡r r❡str✐❝t✐✈❡ ❝♦♥❞✐t✐♦♥s ✇❤❡r❡ ♦♥❧② ❛♣♣r♦①✐♠❛t❡❞ ❡①♣❛♥s✐♦♥ ❝♦❡✣❝✐❡♥ts ❢♦r t❤❡ tr❛♥s❢♦r♠❡❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❛♣✲ ♣r♦①✐♠❛t❡❞ ❥✉♠♣ ❧♦❝❛t✐♦♥s ❛r❡ ✉s❡❞✱ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣♦st♣r♦❝❡ss✐♥❣ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❛t❛✱ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts st✐❧❧ r❡✈❡❛❧ t❤❛t t❤❡ ❙P❈ ❛♣♣r♦①✐♠❛♥t s✉❝❝❡ss❢✉❧❧② r❡✲ ✈♦❧✈❡s t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥ t❤❛t ♦❝❝✉rs ✐♥ t❤❡ ♣r♦❝❡ss ♦❢ r❡❝♦✈❡r✐♥❣ t❤❡ ❢✉♥❝t✐♦♥✳

❘❡❢❡r❡♥❝❡s

❬✶❪ ❉♦♥✱ ❲✳❙✳✱ ❑❛❜❡r✱ ❙✳▼✳✱ ▼✐♥✱ ▼✳❙✳✱ ❋♦✉r✐❡r✲P❛❞é ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ✜❧t❡r✐♥❣ ❢♦r t❤❡ s♣❡❝tr❛❧ s✐♠✉❧❛✲ t✐♦♥s ♦❢ ✐♥❝♦♠♣r❡ss✐❜❧❡ ❇♦✉ss✐♥❡sq ❝♦♥✈❡❝t✐♦♥ ♣r♦❜✲ ❧❡♠✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❈♦♠♣✉t❛t✐♦♥✱ ✼✻ ✭✷✵✵✼✮✱ ♣♣✳ ✶✷✼✺✲✶✷✾✵

❬✷❪ ❉r✐s❝♦❧❧✱ ❚✳❆✳✱ ❋♦r♥❜❡r❣✱ ❇✳✱ ❆ P❛❞❡✲❜❛s❡❞ ❛❧❣♦r✐t❤♠ ❢♦r ♦✈❡r❝♦♠✐♥❣ t❤❡ ●✐❜❜s ♣❤❡♥♦♠❡♥♦♥✱ ◆✉♠❡r✐❝❛❧ ❆❧✲ ❣♦r✐t❤♠s✱ ✷✻ ✭✷✵✵✶✮✱ ♥♦✳✶✱ ♣♣✳✼✼ ✲ ✾✷

❬✸❪ ●❡❧❜✱ ❆ ❛♥❞ ❚❛❞♠♦r✱ ❊✳✱ ❉❡t❡❝t✐♦♥ ♦❢ ❡❞❣❡s ✐♥ s♣❡❝tr❛❧ ❞❛t❛✱ ❆♣♣❧✐❡❞ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s✱ ✼ ✭✶✾✾✾✮✱ ♣♣✳ ✶✵✶✕✶✸✺✳

❬✹❪ ❍❡st❤❛✈❡♥✱ ❏✳❙✳✱ ●♦tt❧✐❡❜✱ ❙✳✱ ●♦tt❧✐❡❜✱ ❉✳✱ ❙♣❡❝tr❛❧ ▼❡t❤♦❞s ❢♦r ❚✐♠❡✲❉❡♣❡♥❞❡♥t Pr♦❜❧❡♠s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❯✳❑✳✱ ✷✵✵✼

❬✺❪ ❍❡st❤❛✈❡♥✱ ❏✳❙✳✱ ❑❛❜❡r✱ ❙✳▼✳✱ ▲✉r❛t✐✱ ▲✳✱ P❛❞é✲ ▲❡❣❡♥❞r❡ ✐♥t❡r♣♦❧❛♥ts ❢♦r ●✐❜❜s r❡❝♦♥str✉❝t✐♦♥✱ ❏♦✉r✲ ♥❛❧ ♦❢ ❙❝✐❡♥t✐✜❝ ❈♦♠♣✉t✐♥❣✱ ✷✽ ✭✷✵✵✻✮✱ ♣♣✳ ✸✸✼✲✸✺✾✳ ❬✻❪ ❏❡rr✐✱ ❆✳❏✳✱ ❚❤❡ ●✐❜❜s P❤❡♥♦♠❡♥♦♥ ✐♥ ❋♦✉r✐❡r ❆♥❛❧②✲

s✐s✱ ❙♣❧✐♥❡s ❛♥❞ ❲❛✈❡❧❡t ❆♣♣r♦①✐♠❛t✐♦♥✱ ❑❧✉✇❡r ❉♦r✲ ❞r❡❝❤t✱ ✶✾✾✽

❬✼❪ ❑✈❡r♥❛❞③❡✱●✳❑✳✱ ❆♣♣r♦①✐♠❛t✐♥❣ t❤❡ ❥✉♠♣ ❞✐s❝♦♥t✐♥✉✲ ✐t✐❡s ♦❢ ❛ ❢✉♥❝t✐♦♥ ❜② ✐ts ❋♦✉r✐❡r✲❏❛❝♦❜✐ ❝♦❡✣❝✐❡♥ts✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❈♦♠♣✉t❛t✐♦♥s✱ ✼✸ ✭✷✵✵✸✮✱ ♣♣✳ ✼✸✶ ✕ ✼✺✶✳

❬✽❪ ❙❛rr❛✱ ❙✳❆✳✱ ❙♣❡❝tr❛❧ ♠❡t❤♦❞s ✇✐t❤ ♣♦st♣r♦❝❡ss✐♥❣ ❢♦r ♥✉♠❡r✐❝❛❧ ❤②♣❡r❜♦❧✐❝ ❤❡❛t tr❛♥s❢❡r✱ ◆✉♠❡r✐❝❛❧ ❍❡❛t ❚r❛♥s❢❡r✱ P❛rt ❆✱ ✹✸✱ ♥♦✳✼ ✭✷✵✵✸✮✱ ♣♣✳✼✶✼✲✼✸✵ ❬✾❪ ❚❛♠♣♦s✱ ❆✳▲✳✱ ▲♦♣❡✱ ❏✳❊✳❈✳✱ ❆ P❛❞é✲❈❤❡❜②s❤❡✈ ❘❡✲