✷ ❍❛♥♥❛ ❆r✐♥✐ P❛r❤✉s✐♣
x(t) = (a−b)cos(t) +ccosa b −1)t
;y(t) = (a−b)sin(t)−csina b −1)t
. ✭✸✮
❚❤❡ ❤②♣♦❝②❝❧♦✐❞ ❝✉r✈❡ ♣r♦❞✉❝❡❞ ❜② ✜①❡❞ ♣♦✐♥t P ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ s♠❛❧❧ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s b r♦❧❧✐♥❣ ❛r♦✉♥❞ t❤❡ ✐♥s✐❞❡ ♦❢ ❛ ❧❛r❣❡ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s a ✭a > b ✮✱
✇❤❡r❡ tt❤❡ ❛♥❣✉❧❛r ❞✐s♣❧❛❝❡♠❡♥t ♦❢ t❤❡ ❝❡♥t❡r ♦❢ s♠❛❧❧ ❝✐r❝❧❡ ❬✸❪✳ ▼❛♥② ❛✉t❤♦rs
❛r❡ ❝♦♥s✐❞❡r✐♥❣ t❤❡ r❡s✉❧t✐♥❣ ❝✉r✈❡s ❞✉❡ t♦ ✐♥t❡❣❡rs ✈❛❧✉❡ ♦❢a/b✳ ❉❡s✐❣♥✐♥❣ ❝②❝❧♦✐❞
r❡❞✉❝❡rs ❛♥❞ ❞❡s✐❣♥✐♥❣ ❛♥ ❡①❛♠♣❧❡ ❢♦r ❛ r♦t❛t✐♥❣ r✐♥❣ ❣❡❛r t②♣❡ ❡♣✐❝②❝❧♦✐❞ r❡❞✉❝❡r ❬✶✵❪ ❛♥❞ ❞❡s✐❣♥✐♥❣ ❤②♣♦❝②❝❧♦✐❞ ❣❡❛r ❛ss❡♠❜❧② ✉s❡❞ ✐♥ ✐♥t❡r♥❛❧ ❝♦♠❜✉s✐♦♥ ❡♥❣✐♥❡ ❛r❡ ♣❛rt✐❝✉❧❛r ❡①❛♠♣❧❡s✳ ❇② ❝♦♥s✐❞❡r✐♥❣ t❤❡ r❛t✐♦♥ ♦❢ a/b ❝❛♥ ❜❡ ❛♥② ✐rr❛t✐♦♥❛❧
✈❛❧✉❡s✱ ♦♥❡ ♠❛② ♦❜s❡r✈❡ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ❝✉r✈❡s ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✲ ✺✳ ❚❤❡s❡ s♣❡❝✐❛❧ ❝✉r✈❡s ❛r❡ ❞r❛✇♥ ✇✐t❤ ❞✐✛❡r❡♥t st②❧❡s ♦❢ ❧✐♥❡ ✱❝♦❧♦r ❛♥❞ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ ♣❛r❛♠❡t❡r t♦ ♦❜t❛✐♥ s♦♠❡ ✈❛r✐❛♥t ♦❢ t❤❡s❡ ❝✉r✈❡s✳ ◆♦✇✱ ♦♥❡ ♠❛② ✉s❡ ❞✐✛❡r❡♥t ❝♦❧♦r ❢♦r t❤❡ ✉s❡❞ ❧✐♥❡s ❛♥❞ t②♣❡s ♦❢ ❧✐♥❡s t♦ ❞r❛✇ t❤❡ ❜❛s✐❝ ♣❛tt❡r♥ ❛s s❤♦✇♥ ✐♥ t❤❡s❡ ❡①❛♠♣❧❡s✳
❋✐❣✉r❡ ✶✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿❤②♣♦❝②❝❧♦✐❞ ✇✐t❤a=π, b=√2,0≤t≤200π❀
❡♣②tr♦❝❤♦✐❞ a = π, b = √2, c = 5,0 ≤t ≤15π❀ ❡♣②tr♦❝❤♦✐❞ a = 1, b = √2, c = 5,0≤t≤20π✳
❚❤❡ r❡s❡❛r❝❤ ❞❡s✐❣♥s ❞✐✛❡r❡♥t ♠♦t✐❢s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛❜♦✈❡ ❛♥❞ ✉s✐♥❣ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ♠❛♣♣✐♥❣ s✉❝❤ ❛s ❝♦♠♣❧❡① ♠❛♣♣✐♥❣ ❛♥❞ ❱♦r♦♥♦✐ ♠❛♣✲ ♣✐♥❣✳ ❙♦♠❡ ✈❛r✐❛♥ts ✐♥ ❝♦❧♦✉rs✱ ❧✐♥❡s ❛♥❞ ♦t❤❡r ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❛❧s♦ ✉s❡❞✳ ❑✐♥❞s ♦❢ ✢♦✇❡r ❛♥❞ ❛♥✐♠❛❧s ❛♣♣❡❛r ✐♥ t❤❡s❡ ♠♦t✐❢s ✐♥s♣✐r✐♥❣ t♦ ❞❡s✐❣♥ t❤r❡❡ ❞✐♠❡♥s✐♦♥❛❧ ♦❜❥❡❝ts ✉s❡❞ ❛s ♣r♦♣❡rt✐❡s ♦❢ ❤②♣♦❝②❝❧♦✐❞ ❞❛♥❝❡ ✇❤✐❝❤ ✐s ❛❧r❡❛❞② ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ②♦✉ t✉❜❡ ✭✐ts ❧✐♥❦ ❦❡②✇♦r❞ ✿ ❤②♣♦❝②❝❧♦✐❞ ❞❛♥❝❡✮✳ ❚❤✐s ♣❛♣❡r s❤♦✇s s♦♠❡ st❛♥❞❛r❞ ✐❞❡❛s ❢♦r ♣r❡s❡♥t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ♠❛♣♣❡❞ ❜② ❝♦♠♣❧❡① ♠❛♣♣✐♥❣ ❛♥❞ ❱♦r♦♥♦✐ ♠❛♣♣✐♥❣ ❛♥❞ t❤❡ ✉s❡❞ ❜❛s✐❝ t❤❡♦r✐❡s ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ s❡❝♦♥❞ ❝❤❛♣t❡r✳
❋✐❣✉r❡ ✷✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿❤②♣♦❝②❝❧♦✐❞ ✇✐t❤a=π, b=√2,0≤t≤200π❀
❡♣②tr♦❝❤♦✐❞ a = π, b = √2, c = 15,0 ≤ t ≤ 15π❀ ❤②♣♦❝②❝❧♦✐❞ ✇✐t❤ a = 0, b =
√
❆rts ❘❡✈❡❛❧❡❞ ✐♥ ❈❛❧❝✉❧✉s ❛♥❞ ■ts ❊①t❡♥s✐♦♥ ✺
❋✐❣✉r❡ ✸✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿✻✲❝✉s♣ ❤②♣♦❝②❝❧♦✐❞ ✇✐t❤ a= 1, a/b= 5,0≤t ≤2π❀
♠❛♣♣✐♥❣ ♦❢ w✐♥ ❊q✳ ✾ ✭♠✐❞❞❧❡✮ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ✭r✐❣❤t✮✳
❋✐❣✉r❡ ✹✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿✻✲❝✉s♣ ❤②♣♦❝②❝❧♦✐❞ ✇✐t❤a= 1, a/b= 5,0≤t≤200π❀
♠❛♣♣✐♥❣ ♦❢ w✐♥ ❊q✳ ✾ t❤❡♥ ✐ts ❞❡r✐✈❛t✐✈❡ ✭r✐❣❤t✮✳
❚❛❜❧❡ ✶✿ Pr♦❣r❛♠ ▼❆❚▲❆❇ t♦ ✈✐s✉❛❧✐③❡ ❤②♣♦❝②❝❧♦✐❞ ❛♥❞ ✐ts ❝♦♠♣❧❡① ♠❛♣♣✐♥❣ ❜② ✶✴③
Pr♦❣r❛♠ ♦❢ ❤②♣♦❝②❝❧♦✐❞ Pr♦❣r❛♠ ♦❢f(z) =cos(z)
✭❝♦♥t✐♥✉✐♥❣ ❧❡❢t ♣r♦❣r❛♠✮ ❛❤❂♣✐❀❜❤❂sqrt✭✷✮❀ ❜❡❧♦✇❂①❤✳Γ2 +yh.✷❀
❛✸❂✵❀❜✸❂✷✵✵✯♣✐❀ ✉❂①❤✳/❜❡❧♦✇❀
t✸❂❧✐♥s♣❛❝❡✭❛✸✱❜✸✱✺✵✵✮❀ ✈❂✲②❤✳/❜❡❧♦✇❀
r❛s✐♦❂✭❛❤✲❜❤✮✴❜❤❀ ❋✐❣✳
①❤❂✭❛❤✲❜❤✮✯❝♦s✭t✸✮✰ ❜❤✯❝♦s✭r❛s✐♦✯t✸✮❀ ♣❧♦t✭✉✱✈✱✬✕r♦✬✱✬▲✐♥❡❲✐❞t❤✬✱✷✱✳✳✳ ②❤❂✭❛❤✲❜❤✮✯s✐♥✭t✸✮✲❜❤✯s✐♥✭r❛s✐♦✯t✸✮❀ ✬▼❛r❦❡r❊❞❣❡❈♦❧♦r✬✱✬r✬✱✳✳✳ ❋✐❣✳ ✬▼❛r❦❡r❋❛❝❡❈♦❧♦r✬✱✬❦✬✱✳✳✳ ♣❧♦t✭①❤✱②❤✱✬✕rs✬✱✬▲✐♥❡❲✐❞t❤✬✱✷✱✳✳✳ ✬▼❛r❦❡r❙✐③❡✬✱✶✵✮
❆rts ❘❡✈❡❛❧❡❞ ✐♥ ❈❛❧❝✉❧✉s ❛♥❞ ■ts ❊①t❡♥s✐♦♥ ✼
❋✐❣✉r❡ ✺✿ ❊❛❝❤ ♣❛✐r ❝♦♥t❛✐♥s ❛ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ ✐ts ❝♦♠♣❧❡① ♠❛♣♣✐♥❣
❋✐❣✉r❡ ✻✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿❍②♣♦❝②❝♦❧♦✐❞ ✇❤✐❝❤ ✐ts r❛t✐♦ s❛t✐s✜❡s ●♦❧❞❡♥ r❛t✐♦✱ ✇✐t❤ b=√2 ✐s t❤❡♥ ♠❛♣♣❡❞ ❜②f(z) = 1/z ✭✷✲♥❞✮✱ ❝♦♠♣♦s✐t❡s ✇✐t❤ ❝♦s✭z✮ ✭✸✲r❞✮
❛♥❞ ❛❣❛✐♥ ✇✐t❤ ❝♦s✭z✮ ✭✹✲t❤✮✳
❋✐❣✉r❡ ✼✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿❊❛❝❤ ♣❛✐r ✭❛❜♦✈❡ ❛♥❞ ❜❡❧♦✇✮ ✐❧❧✉str❛t❡s t❤❡ ♠♦t✐❢ ❛♥❞ ✐ts ♦r♥❛♠❡♥t✳
✽ ❍❛♥♥❛ ❆r✐♥✐ P❛r❤✉s✐♣
❋✐❣✉r❡ ✾✿ ❋r♦♠ ❧❡❢t t♦ r✐❣❤t ✿ ❋✐rst ♣❛✐r ✐s ❛ ❤②♣♦❝②❝❧♦✐❞ ✭❊q✳ ✷✮ ♠❛♣♣❡❞ ❜② ❢✭③✮❂❝♦s✭③✮ ✳ ❙❡❝♦♥❞ ♣❛✐r ✿x(t) = rcos(2πt);y(t) = rsin(2πt);r = 2 +
sin(20πt); 0≤t≤π/2✳
✢♦✇❡rs ✇❤✐❝❤ ❛r❡ ♠♦st❧② ❢r♦♠ ❤②♣♦❝②❝❧♦✐❞ ❛♥❞ ✐ts ✈❛r✐❛♥ts ❞✉❡ t♦ s♦♠❡ ❝♦♠♣❧❡① ♠❛♣♣✐♥❣s✳ ❆❢t❡r ❜❡❝♦♠✐♥❣ s♦♠❡ ♦r♥❛♠❡♥ts✱ t❤❡s❡ ♦r♥❛♠❡♥ts ❛r❡ ✉s❡❞ t♦ ❜❡ ❞❡❝♦✲ r❛t✐♦♥ ♦❢ ❍②♣♦❝②❝❧♦✐❞ ❞❛♥❝❡ ✇❤✐❝❤ ✐s s❤♦✇♥ ✐♥ ②♦✉ t✉❜❡ ❛♥❞ t❤❡ ❞❛♥❝❡ ✐s ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❞✐s❝✉ss✐♦♥✳ ❚❤❡ ♣❤♦t♦s ❛♥❞ s♦♠❡ ♦r♥❛♠❡♥ts ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✳ ✽✲ ✶✶✳
❋✐❣✉r❡ ✶✵✿ ❖r♥❛♠❡♥ts ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❍②♣♦❝②❝❧♦✐❞ ❞❛♥❝❡
❋✐❣✉r❡ ✶✶✿ ▲❡❢t t♦ r✐❣❤t ✿ ▼♦t✐❢ ♦❜t❛✐♥❡❞ ❜② f(z) = cos(z) ✇❤❡r❡ z✲❞♦♠❛✐♥ ✐s
❝r❡❛t❡❞ ❜② ❤②♣♦❝②❝❧♦✐❞ ❝✉r✈❡✱ a=π, b=√2❀ ❚❤❡ ❝r❡❛t❡❞ ♦r♥❛♠❡♥t ✐s ❜❛s❡❞ ♦♥
t❤❡ ♠♦t✐❢ ❛♥❞ ✉s❡❞ ✐♥ ❍②♣♦❝②❝❧♦✐❞ ❞❛♥❝❡ ✭♦♥ t❤❡ ❜❛❝❦ ♦❢ t❤❡ ❞❛♥❝❡r✮✳
✸✳✷✳ ❱✐s✐t ♦♥ ❍②♣♦❝②❝❧♦✐❞✳
❉❡❧t♦✐❞ ✱ ❛str♦✐❞ ❛♥❞ ✺✲❝✉s♣ ❤②♣♦❝②❝❧♦✐❞ ❛r❡ ♣❛rt✐❝✉❧❛r ❤②♣♦❝②❝❧♦✐❞s ❝r❡❛t❡❞ ❜② ♣♦s✐t✐❢ ✐♥t❡❣❡r ✈❛❧✉❡s ♦❢ n = a/b✐♥ ❊q✳ ✷ ❢♦r n = 3,4 ❛♥❞ n = 5 r❡s♣❡❝t✐✈❡❧②✳ ❆♥❛❧②s❡s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ t♦♣♦❧♦❣✐❝❛❧str✉❝t✉r❡s ♦❢ ❡①✐st✐♥❣ ♣❧❛♥❡t❛r② ❣❡❛r tr❛✐♥ t②♣❡ ❤②♣♦❝②❝❧♦✐❞ ❛♥❞ ❡♣✐❝②❝❧♦✐❞ ♠❡❝❤✲ ❛♥✐s♠s ✇✐t❤ ♦♥❡ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ✐s ❛❧s♦ s❤♦✇♥ ❬✷❪✳ ❉❡❧t♦✐❞ ✐s ♠❛♣♣❡❞ ❜②
❆rts ❘❡✈❡❛❧❡❞ ✐♥ ❈❛❧❝✉❧✉s ❛♥❞ ■ts ❊①t❡♥s✐♦♥ ✾
❋✐❣✉r❡ ✶✷✿ ▲❡❢t t♦ r✐❣❤t ✿ ▼♦t✐❢ ♦❜t❛✐♥❡❞ ❜② ❉❡❧t♦✐❞ ❛♥❞ ♠❛♣♣❡❞ ❜②f(z) = 1/z+z
❛♥❞f(z) =z+iz2 ✳
s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✷✳ ❋✐♥❛❧❧② ✱ ♦♥❡ ♠❛② ✐♥♥♦✈❛t❡ t❤❡ ✜❣✉r❡ ❜② ✐♥❝r❡❛s✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢n❛♥❞ ♣❧✉❣ t❤❡ r❡s✉❧t✐♥❣ ❝✉r✈❡ ♦❢ ❡❛❝❤n♦♥ t❤❡ s❛♠❡ ✜❣✉r❡✳ ❚❤✐s ✐s ❞❡♣✐❝t❡❞ ✐♥
❋✐❣✳ ✶✹✳ ◆♦t❡ t❤❛t ❞♦✐♥❣ t❤✐s ❛❝t✐✈✐t② ✐s ❜❡❝♦♠✐♥❣ ♠♦r❡ ❡❛s✐❧② ❞♦♥❡ ❜② ♣r♦❣r❛♠✲ ♠✐♥❣ t❤❛♥ ❞♦♥❡ ❜② ♠❛♥✉❛❧ ❞r❛✇✐♥❣ ❛s ✉s✉❛❧❧② tr❛❞✐t✐♦♥❛❧ ♠❛t❤❡♠❛t✐❝✐❛♥s ❞♦✳ ❖♥❡ ♠❛② ✈❛r② ❞✐r❡❝t❧② t❤❡ ♦❜t❛✐♥❡❞ ♠♦t✐❢s ❜② ❝❤❛♥❣✐♥❣ ♣❛r❛♠❡t❡rs t❤♦✉❣❤ t❤❡r❡ ❡①✲ ✐sts ♥♦ ❛♥❛❧②t✐❝❛❧ ❡①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ♦❜t❛✐♥❡❞ ♠♦t✐❢s✳ ❚❤❡r❡❢♦r❡ ❦♥♦✇✐♥❣ ❛ ❜✐t ♦❢ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ ✐♥ ♠❛t❤❡♠❛t✐❝s ❡❞✉❝❛t✐♦♥ ✇✐❧❧ ❣✐✈❡ ❤✐❣❤❡r ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♠❛t❤❡♠❛t✐❝s✳
❋✐❣✉r❡ ✶✸✿ ▲❡❢t t♦ r✐❣❤t ✿❊❛❝❤ ♣❛✐r ♦❜t❛✐♥❡❞ ❜② ❊q✳✷ ❛♥❞ ♠❛♣♣❡❞ ❜②f(z) =cos(z)
❛♥❞ ❜♦t❤ ❛r❡ ♠❛♣♣❡❞ ❜② ✈♦r♦♥♦✐ ♠❛♣♣✐♥❣✳
❋✐❣✉r❡ ✶✹✿ ▲❡❢t t♦ r✐❣❤t ✿ ▼♦t✐❢ ♦❜t❛✐♥❡❞ ❜② ❉❡❧t♦✐❞ ❛♥❞ ♠❛♣♣❡❞ ❜②f(z) = 1/z+z
❛♥❞f(z) =z+iz2 ❜② ✈❛r②✐♥❣n❂✷✱✳✳✳✱✶✵✳
❆s s❤♦✇♥ ✐♥ t❤❡♦r❡t✐❝❛❧ ♣❛rt✱ ♦♥❡ ♥❡❡❞s t♦ ✈✐s✉❛❧✐③❡ ❛ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✳ ▲❡t ✉s ❝♦♥s✐❞❡rf(z) =cos(z)❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡✱✐✳❡ df
dz =−sin(z)❛♣♣❧✐❡❞
t♦ t❤❡ z✲❞♦♠❛✐♥ ❝r❡❛t❡❞ ❜② ❛ ❤②♣♦❝②❝❧♦✐❞ ❝✉r✈❡ ✇✐t❤ ❛♥ ✐rr❛t✐♦♥❛❧ ✈❛❧✉❡ ♦❢ a/b✳
❆♥ ❡①❛♠♣❧❡ ✐s ❞❡♣✐❝t❡❞ ✐♥ ❋✐❣✳ ✶✺✳
✸✳✸✳ ❘❡s✉❧t ♦♥ ❱♦r♦♥♦✐ ♠❛♣♣✐♥❣✳
❉❡s✐❣♥ ♦❢ ✈♦r♦♥♦✐ ♠❛♣♣✐♥❣ ✐s ♥♦t ❞✐s❝✉ss❡❞ ✐♥t♦ ❞❡t❛✐❧ ❤❡r❡ s✐♥❝❡ t❤❡ r❡s❡❛r❝❤ ❝♦♥❝❡r♥s ♦♥❧② ✐ts ♦✉t♣✉t ❞✉❡ t♦ t❤❡ ❣✐✈❡♥ ❞♦♠❛✐♥✱ ✐✳❡ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡ ❛♥❞ t❤❡ r❡s✉❧t ♦❢ ❝♦♠♣❧❡① ♠❛♣♣✐♥❣✳ ❙✉♣♣♦s❡✱ ✇❡ ❤❛✈❡ t❤❡ ❤②♣♦❝②❝❧♦✐❞ ❡q✉❛t✐♦♥ s❤♦✇♥ ❊q✳ ✷✳ ❇② ❡♠♣❧♦②✐♥❣ t❤❡ f(z) = cos(z)✶✵ ❍❛♥♥❛ ❆r✐♥✐ P❛r❤✉s✐♣
❋✐❣✉r❡ ✶✺✿ ▲❡❢t t♦ r✐❣❤t ✿❤②♣♦❝②❝❧♦✐❞ ✇✐t❤ a/b = 1.6✱ 0 ≤ t ≤200π ✭r✐❣❤t✮✱ ✐ts
♠❛♣♣❡❞ ❜②f(z) =cos(z)✭♠✐❞❞❧❡✮✱ ❛♥❞ ✐ts ❞❡r✐✈❛t✐✈❡ ♦❢f(z)✐✳❡ ✿✲sin(z)✭r✐❣❤t✮✳
♠❛♣♣✐♥❣✳ ❚❤❡ ✈♦r♦♥♦✐ ❢✉♥❝t✐♦♥ ♣r♦✈✐❞❡❞ ❜② ▼❆❚▲❆❇ ✐s ❛♣♣❧✐❡❞ t♦ t❤❡s❡ t✇♦ ❞♦♠❛✐♥s ❛♥❞ ♦♥❡ ②✐❡❧❞s t❤❡ ✈♦r♦♥♦✐ ❞✐❛❣r❛♠s ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✸✳ ❚❤❡ ♦❜t❛✐♥❡❞ ♣❛tt❡r♥s ❛r❡ ♣♦ss✐❜❧❡ t♦ ❜❡ ❝❧♦t❤❡s ♠♦t✐❢s ✐♥ ❢✉t✉r❡ r❡s❡❛r❝❤✳
❋✐❣✉r❡ ✶✻✿ ▲❡❢t t♦ r✐❣❤t ✿❍②♣♦❝②❝❧♦✐❞ ✇✐t❤a =π, b =p
(2)✱ ♠❛♣♣❡❞ ❜② ✶✴z✱ ✸✲
❞✐♠❡♥s✐♦♥❛❧ ❉❡❧❛✉♥❛② t❡ss❡❧❧❛t✐♦♥ ♦❢ ✭x, y, u✮ ✇❤❡r❡ ✭x, y✮ t❛❦❡♥ ❢r♦♠ ❍②♣♦❝②❝❧♦✐❞
❛♥❞u✐s t❤❡ r❡❛❧ ♣❛rt ♦❢ ✶✴z✳
❈r❡❛t✐♦♥ ♦♥ ♠♦t✐❢s ❛♥❞ t❤❡ r❡s✉❧t✐♥❣ ♦r♥❛♠❡♥ts ❝❛♥ ❜❡ ❛❧s♦ ✐♠♣r♦✈❡❞ ✐♥ t❤r❡❡ ❞✐♠❡♥s✐♦♥❛❧ ♦❜❥❡❝ts✳ ❚❤❛♥❦ t♦ s♦♠❡ ✉s❡r ❢r✐❡♥❞❧② ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ ▼❆❚▲❆❇✳ ❍②♣♦❝②❝❧♦✐❞ ✇✐t❤ ✐rr❛t✐♦♥❛❧ ✈❛❧✉❡ ♦❢ ✭a/b✮ ♠❛♣♣❡❞ ❜②1/z✇✐❧❧ ❣✐✈❡ s❡ts
♦❢ ✭x, y, u✮ ❛♥❞ ✭x, y, v✮✳ ❚❤❡s❡ s❡ts ❛r❡ ♣♦ss✐❜❧❡ t♦ ♣r❡s❡♥t ✉s✐♥❣ ❞✉❛❧ ♦❢ ❱♦r♦♥♦✐
❞✐❛❣r❛♠ ❣✐✈❡♥ ❜② ❞❡❧❛✉♥❛②✸✳♠ ✐♥ ▼❆❚▲❆❇✳ ❖♥❡ ♦❢ ❡①❛♠♣❧❡s ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✻✳ ❙♦♠❡ ♣♦ss✐❜❧❡ ✸ ❞✐♠❡♥s✐♦♥❛❧ ♦❜❥❡❝ts ❝❛♥ st✐❧❧ ❜❡ ❝r❡❛t❡❞✳
✹✳ ❈♦♥❝❧✉s✐♦♥
P❛r❛♠❡tr✐❝ ❝✉r✈❡s s✉❝❤ ❛s ❤②♣♦❝②❝❧♦✐❞ ✱ ❡♣②❝②❝❧♦✐❞ ❛♥❞ ❡♣②tr♦❝♦✐❞ ❛r❡ r❡✈✐❡✇❡❞ ❤❡r❡ t♦ ❜❡ t❤❡z✲❞♦♠❛✐♥s✳ ❚❤❡ r❡s✉❧t✐♥❣ ❝✉r✈❡s ❛r❡ t❤❡♥ ❝❛♥ ❜❡ ♠❛♣♣❡❞ ❜② ❝♦♠♣❧❡①
♠❛♣♣✐♥❣ ❛♥❞ ❱♦r♦♥♦✐ ♠❛♣♣✐♥❣✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❝♦♠♣❧❡① ♠❛♣♣✐♥❣s✱ ♦♥❡ ♥❡❡❞s t♦ ❡①✲ ♣r❡ss ✐♥t♦ r❡❛❧ ♣❛rt ❛♥ ✐♠❛❣✐♥❛r② ♣❛rt ❢♦r ❡❛❝❤ ❢✉♥❝t✐♦♥✳ ❙❡✈❡r❛❧ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥s ❛r❡ s❤♦✇♥ ❤❡r❡✱ ❡✳❣ t❤❡ r❡❛❧ ♣❛rt ❛♥❞ ✐♠❛❣✐♥❛r② ♣❛rts ♦❢ ❝♦s✭z✮✱ s✐♥✭z✮✱✶✴z✱ z2 ✱
❛♥❞ez✱ 1+z
1−z✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❡❛❝❤ ❢✉♥❝t✐♦♥ ❝❛♥ ❛❧s♦ ❜❡ ❞❡r✐✈❡❞ ❛♥❞ ❝♦♥s✐❞❡r❡❞ t♦
❆rts ❘❡✈❡❛❧❡❞ ✐♥ ❈❛❧❝✉❧✉s ❛♥❞ ■ts ❊①t❡♥s✐♦♥ ✶✶
✺✳ ❘❡❢❡r❡♥❝❡s
❘❡❢❡r❡♥❝❡s
❬✶❪ ❆✉r❡♥❤❛♠♠❡r✱❋✳✱ ❱♦r♦♥♦✐ ❉✐❛❣r❛♠s â⑨➈ ❆ ❙✉r✈❡② ♦❢ ❛ ❋✉♥❞❛♠❡♥t❛❧ ●❡♦♠❡tr✐❝ ❉❛t❛ ❙tr✉❝t✉r❡✱❆❈▼ ❈♦♠♣✉t✐♥❣ ❙✉r✈❡②s✱✭✶✾✾✶✮✱✭✷✸✮✱◆♦✳ ✸✳
❬✷❪ ❇❛rt♦❧♦✱❊✳❆✳✱❆❣✉st➹➢♥✱ ❏✳■ ✳❈✳✱❖♥ t❤❡ ❚♦♣♦❧♦❣② ♦❢ ❍②♣♦❝②❝❧♦✐❞s✱ ❋➹➢s✐❝❛ ❚❡➹➩r✐❝❛✱ ❏✉❧✐♦ ❆❜❛❞✱✭✷✵✵✽✮✱✶✲✶✻✳
❬✸❪ ❍s✉ ▼✳❍✳✱ ❨❛♥✱❍✳❙✳✱ ▲✐✉✱❏✳❨✳✱ ❛♥❞ ❍s✐❡❤ ▲✳❈✳✱ ❊♣✐❝②❝❧♦✐❞ ✭❍②♣♦❝②❝❧♦✐❞✮ ▼❡❝❤❛♥✐s♠s ❉❡s✐❣♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ▼✉❧t✐❈♦♥❢❡r❡♥❝❡ ♦❢ ❊♥❣✐♥❡❡rs ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥t✐sts✱ ■▼❊❈❙✱❍♦♥❣ ❑♦♥❣✭✷✵✵✽✮✱✭✷✮✳
❬✹❪ ▲❡❞♦✉①✱ ❍✳✱ ●♦❧❞ ❈✳✱ ❆ ❱♦r♦♥♦✐✲❇❛s❡❞ ▼❛♣ ❆❧❣❡❜r❛✱ Pr♦❣r❡ss ✐♥ ❙♣❛t✐❛❧ ❉❛t❛ ❍❛♥✲ ❞❧✐♥❣✱ ✭✷✵✵✻✮✱ ✭✶✶✼✲✶✸✶✮
❬✺❪ P❛r❤✉s✐♣✱❍✳❆✳✱▲❡❛r♥✐♥❣ ❈♦♠♣❧❡① ❋✉♥❝t✐♦♥ ❆♥❞ ■ts ❱✐s✉❛❧✐③❛t✐♦♥ ❲✐t❤ ▼❛t❧❛❜✱ Pr♦✲ ❝❡❡❞✐♥❣s ❙♦✉t❤ ❊❛st ❆s✐❛♥ ❈♦♥❢❡r❡♥❝❡ ❖♥ ▼❛t❤❡♠❛t✐❝s ❆♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s ✐♥ ■♥st✐t✉t ❚❡❦♥♦❧♦❣✐ ❙❡♣✉❧✉❤ ◆♦♣❡♠❜❡r✱ ❙✉r❛❜❛②❛ ✱✭✷✵✶✵✮✭✶✮✱✸✼✸✕✸✽✹✳
❬✻❪ P❛r❤✉s✐♣✱ ❍✳ ❆✳✱ ❙✉❧✐st②♦♥♦✱ P❡♠❡t❛❛♥ ❞❛♥ ❤❛s✐❧ ♣❡♠❡t❛❛♥♥②❛✱Pr♦s✐❞✐♥❣✱ ❙❡♠✐♥❛r ◆❛s✐♦♥❛❧ ▼❛t❡♠❛t✐❦❛ ❞❛♥ P❡♥❞✐❞✐❦❛♥ ▼❛t❡♠❛t✐❦❛✱❋▼■P❆ ❯◆❨✱✭✷✵✵✾✮✭❚✲✶✻✮✱✶✶✷✼✕ ✶✶✸✽ ✭✐♥ ■♥❞♦♥❡s✐❛♥✮✳
❬✼❪ P❛r❤✉s✐♣ ❍✳ ❆✳✱ ❞❛♥ ❙✉❧✐st②♦♥♦✱ P❡♠❡t❛❛❛♥ ❑♦♥❢♦r♠❛❧ ❞❛♥ ▼♦❞✐✜❦❛s✐♥②❛ ✉♥t✉❦ s✉❛t✉ ❇✐❞❛♥❣ P❡rs❡❣✐✱Pr♦s✐❞✐♥❣ ❙❡♠✐♥❛r ◆❛s✐♦❛♥❛❧ ▼❛t❡♠❛t✐❦❛ ❯◆✲ P❆❘✱✭✷✵✵✾✮✱✭✹✮✱▼❚ ✷✺✵✕✷✺✾ ✭✐♥ ■♥❞♦♥❡s✐❛♥✮✳
❬✽❪ ❙✉r②❛♥✐♥❣s✐❤✱ ❱✳✱ P❛r❤✉s✐♣✱❍✳❆✳✱ ▼❛❤❛t♠❛✱ ❚✳✱ ❑✉r✈❛ P❛r❛♠❡tr✐❦ ❞❛♥ ❚r❛♥s❢♦r✲ ♠❛s✐♥②❛ ✉♥t✉❦ P❡♠❜❡♥t✉❦❛♥ ▼♦t✐❢ ❉❡❦♦r❛t✐❢✱ Pr♦s✐❞✐♥❣✱ ❙❡♠✐♥❛r ◆❛s✐♦♥❛❧ ▼❛t❡♠✲ ❛t✐❦❛ ❞❛♥ P❡♥❞✐❞✐❦❛♥ ▼❛t❡♠❛t✐❦❛ ❯◆❨✱✭✷✵✶✸✮✱✭✹✮✱ ▼❚✷✹✾✕✷✺✽ ✭✐♥ ■♥❞♦♥❡s✐❛♥✮✳ ❬✾❪ P❛t❡❧✱◆✳✱❱♦r♦♥♦✐ ❉✐❛❣r❛♠s ❘♦❜✉st ❛♥❞ ❊✣❝✐❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ t❤❡s✐s✱✭✷✵✵✺✮✱
❇✐♥❣❤❛♠t♦♥ ❯♥✐✈❡rs✐t② ❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❚❤♦♠❛s ❏✳ ❲❛ts♦♥ ❙❝❤♦♦❧ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❆♣♣❧✐❡❞ ❙❝✐❡♥❝❡✳
❬✶✵❪ ❙❤✐♥✱ ❏✳❍✳✱ ❑✇♦♥✱ ❙✳▼✳✱ ❖♥ t❤❡ ❧♦❜❡ ♣r♦✜❧❡ ❞❡s✐❣♥ ✐♥ ❛ ❝②❝❧♦✐❞ r❡❞✉❝❡r ✉s✐♥❣ ✐♥st❛♥t ✈❡❧♦❝✐t② ❝❡♥t❡r✱ ▼❡❝❤❛♥✐s♠ ❛♥❞ ▼❛❝❤✐♥❡ ❚❤❡♦r②✱✭✷✵✵✻✮✱✭✹✶✮✱✺✾✻â⑨➇✻✶✻✳
▼❛t❤❡♠❛t✐❝s ❉❡♣❛rt♠❡♥t✱ ❙❝✐❡♥❝❡ ❛♥❞ ▼❛t❤❡♠❛t✐❝s ❋❛❝✉❧t② ❙❲❈❯✲❏❧✳❉✐♣♦♥❡❣♦r♦ ✺✷✲✻✵ ❙❛❧❛t✐❣❛✱ ■♥❞♦♥❡s✐❛