A first coures in electrical and computer engineering by louis scharf

A First Course in Electrical
and Computer Engineering

By:
Louis Scharf



A First Course in Electrical
and Computer Engineering

By:
Louis Scharf

Online:
< >

CONNEXIONS
Rice University, Houston, Texas


❚❤✐s s❡❧❡❝t✐♦♥ ❛♥❞ ❛rr❛♥❣❡♠❡♥t ♦❢ ❝♦♥t❡♥t ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ✐s ❝♦♣②r✐❣❤t❡❞ ❜②
▲♦✉✐s ❙❝❤❛r❢✳ ■t ✐s ❧✐❝❡♥s❡❞ ✉♥❞❡r t❤❡ ❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥ ✸✳✵ ❧✐❝❡♥s❡
✭❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴✮✳
❈♦❧❧❡❝t✐♦♥ str✉❝t✉r❡ r❡✈✐s❡❞✿ ❙❡♣t❡♠❜❡r ✶✹✱ ✷✵✵✾
P❉❋ ❣❡♥❡r❛t❡❞✿ ❖❝t♦❜❡r ✷✻✱ ✷✵✶✷
❋♦r ❝♦♣②r✐❣❤t ❛♥❞ ❛ttr✐❜✉t✐♦♥ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♠♦❞✉❧❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤✐s
❝♦❧❧❡❝t✐♦♥✱ s❡❡ ♣✳ ✸✵✶✳


❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

❉❡❞✐❝❛t✐♦♥ ♦❢ ✧❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✧
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✶

Pr❡❢❛❝❡ t♦ ✧❆ ❋✐rst ❈♦✉rs❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✧
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✸

✶ ❈♦♠♣❧❡① ◆✉♠❜❡rs
✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺

❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
❈♦♠♣❧❡①

◆✉♠❜❡rs✿

●❡♦♠❡tr②

♦❢

❈♦♠♣❧❡①

◆✉♠❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♠♣❧❡① ◆✉♠✲
❜❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ❘♦♦ts ♦❢ ◗✉❛❞r❛t✐❝ ❊q✉❛✲
t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
❈♦♠♣❧❡①

◆✉♠❜❡rs✿

❘❡♣r❡s❡♥t✐♥❣

❈♦♠♣❧❡①

◆✉♠❜❡rs ✐♥ ❛ ❱❡❝t♦r ❙♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✶✳✻
✶✳✼

❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ❆♥ ❊❧❡❝tr✐❝ ❋✐❡❧❞ ❈♦♠♣✉✲
t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
❈♦♠♣❧❡①

◆✉♠❜❡rs✿

◆✉♠❡r✐❝❛❧

❊①♣❡r✐♠❡♥t

✭◗✉❛❞r❛t✐❝ ❘♦♦ts✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✷ ❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥Θ
✷✳✶
✷✳✷
✷✳✸

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿ ■♥tr♦❞✉❝t✐♦♥

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿
❡❫①

✷✳✹
✷✳✺

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿
❡❫❥

θ

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

❚❤❡ ❋✉♥❝t✐♦♥
❚❤❡ ❋✉♥❝t✐♦♥

❛♥❞ t❤❡ ❯♥✐t ❈✐r❝❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿

❉❡ ▼♦✐✈r❡ ■❞❡♥t✐t✐❡s

❚❤❡ ❊✉❧❡r ❛♥❞

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿

❘♦♦ts ♦❢ ❯♥✐t②

❛♥❞ ❘❡❧❛t❡❞ ❚♦♣✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✷✳✻

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿

❙❡❝♦♥❞✲❖r❞❡r

❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ❉✐✛❡r❡♥❝❡ ❊q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✷✳✼

θ

❚❤❡ ❋✉♥❝t✐♦♥s ❡❫① ❛♥❞ ❡❫❥ ✿

◆✉♠❡r✐❝❛❧ ❊①✲

θ

♣❡r✐♠❡♥t ✭❆♣♣r♦①✐♠❛t✐♥❣ ❡❫❥ ✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

✸ P❤❛s♦rs
✸✳✶ P❤❛s♦rs✿
✸✳✷ P❤❛s♦rs✿
✸✳✸ P❤❛s♦rs✿
✸✳✹ P❤❛s♦rs✿
✸✳✺ P❤❛s♦rs✿
✸✳✻ P❤❛s♦rs✿

■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
P❤❛s♦r ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❙✐❣♥❛❧s
❇❡❛t✐♥❣ ❜❡t✇❡❡♥ ❚♦♥❡s
▼✉❧t✐♣❤❛s❡ P♦✇❡r

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸

▲✐ss❛ ❥♦✉s ❋✐❣✉r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻
❙✐♥✉s♦✐❞❛❧ ❙t❡❛❞② ❙t❛t❡ ❛♥❞ t❤❡ ❙❡✲

r✐❡s ❘▲❈ ❈✐r❝✉✐t

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾


✐✈

✸✳✼
✸✳✽

P❤❛s♦rs✿ ▲✐❣❤t ❙❝❛tt❡r✐♥❣ ❜② ❛ ❙❧✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾
P❤❛s♦rs✿

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥t ✭■♥t❡r❢❡r❡♥❝❡

P❛tt❡r♥s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻

✹ ▲✐♥❡❛r ❆❧❣❡❜r❛
✹✳✶
✹✳✷
✹✳✸

▲✐♥❡❛r ❆❧❣❡❜r❛✿ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾
▲✐♥❡❛r ❆❧❣❡❜r❛✿ ❱❡❝t♦rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷
▲✐♥❡❛r ❆❧❣❡❜r❛✿ ■♥♥❡r Pr♦❞✉❝t ❛♥❞ ❊✉❝❧✐❞❡❛♥
◆♦r♠

✹✳✹
✹✳✺
✹✳✻
✹✳✼
✹✳✽

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✼

▲✐♥❡❛r ❆❧❣❡❜r❛✿ ❉✐r❡❝t✐♦♥ ❈♦s✐♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵
▲✐♥❡❛r ❆❧❣❡❜r❛✿ Pr♦ ❥❡❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✼
▲✐♥❡❛r ❆❧❣❡❜r❛✿ ❖t❤❡r ◆♦r♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✸
▲✐♥❡❛r ❆❧❣❡❜r❛✿ ▼❛tr✐❝❡s
▲✐♥❡❛r

❆❧❣❡❜r❛✿

❙♦❧✈✐♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✼
▲✐♥❡❛r

❙②st❡♠s

♦❢

❊q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✻

✹✳✾
✹✳✶✵

▲✐♥❡❛r ❆❧❣❡❜r❛✿ ❈✐r❝✉✐t ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✺
▲✐♥❡❛r ❆❧❣❡❜r❛✿ ◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥t ✭❈✐r✲
❝✉✐t ❉❡s✐❣♥✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✶

✺ ❱❡❝t♦r ●r❛♣❤✐❝s

✺✳✶
✺✳✷

❱❡❝t♦r ●r❛♣❤✐❝s✿ ■♥tr♦❞✉❝t✐♦♥
❱❡❝t♦r

●r❛♣❤✐❝s✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✸

❚✇♦✲❉✐♠❡♥s✐♦♥❛❧

■♠❛❣❡

❘❡♣r❡s❡♥t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✻

✺✳✸
✺✳✹

❱❡❝t♦r

●r❛♣❤✐❝s✿

❚✇♦✲❉✐♠❡♥s✐♦♥❛❧

■♠❛❣❡

❚r❛♥s❢♦r♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✷
❱❡❝t♦r ●r❛♣❤✐❝s✿ ❈♦♠♣♦s✐t✐♦♥ ♦❢ ❚r❛♥s❢♦r♠❛✲
t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✹

✺✳✺
✺✳✻

❱❡❝t♦r ●r❛♣❤✐❝s✿ ❍♦♠♦❣❡♥❡♦✉s ❈♦♦r❞✐♥❛t❡s ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼✻
❱❡❝t♦r ●r❛♣❤✐❝s✿ ❚❤r❡❡✲❉✐♠❡♥s✐♦♥❛❧ ❍♦♠♦❣❡✲
♥❡♦✉s ❈♦♦r❞✐♥❛t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✵

✺✳✼
✺✳✽

❱❡❝t♦r ●r❛♣❤✐❝s✿ Pr♦ ❥❡❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽✽
❱❡❝t♦r ●r❛♣❤✐❝s✿ ◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥t ✭❙t❛r
❋✐❡❧❞✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾✺

✻ ❋✐❧t❡r✐♥❣
✻✳✶
✻✳✷
✻✳✸
✻✳✹
✻✳✺

❋✐❧t❡r✐♥❣✿ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✸
❋✐❧t❡r✐♥❣✿ ❙✐♠♣❧❡ ❆✈❡r❛❣❡s

❋✐❧t❡r✐♥❣✿ ▼♦✈✐♥❣ ❆✈❡r❛❣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✽
❋✐❧t❡r✐♥❣✿ ❊①♣♦♥❡♥t✐❛❧ ❆✈❡r❛❣❡s ❛♥❞ ❘❡❝✉rs✐✈❡
❋✐❧t❡rs

✻✳✻

✻✳✼

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵✾

❋✐❧t❡r✐♥❣✿ ❲❡✐❣❤t❡❞ ❆✈❡r❛❣❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✶

❋✐❧t❡r✐♥❣✿ ❚❡st ❙❡q✉❡♥❝❡s
❋✐❧t❡r✐♥❣✿

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷✸

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥t ✭❋r❡q✉❡♥❝②

❘❡s♣♦♥s❡ ♦❢ ❋✐rst✲❖r❞❡r ❋✐❧t❡r✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✵

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


✈

✼ ❇✐♥❛r② ❈♦❞❡s
✼✳✶
✼✳✷
✼✳✸
✼✳✹

❇✐♥❛r② ❈♦❞❡s✿ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✸

❇✐♥❛r② ❈♦❞❡s✿ ❚❤❡ ❈♦♠♠✉♥✐❝❛t✐♦♥ P❛r❛❞✐❣♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✺
❇✐♥❛r② ❈♦❞❡s✿ ❋r♦♠ ❙②♠❜♦❧s t♦ ❇✐♥❛r② ❈♦❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸✽
❇✐♥❛r② ❈♦❞❡s✿ ❍✉✛♠❛♥ ❈♦❞❡s ❢♦r ❙♦✉r❝❡ ❈♦❞✲
✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹✻

✼✳✺

❇✐♥❛r② ❈♦❞❡s✿

❍❛♠♠✐♥❣ ❈♦❞❡s ❢♦r ❈❤❛♥♥❡❧

❈♦❞✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺✹

✼✳✻

❇✐♥❛r② ❈♦❞❡s✿

◆✉♠❡r✐❝❛❧ ❊①♣❡r✐♠❡♥t ✭❍✉✛✲

♠❛♥ ❈♦❞❡s✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✸

✽ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇
✽✳✶ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ■♥tr♦❞✉❝t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✺
✽✳✷ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❘✉♥♥✐♥❣ ▼❆❚✲
▲❆❇ ✭▼❛❝✐♥t♦s❤✮

✽✳✸

▲❆❇ ✭P❈✮

✽✳✹

❆♥

✽✳✼

t♦

▼❆❚▲❆❇✿

■♥t❡r❛❝t✐✈❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✾

❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❱❛r✐❛❜❧❡s

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✵

❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❈♦♠♣❧❡① ❱❛r✐✲
❛❜❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✶
❆♥

■♥tr♦❞✉❝t✐♦♥

▼❛tr✐❝❡s

✽✳✽
✽✳✾
✽✳✶✵

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✽

■♥tr♦❞✉❝t✐♦♥

▼♦❞❡

✽✳✺
✽✳✻

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻✻

❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❘✉♥♥✐♥❣ ▼❆❚✲

t♦

▼❆❚▲❆❇✿

❱❡❝t♦rs

❛♥❞

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✸

❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❚❤❡ ❈♦❧♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✽
❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ●r❛♣❤✐❝s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼✾
❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❊❞✐t✐♥❣ ❋✐❧❡s
❛♥❞ ❈r❡❛t✐♥❣ ❋✉♥❝t✐♦♥s ✭▼❛❝✐♥t♦s❤✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✸

✽✳✶✶

❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✿ ❊❞✐t✐♥❣ ❋✐❧❡s
❛♥❞ ❈r❡❛t✐♥❣ ❋✉♥❝t✐♦♥s ✭P❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✻

✽✳✶✷

❆♥

■♥tr♦❞✉❝t✐♦♥

❈♦♥tr♦❧

t♦

▼❆❚▲❆❇✿

▲♦♦♣s

❛♥❞

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽✽

✾ ❚❤❡ ❊❞✐① ❊❞✐t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✸
✶✵ ❯s❡❢✉❧ ▼❛t❤❡♠❛t✐❝❛❧ ■❞❡♥t✐t✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✺
■♥❞❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾✽
❆ttr✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵✶

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✈✐

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


❉❡❞✐❝❛t✐♦♥ ♦❢ ✧❆ ❋✐rst
❈♦✉rs❡ ✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞
❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✧
✶

▲♦✉✐s ❙❝❤❛r❢ ❞❡❞✐❝❛t❡s t❤✐s ❜♦♦❦ t♦
❤✐s ✇✐❢❡ ❈❛r♦❧✱ s♦♥ ●r❡❣✱ ❛♥❞ ❞❛✉❣❤t❡r ❍❡✐❞✐✱
❢♦r ❧♦✈❡ ❛♥❞ ✐♥s♣✐r❛t✐♦♥❀
❤✐s ♣❛r❡♥ts ▲♦✉✐s ❙r✳ ❛♥❞ ❆♥♥✱
✐♥ ❝❡❧❡❜r❛t✐♦♥ ♦❢ t❤❡✐r ✺✵t❤ ✇❡❞❞✐♥❣ ❛♥♥✐✈❡rs❛r②✳

❘✐❝❤❛r❞ ❇❡❤r❡♥s ❞❡❞✐❝❛t❡s t❤✐s ❜♦♦❦ t♦
❤✐s ✇✐❢❡ ❉❡❜❜✐❡✱ ❛♥❞ ❝❤✐❧❞ ❛s ②❡t ✉♥❜♦r♥✱
❢♦r ❧♦✈❡ ❛♥❞ ❡♥❝♦✉r❛❣❡♠❡♥t❀
❤✐s ♣❛r❡♥ts ❘✐❝❤❛r❞ ❛♥❞ ❊❧s✐❡✱
✐♥ ❣r❛t✐t✉❞❡ ❢♦r ❛ ❣♦♦❞ ❡❞✉❝❛t✐♦♥✳

✶ ❚❤✐s

❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✷✶✼✾✻✴✶✳✸✴❃✳
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✶

✷

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


Pr❡❢❛❝❡ t♦ ✧❆ ❋✐rst ❈♦✉rs❡
✐♥ ❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r
❊♥❣✐♥❡❡r✐♥❣✧
✷

♥♦t❡✿ ❚❤✐s ♠♦❞✉❧❡ ✐s ♣❛rt ♦❢ t❤❡ ❝♦❧❧❡❝t✐♦♥✱

❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✳

❆ ❋✐rst ❈♦✉rs❡ ✐♥

❚❤❡ ▲❛❚❡❳ s♦✉r❝❡ ✜❧❡s

❢♦r t❤✐s ❝♦❧❧❡❝t✐♦♥ ✇❡r❡ ❝r❡❛t❡❞ ✉s✐♥❣ ❛♥ ♦♣t✐❝❛❧ ❝❤❛r❛❝t❡r r❡❝♦❣✲
♥✐t✐♦♥ t❡❝❤♥♦❧♦❣②✱ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤✐s ♣r♦❝❡ss t❤❡r❡ ♠❛② ❜❡ ♠♦r❡
❡rr♦rs t❤❛♥ ✉s✉❛❧✳ P❧❡❛s❡ ❝♦♥t❛❝t ✉s ✐❢ ②♦✉ ❞✐s❝♦✈❡r ❛♥② ❡rr♦rs✳

❚❤✐s ❜♦♦❦ ✇❛s ✇r✐tt❡♥ ❢♦r ❛♥ ❡①♣❡r✐♠❡♥t❛❧ ❢r❡s❤♠❛♥ ❝♦✉rs❡ ❛t t❤❡ ❯♥✐✈❡r✲
s✐t② ♦❢ ❈♦❧♦r❛❞♦✳ ❚❤❡ ❝♦✉rs❡ ✐s ♥♦✇ ❛♥ ❡❧❡❝t✐✈❡ t❤❛t t❤❡ ♠❛❥♦r✐t② ♦❢ ♦✉r
❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣ st✉❞❡♥ts t❛❦❡ ✐♥ t❤❡ s❡❝♦♥❞ s❡♠❡st❡r
♦❢ t❤❡✐r ❢r❡s❤♠❛♥ ②❡❛r✱ ❥✉st ❜❡❢♦r❡ t❤❡✐r ✜rst ❝✐r❝✉✐ts ❝♦✉rs❡✳ ❖✉r ❞❡♣❛rt✲
♠❡♥t ❞❡❝✐❞❡❞ t♦ ♦✛❡r t❤✐s ❝♦✉rs❡ ❢♦r s❡✈❡r❛❧ r❡❛s♦♥s✿

✶✳ ✇❡ ✇❛♥t❡❞ t♦ ♣✐q✉❡ st✉❞❡♥t✬ ✐♥t❡r❡st ✐♥ ❡♥❣✐♥❡❡r✐♥❣ ❜② ❛❝q✉❛✐♥t✐♥❣
t❤❡♠ ✇✐t❤ ❡♥❣✐♥❡❡r✐♥❣ t❡❛❝❤❡rs ❡❛r❧② ✐♥ t❤❡✐r ✉♥✐✈❡rs✐t② ❝❛r❡❡rs ❛♥❞
❜② ♣r♦✈✐❞✐♥❣ ✇✐t❤ ❡①♣♦s✉r❡ t♦ t❤❡ t②♣❡s ♦❢ ♣r♦❜❧❡♠s t❤❛t ❡❧❡❝tr✐❝❛❧
❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡rs ❛r❡ ❛s❦❡❞ t♦ s♦❧✈❡❀
✷✳ ✇❡ ✇❛♥t❡❞ st✉❞❡♥ts ❡♥t❡r✐♥❣ t❤❡ ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣
♣r♦❣r❛♠s t♦ ❜❡ ♣r❡♣❛r❡❞ ✐♥ ❝♦♠♣❧❡① ❛♥❛❧②s✐s✱ ♣❤❛s♦rs✱ ❛♥❞ ❧✐♥❡❛r
❛❧❣❡❜r❛✱ t♦♣✐❝s t❤❛t ❛r❡ ♦❢ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rt❛♥❝❡ ✐♥ ♦✉r ❞✐s❝✐♣❧✐♥❡❀

✸✳ ✇❡ ✇❛♥t❡❞ st✉❞❡♥ts t♦ ❤❛✈❡ ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ❛ s♦❢t✇❛r❡ ❛♣♣❧✐❝❛t✐♦♥
t♦♦❧✱ s✉❝❤ ❛s ▼❆❚▲❆❇✱ t♦ ❝♦♠♣❧❡t❡ t❤❡✐r ♣r❡♣❛r❛t✐♦♥ ❢♦r ♣r❛❝t✐❝❛❧
❛♥❞ ❡✣❝✐❡♥t ❝♦♠♣✉t✐♥❣ ✐♥ t❤❡✐r s✉❜s❡q✉❡♥t ❝♦✉rs❡s ❛♥❞ ✐♥ t❤❡✐r
♣r♦❢❡ss✐♦♥❛❧ ❝❛r❡❡rs❀

✷ ❚❤✐s

❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✷✶✹✼✻✴✶✳✼✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✸


✹

✹✳ ✇❡ ✇❛♥t❡❞ st✉❞❡♥ts t♦ ♠❛❦❡ ❡❛r❧② ❝♦♥t❛❝t ✇✐t❤ ❛❞✈❛♥❝❡❞ t♦♣✐❝s
❧✐❦❡ ✈❡❝t♦r ❣r❛♣❤✐❝s✱ ✜❧t❡r✐♥❣✱ ❛♥❞ ❜✐♥❛r② ❝♦❞✐♥❣ s♦ t❤❛t t❤❡② ✇♦✉❧❞
❣❛✐♥ ❛ ♠♦r❡ r♦✉♥❞❡❞ ♣✐❝t✉r❡ ♦❢ ♠♦❞❡r♥ ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r
❡♥❣✐♥❡❡r✐♥❣✳

■♥ ♦r❞❡r t♦ ✐♥tr♦❞✉❝❡ t❤✐s ❝♦✉rs❡✱ ✇❡ ❤❛❞ t♦ s❛❝r✐✜❝❡ ❛ s❡❝♦♥❞ s❡♠❡st❡r ♦❢
P❛s❝❛❧ ♣r♦❣r❛♠♠✐♥❣✳ ❲❡ ❝♦♥❝❧✉❞❡❞ t❤❛t t❤❡ s❛❝r✐✜❝❡ ✇❛s ✇♦rt❤ ♠❛❦✐♥❣
❜❡❝❛✉s❡ ✇❡ ❢♦✉♥❞ t❤❛t ♠♦st ♦❢ ♦✉r st✉❞❡♥ts ✇❡r❡ ♣r❡♣❛r❡❞ ❢♦r ❤✐❣❤✲❧❡✈❡❧
❧❛♥❣✉❛❣❡ ❝♦♠♣✉t✐♥❣ ❛❢t❡r ❥✉st ♦♥❡ s❡♠❡st❡r ♦❢ ♣r♦❣r❛♠♠✐♥❣✳
❲❡ ❜❡❧✐❡✈❡ ❡♥❣✐♥❡❡r✐♥❣ ❡❞✉❝❛t♦rs ❡❧s❡✇❤❡r❡ ❛r❡ r❡❛❝❤✐♥❣ s✐♠✐❧❛r ❝♦♥✲
❝❧✉s✐♦♥s ❛❜♦✉t t❤❡✐r ♦✇♥ st✉❞❡♥ts ❛♥❞ ❝✉rr✐❝✉❧✉♠s✳

❲❡ ❤♦♣❡ t❤✐s ❜♦♦❦

❤❡❧♣s ❝r❡❛t❡ ❛ ♠✉❝❤ ♥❡❡❞❡❞ ❞✐❛❧♦❣✉❡ ❛❜♦✉t ❝✉rr✐❝✉❧✉♠ r❡✈✐s✐♦♥ ❛♥❞ t❤❛t
✐t ❧❡❛❞s t♦ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ s✐♠✐❧❛r ✐♥tr♦❞✉❝t♦r② ❝♦✉rs❡s t❤❛t ❡♥❝♦✉r❛❣❡
st✉❞❡♥ts t♦ ❡♥t❡r ❛♥❞ ♣r❛❝t✐❝❡ ♦✉r ❝r❛❢t✳
❙t✉❞❡♥ts ❡❧❡❝t✐♥❣ t♦ t❛❦❡ t❤✐s ❝♦✉rs❡ ❤❛✈❡ ❝♦♠♣❧❡t❡❞ ♦♥❡ s❡♠❡st❡r ♦❢
❝❛❧❝✉❧✉s✱ ❝♦♠♣✉t❡r ♣r♦❣r❛♠♠✐♥❣✱ ❝❤❡♠✐str②✱ ❛♥❞ ❤✉♠❛♥✐t✐❡s✳

❈♦♥❝✉r✲

r❡♥t❧② ✇✐t❤ t❤✐s ❝♦✉rs❡✱ st✉❞❡♥ts t❛❦❡ ♣❤②s✐❝s ❛♥❞ ❛ s❡❝♦♥❞ s❡♠❡st❡r ♦❢
❝❛❧❝✉❧✉s✱ ❛s ✇❡❧❧ ❛s ❛ s❡❝♦♥❞ s❡♠❡st❡r ✐♥ t❤❡ ❤✉♠❛♥✐t✐❡s✳

❇② ♦♠✐tt✐♥❣

t❤❡ ❛❞✈❛♥❝❡❞ t♦♣✐❝s ♠❛r❦❡❞ ❜② ❛st❡r✐s❦s✱ ✇❡ ❛r❡ ❛❜❧❡ t♦ ❝♦✈❡r ❈♦♠♣❧❡①
◆✉♠❜❡rs ✭❙❡❝t✐♦♥ ✶✳✶✮ t❤r♦✉❣❤ ▲✐♥❡❛r ❆❧❣❡❜r❛ ✭❙❡❝t✐♦♥ ✹✳✶✮✱ ♣❧✉s t✇♦
♦❢ t❤❡ t❤r❡❡ r❡♠❛✐♥✐♥❣ ❝❤❛♣t❡rs✳

❚❤❡ ❜♦♦❦ ✐s ♦r❣❛♥✐③❡❞ s♦ t❤❛t t❤❡ ✐♥✲

str✉❝t♦r ❝❛♥ s❡❧❡❝t ❛♥② t✇♦ ♦❢ t❤❡ t❤r❡❡✳ ■❢ ❡✈❡r② ❝❤❛♣t❡r ♦❢ t❤✐s ❜♦♦❦ ✐s

❝♦✈❡r❡❞✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❛❞✈❛♥❝❡❞ t♦♣✐❝s✱ t❤❡♥ ❡♥♦✉❣❤ ♠❛t❡r✐❛❧ ❡①✐sts ❢♦r ❛
t✇♦✲s❡♠❡st❡r ❝♦✉rs❡✳
❚❤❡ ✜rst t❤r❡❡ ❝❤❛♣t❡rs ♦❢ t❤✐s ❜♦♦❦ ♣r♦✈✐❞❡ ❛ ❢❛✐r❧② ❝♦♠♣❧❡t❡ ❝♦✈❡r❛❣❡
♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✱ t❤❡ ❢✉♥❝t✐♦♥s

❡①

❛♥❞

ejθ ✱

❛♥❞ ♣❤❛s♦rs✳ ❖✉r ❞❡♣❛rt✲

♠❡♥t ♣❤✐❧♦s♦♣❤② ✐s t❤❛t t❤❡s❡ t♦♣✐❝s ♠✉st ❜❡ ✉♥❞❡rst♦♦❞ ✐❢ ❛ st✉❞❡♥t ✐s
t♦ s✉❝❝❡❡❞ ✐♥ ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣✳

❚❤❡s❡ t❤r❡❡ ❝❤❛♣t❡rs

♠❛② ❛❧s♦ ❜❡ ✉s❡❞ ❛s ❛ s✉♣♣❧❡♠❡♥t t♦ ❛ ❝✐r❝✉✐ts ❝♦✉rs❡✳ ❆ ♠❡❛s✉r❡❞ ♣❛❝❡
♦❢ ♣r❡s❡♥t❛t✐♦♥✱ t❛❦✐♥❣ ❜❡t✇❡❡♥ s✐①t❡❡♥ ❛♥❞ ❡✐❣❤t❡❡♥ ❧❡❝t✉r❡s✱ ✐s s✉✣❝✐❡♥t
t♦ ❝♦✈❡r ❛❧❧ ❜✉t t❤❡ ❛❞✈❛♥❝❡❞ s❡❝t✐♦♥s ✐♥ ❈♦♠♣❧❡① ◆✉♠❜❡rs ✭❙❡❝t✐♦♥ ✶✳✶✮
t❤r♦✉❣❤ P❤❛s♦rs ✭❙❡❝t✐♦♥ ✸✳✶✮✳
❚❤❡ ❝❤❛♣t❡r ♦♥ ✧❧✐♥❡❛r ❛❧❣❡❜r❛✧ ✭❙❡❝t✐♦♥ ✹✳✶✮ ✐s ♣r❡r❡q✉✐s✐t❡ ❢♦r ❛❧❧
s✉❜s❡q✉❡♥t ❝❤❛♣t❡rs✳

❲❡ ✉s❡ ❡✐❣❤t t♦ t❡♥ ❧❡❝t✉r❡s t♦ ❝♦✈❡r ✐t✳

❲❡ ❞❡✲

✈♦t❡ t✇❡❧✈❡ t♦ s✐①t❡❡♥ ❧❡❝t✉r❡s t♦ ❝♦✈❡r t♦♣✐❝s ❢r♦♠ ❱❡❝t♦r ●r❛♣❤✐❝s ✭❙❡❝✲

t✐♦♥ ✺✳✶✮ t❤r♦✉❣❤ ❇✐♥❛r② ❈♦❞❡s ✭❙❡❝t✐♦♥ ✼✳✶✮✳

✭❲❡ ❛ss✉♠❡ ❛ s❡♠❡st❡r

❝♦♥s✐st✐♥❣ ♦❢ ✹✷ ❧❡❝t✉r❡s ❛♥❞ t❤r❡❡ ❡①❛♠s✳✮ ❚❤❡ ❝❤❛♣t❡r ♦♥ ✈❡❝t♦r ❣r❛♣❤✲
✐❝s ✭❙❡❝t✐♦♥ ✺✳✶✮ ❛♣♣❧✐❡s t❤❡ ❧✐♥❡❛r ❛❧❣❡❜r❛ ❧❡❛r♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r
✭❙❡❝t✐♦♥ ✹✳✶✮ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ tr❛♥s❧❛t✐♥❣✱ s❝❛❧✐♥❣✱ ❛♥❞ r♦t❛t✐♥❣ ✐♠❛❣❡s✳
✧❋✐❧t❡r✐♥❣✧ ✭❙❡❝t✐♦♥ ✻✳✶✮ ✐♥tr♦❞✉❝❡s t❤❡ st✉❞❡♥t t♦ ❜❛s✐❝ ✐❞❡❛s ✐♥ ❛✈❡r❛❣✲
✐♥❣ ❛♥❞ ✜❧t❡r✐♥❣✳

❚❤❡ ❝❤❛♣t❡r ♦♥ ✧❇✐♥❛r② ❈♦❞❡s✧ ✭❙❡❝t✐♦♥ ✼✳✶✮ ❝♦✈❡rs

t❤❡ r✉❞✐♠❡♥ts ♦❢ ❜✐♥❛r② ❝♦❞✐♥❣✱ ✐♥❝❧✉❞✐♥❣ ❍✉✛♠❛♥ ❝♦❞❡s ❛♥❞ ❍❛♠♠✐♥❣

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


✺

❝♦❞❡s✳
■❢ t❤❡ ✉s❡rs ♦❢ t❤✐s ❜♦♦❦ ✜♥❞ ✧❱❡❝t♦r ●r❛♣❤✐❝s✧ ✭❙❡❝t✐♦♥ ✺✳✶✮ t❤r♦✉❣❤
✧❇✐♥❛r② ❈♦❞❡s✧ ✭❙❡❝t✐♦♥ ✼✳✶✮ t♦♦ ❝♦♥✜♥✐♥❣✱ ✇❡ ❡♥❝♦✉r❛❣❡ t❤❡♠ t♦ s✉♣♣❧❡✲
♠❡♥t t❤❡ ❡ss❡♥t✐❛❧ ♠❛t❡r✐❛❧ ✐♥ ✧❈♦♠♣❧❡① ◆✉♠❜❡rs✧ ✭❙❡❝t✐♦♥ ✶✳✶✮ t❤r♦✉❣❤
✧▲✐♥❡❛r ❆❧❣❡❜r❛✧ ✭❙❡❝t✐♦♥ ✹✳✶✮ ✇✐t❤ t❤❡✐r ♦✇♥ ❝♦✉rs❡ ♥♦t❡s ♦♥ ❛❞❞✐t✐♦♥❛❧
t♦♣✐❝s✳ ❲✐t❤✐♥ ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣ t❤❡r❡ ❛r❡ ❡♥❞❧❡ss ♣♦s✲
s✐❜✐❧✐t✐❡s✳ Pr❛❝t✐❝❛❧❧② ❛♥② s❡t ♦❢ t♦♣✐❝s t❤❛t ❝❛♥ ❜❡ t❛✉❣❤t ✇✐t❤ ❝♦♥✈✐❝t✐♦♥
❛♥❞ ❡♥t❤✉s✐❛s♠ ✇✐❧❧ ✇❤❡t t❤❡ st✉❞❡♥t✬s ❛♣♣❡t✐t❡✳

❲❡ ❡♥❝♦✉r❛❣❡ ②♦✉ t♦

✇r✐t❡ t♦ ✉s ♦r t♦ ♦✉r ❡❞✐t♦r✱ ❚♦♠ ❘♦❜❜✐♥s✱ ❛❜♦✉t ②♦✉r ✐❞❡❛s ❢♦r ❛❞❞✐t✐♦♥❛❧
t♦♣✐❝s✳ ❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ t❤✐♥❦ t❤❛t ♦✉r ❜♦♦❦ ❛♥❞ ✐ts s✉❜s❡q✉❡♥t ❡❞✐t✐♦♥s
✇✐❧❧ ❤❛✈❡ ❛♥ ♦♣❡♥ ❛r❝❤✐t❡❝t✉r❡ t❤❛t ❡♥❛❜❧❡s ✉s t♦ ❛❝❝♦♠♠♦❞❛t❡ ❛ ✇✐❞❡
r❛♥❣❡ ♦❢ st✉❞❡♥t ❛♥❞ ❢❛❝✉❧t② ✐♥t❡r❡sts✳
❚❤r♦✉❣❤♦✉t t❤✐s ❜♦♦❦ ✇❡ ❤❛✈❡ ✉s❡❞ ▼❆❚▲❆❇ ♣r♦❣r❛♠s t♦ ✐❧❧✉str❛t❡
❦❡② ✐❞❡❛s✳

▼❆❚▲❆❇ ✐s ❛♥ ✐♥t❡r❛❝t✐✈❡✱ ♠❛tr✐①✲♦r✐❡♥t❡❞ ❧❛♥❣✉❛❣❡ t❤❛t ✐s

✐❞❡❛❧❧② s✉✐t❡❞ t♦ ❝✐r❝✉✐t ❛♥❛❧②s✐s✱ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥tr♦❧ t❤❡♦r②✱ ❝♦♠♠✉✲
♥✐❝❛t✐♦♥s✱ ❧✐♥❡❛r ❛❧❣❡❜r❛✱ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s✳

▼❆❚▲❆❇ ✐s r❛♣✐❞❧②

❜❡❝♦♠✐♥❣ ❛ st❛♥❞❛r❞ s♦❢t✇❛r❡ t♦♦❧ ✐♥ ✉♥✐✈❡rs✐t✐❡s ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣ ❝♦♠♣❛✲
♥✐❡s✳

✭❋♦r ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ▼❆❚▲❆❇✱ r❡t✉r♥ t❤❡ ❛tt❛❝❤❡❞ ❝❛r❞

✐♥ t❤❡ ❜❛❝❦ ♦❢ t❤✐s ❜♦♦❦ t♦ ❚❤❡ ▼❛t❤❲♦r❦s✱ ■♥❝✳✮

▼❆❚▲❆❇ ♣r♦❣r❛♠s

❛r❡ ❞❡s✐❣♥❡❞ t♦ ❞❡✈❡❧♦♣ t❤❡ st✉❞❡♥t✬s ❛❜✐❧✐t② t♦ s♦❧✈❡ ♠❡❛♥✐♥❣❢✉❧ ♣r♦❜❧❡♠s✱
❝♦♠♣✉t❡✱ ❛♥❞ ♣❧♦t ✐♥ ❛ ❤✐❣❤✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥s ❧❛♥❣✉❛❣❡✳ ❖✉r st✉❞❡♥ts ❣❡t
st❛rt❡❞ ✐♥ ▼❆❚▲❆❇ ❜② ✇♦r❦✐♥❣ t❤r♦✉❣❤ ✏❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✱✑
✭❙❡❝t✐♦♥ ✽✳✶✮ ✇❤✐❧❡ s❡❛t❡❞ ❛t ❛♥ ■❇▼ P❈ ✭♦r ❧♦♦❦✲❛❧✐❦❡✮ ♦r ❛♥ ❆♣♣❧❡ ▼❛❝✲
✐♥t♦s❤✳

❲❡ ❛❧s♦ ❤❛✈❡ t❤❡♠ r✉♥ t❤r♦✉❣❤ t❤❡ ❞❡♠♦♥str❛t✐♦♥ ♣r♦❣r❛♠s ✐♥

✧❈♦♠♣❧❡① ◆✉♠❜❡rs✧ ✭❙❡❝t✐♦♥ ✶✳✶✮✳

❊❛❝❤ ✇❡❡❦ ✇❡ ❣✐✈❡ t❤r❡❡ ❝❧❛ssr♦♦♠

❧❡❝t✉r❡s ❛♥❞ ❝♦♥❞✉❝t ❛ ♦♥❡✲❤♦✉r ❝♦♠♣✉t❡r ❧❛❜ s❡ss✐♦♥✳ ❙t✉❞❡♥ts ✉s❡ t❤✐s
❧❛❜ s❡ss✐♦♥ t♦ ❤♦♥❡ ▼❆❚▲❆❇ s❦✐❧❧s✱ t♦ ✇r✐t❡ ♣r♦❣r❛♠s✱ ♦r t♦ ❝♦♥❞✉❝t t❤❡
♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥ts t❤❛t ❛r❡ ❣✐✈❡♥ ❛t t❤❡ ❡♥❞ ♦❢ ❡❛❝❤ ❝❤❛♣t❡r✳ ❲❡ r❡✲
q✉✐r❡ t❤❛t t❤❡s❡ ❡①♣❡r✐♠❡♥ts ❜❡ ❝❛rr✐❡❞ ♦✉t ❛♥❞ t❤❡♥ r❡♣♦rt❡❞ ✐♥ ❛ s❤♦rt
❧❛❜ r❡♣♦rt t❤❛t ❝♦♥t❛✐♥s ✭✐✮ ✐♥tr♦❞✉❝t✐♦♥✱ ✭✐✐✮ ❛♥❛❧②t✐❝❛❧ ❝♦♠♣✉t❛t✐♦♥s✱
✭✐✐✐✮ ❝♦♠♣✉t❡r ❝♦❞❡✱ ✭✐✈✮ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✱ ❛♥❞ ✭✈✮ ❝♦♥❝❧✉s✐♦♥s✳

❚❤❡

q✉❛❧✐t② ♦❢ t❤❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ❛♥❞ t❤❡ ❝♦♠♣✉t❡r ❣r❛♣❤✐❝s ❛st♦♥✐s❤❡s st✉✲
❞❡♥ts✳ ❙♦❧✉t✐♦♥s t♦ t❤❡ ❝❤❛♣t❡r ♣r♦❜❧❡♠s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ♣✉❜❧✐s❤❡r
❢♦r ✐♥str✉❝t♦rs ✇❤♦ ❛❞♦♣t t❤✐s t❡①t ❢♦r ❝❧❛ssr♦♦♠ ✉s❡✳
❲❡ ✇✐s❤ t♦ ❛❝❦♥♦✇❧❡❞❣❡ ♦✉r ❧❛t❡ ❝♦❧❧❡❛❣✉❡ ❘✐❝❤❛r❞ ❘♦❜❡rts✱ ✇❤♦ ❡♥✲
❝♦✉r❛❣❡❞ ✉s t♦ ♣✉❜❧✐s❤ t❤✐s ❜♦♦❦✱ ❛♥❞ ▼✐❝❤❛❡❧ ▲✐❣❤t♥❡r ❛♥❞ ❘✉t❤ ❘❛✈❡♥❡❧✱
✇❤♦ t❛✉❣❤t ✧▲✐♥❡❛r ❆❧❣❡❜r❛✧ ✭❙❡❝t✐♦♥ ✹✳✶✮ ❛♥❞ ✧❱❡❝t♦r ●r❛♣❤✐❝s✧ ✭❙❡❝✲
t✐♦♥ ✺✳✶✮ ❛♥❞ ♦✛❡r❡❞ ❤❡❧♣❢✉❧ s✉❣❣❡st✐♦♥s ♦♥ t❤❡ ♠❛♥✉s❝r✐♣t✳

❲❡ t❤❛♥❦

❈✳ ❚✳ ▼✉❧❧✐s ❢♦r ❛❧❧♦✇✐♥❣ ✉s t♦ ✉s❡ ❤✐s ♥♦t❡s ♦♥ ❜✐♥❛r② ❝♦❞❡s t♦ ❣✉✐❞❡
♦✉r ✇r✐t✐♥❣ ♦❢ ✧❇✐♥❛r② ❈♦❞❡s✧ ✭❙❡❝t✐♦♥ ✼✳✶✮✳ ❲❡ t❤❛♥❦ ❈é❞r✐❝ ❉❡♠❡✉r❡
❛♥❞ P❡t❡r ▼❛ss❡② ❢♦r t❤❡✐r ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ ✇r✐t✐♥❣ ♦❢ ✧❆♥ ■♥tr♦✲
❞✉❝t✐♦♥ t♦ ▼❆❚▲❆❇✧ ✭❙❡❝t✐♦♥ ✽✳✶✮ ❛♥❞ ✧❚❤❡ ❊❞✐① ❊❞✐t♦r✧ ✭❈❤❛♣t❡r ✾✮✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


✻

❲❡ t❤❛♥❦ ❚♦♠ ❘♦❜❜✐♥s✱ ♦✉r ❡❞✐t♦r ❛t ❆❞❞✐s♦♥✲❲❡s❧❡②✱ ❢♦r ❤✐s ❡♥❝♦✉r❛❣❡✲
♠❡♥t✱ ♣❛t✐❡♥❝❡✱ ❛♥❞ ♠❛♥② s✉❣❣❡st✐♦♥s✳ ❲❡ ❛r❡ ❡s♣❡❝✐❛❧❧② ❣r❛t❡❢✉❧ t♦ ❏✉❧✐❡
❋r❡❞❧✉♥❞✱ ✇❤♦ ❝♦♠♣♦s❡❞ t❤✐s t❡①t t❤r♦✉❣❤ ♠❛♥② ❞r❛❢ts ❛♥❞ ✐♠♣r♦✈❡❞ ✐t
✐♥ ♠❛♥② ✇❛②s✳

❲❡ t❤❛♥❦ ❤❡r ❢♦r ♣r❡♣❛r✐♥❣ ❛♥ ❡①❝❡❧❧❡♥t ♠❛♥✉s❝r✐♣t ❢♦r

♣r♦❞✉❝t✐♦♥✳

▲✳ ▲✳ ❙❝❤❛r❢
❘✳ ❚✳ ❇❡❤r❡♥s
❇♦✉❧❞❡r✱ ❈♦❧♦r❛❞♦

❚♦ t❤❡ ❚❡❛❝❤❡r✿
❆♥ ✐♥❝♦♠♣❧❡t❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛♥❞ ♣❤❛s♦rs ❤❛♥❞✐❝❛♣s
st✉❞❡♥ts ✐♥ ❝✐r❝✉✐ts ❛♥❞ ❡❧❡❝tr♦♥✐❝s ❝♦✉rs❡s✱ ❛♥❞ ❡✈❡♥ ♠♦r❡ s♦ ✐♥ ❛❞✈❛♥❝❡❞
❝♦✉rs❡s s✉❝❤ ❛s ❡❧❡❝tr♦♠❛❣♥❡t✐❝s✳

♦♣t✐❝s✱ ❧✐♥❡❛r s②st❡♠s✱ ❝♦♥tr♦❧✱ ❛♥❞

❝♦♠♠✉♥✐❝❛t✐♦♥ s②st❡♠s✳ ❖✉r ❢❛❝✉❧t② ❤❛s ❞❡❝✐❞❡❞ t♦ ❛❞❞r❡ss t❤✐s ♣r♦❜❧❡♠
❛s ❡❛r❧② ❛s ♣♦ss✐❜❧❡ ✐♥ t❤❡ ❝✉rr✐❝✉❧✉♠ ❜② ❞❡s✐❣♥✐♥❣ ❛ ❝♦✉rs❡ t❤❛t ❞r✐❧❧s
❝♦♠♣❧❡① ♥✉♠❜❡rs ❛♥❞ ♣❤❛s♦rs ✐♥t♦ t❤❡ ♠✐♥❞s ♦❢ ❜❡❣✐♥♥✐♥❣ ❡♥❣✐♥❡❡r✐♥❣
st✉❞❡♥ts✳

❲❡ ❤❛✈❡ ✉s❡❞ ♣♦✇❡r s✐❣♥❛❧s✱ ♠✉s✐❝❛❧ t♦♥❡s✱ ▲✐ss❛❥♦✉s ✜❣✉r❡s✱

❧✐❣❤t s❝❛tt❡r✐♥❣✱ ❛♥❞
❝❛❧❝✉❧✉s✳

❘▲❈

❝✐r❝✉✐ts t♦ ✐❧❧✉str❛t❡ t❤❡ ✉s❡❢✉❧♥❡ss ♦❢ ♣❤❛s♦r

✧▲✐♥❡❛r ❆❧❣❡❜r❛✧ ✭❙❡❝t✐♦♥ ✹✳✶✮ t❤r♦✉❣❤ ✧❇✐♥❛r② ❈♦❞❡s✧ ✭❙❡❝✲

t✐♦♥ ✼✳✶✮ ✐♥tr♦❞✉❝❡ st✉❞❡♥ts t♦ ❛ ❤❛♥❞❢✉❧ ♦❢ ♠♦❞❡r♥ ✐❞❡❛s ✐♥ ❡❧❡❝tr✐❝❛❧ ❛♥❞
❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣✳ ❚❤❡ ♠♦t✐✈❛t✐♦♥ ✐s t♦ ✇❤❡t st✉❞❡♥ts✬ ❛♣♣❡t✐t❡s ❢♦r
♠♦r❡ ❛❞✈❛♥❝❡❞ ♣r♦❜❧❡♠s✳

❚❤❡ t♦♣✐❝s ✇❡ ❤❛✈❡ ❝❤♦s❡♥ ✕ ❧✐♥❡❛r ❛❧❣❡❜r❛✱

✈❡❝t♦r ❣r❛♣❤✐❝s✱ ✜❧t❡r✐♥❣✱ ❛♥❞ ❜✐♥❛r② ❝♦❞❡s ✕ ❛r❡ ♦♥❧② r❡♣r❡s❡♥t❛t✐✈❡✳
❆t ✜rst ❣❧❛♥❝❡✱ ♠❛♥② ♦❢ t❤❡ ❡q✉❛t✐♦♥s ✐♥ t❤✐s ❜♦♦❦ ❧♦♦❦ ✐♥t✐♠✐❞❛t✐♥❣
t♦ ❜❡❣✐♥♥✐♥❣ st✉❞❡♥ts✳

❋♦r t❤✐s r❡❛s♦♥✱ ✇❡ ♣r♦❝❡❡❞ ❛t ❛ ✈❡r② ♠❡❛s✉r❡❞

♣❛❝❡✳ ■♥ ♦✉r ❧❡❝t✉r❡s✱ ✇❡ ✇r✐t❡ ♦✉t ✐♥ ❛❣♦♥✐③✐♥❣ ❞❡t❛✐❧ ❡✈❡r② ❡q✉❛t✐♦♥ t❤❛t

N −1
n=0

✐♥✈♦❧✈❡s ❛ s❡q✉❡♥❝❡ ♦r s❡r✐❡s✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s✉♠

zn

✐s ✇r✐tt❡♥

♦✉t ❛s

1 + z + z 2 + · · · + z N −1 ,
❛♥❞ t❤❡♥ ✐t ✐s ❡✈❛❧✉❛t❡❞ ❢♦r s♦♠❡ s♣❡❝✐✜❝ ✈❛❧✉❡ ♦❢
❛♥❛❧②t✐❝❛❧ r❡s✉❧t

1−z N
1−z

✭✶✮

③

❜❡❢♦r❡ ✇❡ ❞❡r✐✈❡ t❤❡

❙✐♠✐❧❛r❧②✱ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ❧✐❦❡

n→∞

x n
n

, ...,

✭✷✮

lim 1 +

✐s ✇r✐tt❡♥ ♦✉t ❛s

(1 + x) ,

1+

x
2

2

,

1+

x
3

3

, ...,

❛♥❞ t❤❡♥ ✐t ✐s ❡✈❛❧✉❛t❡❞ ❢♦r s♦♠❡ s♣❡❝✐✜❝
❜❡❢♦r❡ t❤❡ ❧✐♠✐t ✐s ❞❡r✐✈❡❞✳

①

1+

x
100

100

❛♥❞ ❢♦r s❡✈❡r❛❧ ✈❛❧✉❡s ♦❢

♥

❲❡ tr② t♦ ♣r❡s❡r✈❡ t❤✐s ♣r❛❝t✐❝❡ ♦❢ ♣❡❞❛♥t✐❝

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


✼

❡①❝❡ss ✉♥t✐❧ ✐t ✐s ❝❧❡❛r t❤❛t ❡✈❡r② st✉❞❡♥t ✐s ❝♦♠❢♦rt❛❜❧❡ ✇✐t❤ ❛♥ ✐❞❡❛ ❛♥❞
t❤❡ ♥♦t❛t✐♦♥ ❢♦r ❝♦❞✐♥❣ t❤❡ ✐❞❡❛✳

❚♦ t❤❡ ❙t✉❞❡♥t✿
❚❤❡s❡ ❛r❡ ❡①❝✐t✐♥❣ t✐♠❡s ❢♦r ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣✳ ❚♦ ❝❡❧✲
❡❜r❛t❡ ✐ts s✐❧✈❡r ❛♥♥✐✈❡rs❛r②✱ t❤❡ ◆❛t✐♦♥❛❧ ❆❝❛❞❡♠② ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❛♥✲
♥♦✉♥❝❡❞ ✐♥ ❋❡❜r✉❛r② ♦❢ ✶✾✾✵ t❤❡ t♦♣ t❡♥ ❡♥❣✐♥❡❡r✐♥❣ ❢❡❛ts ♦❢ t❤❡ ♣r❡✈✐✲
♦✉s t✇❡♥t②✲✜✈❡ ②❡❛rs✳

❚❤❡ ❆♣♦❧❧♦ ♠♦♦♥ ❧❛♥❞✐♥❣✱ ❛ tr✉❧② ❖❧②♠♣✐❛♥ ❛♥❞

♣r♦t❡❛♥ ❛❝❤✐❡✈❡♠❡♥t✱ r❛♥❦❡❞ ♥✉♠❜❡r ♦♥❡✳

❍♦✇❡✈❡r✱ ❛ ♥✉♠❜❡r ♦❢ ♦t❤❡r

❛❝❤✐❡✈❡♠❡♥ts ✐♥ t❤❡ t♦♣ t❡♥ ✇❡r❡ ❛❧s♦ r❡❛❞✐❧② ✐❞❡♥t✐✜❛❜❧❡ ❛s t❤❡ ♣r♦❞✉❝ts
♦❢ ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡rs✿

✶✳ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ r❡♠♦t❡ s❡♥s✐♥❣ s❛t❡❧❧✐t❡s✱
✷✳ t❤❡ ♠✐❝r♦♣r♦❝❡ss♦r✱
✸✳ ❝♦♠♣✉t❡r✲❛✐❞❡❞ ❞❡s✐❣♥ ❛♥❞ ♠❛♥✉❢❛❝t✉r✐♥❣ ✭❈❆❉❈❆▼✮✱
✹✳ ❝♦♠♣✉t❡r✐③❡❞ ❛①✐❛❧ t♦♠♦❣r❛♣❤② ✭❈❆❚ s❝❛♥✮✱
✺✳ ❧❛s❡rs✱ ❛♥❞
✻✳ ✜❜❡r ♦♣t✐❝ ❝♦♠♠✉♥✐❝❛t✐♦♥✳

❆s ❡♥❣✐♥❡❡r✐♥❣ st✉❞❡♥ts✱ ②♦✉ r❡❝♦❣♥✐③❡ t❤❡s❡ ❛❝❤✐❡✈❡♠❡♥ts t♦ ❜❡ ✐♠♣♦r✲
t❛♥t ♠✐❧❡st♦♥❡s ❢♦r ❤✉♠❛♥✐t②❀ ②♦✉ t❛❦❡ ♣r✐❞❡ ✐♥ t❤❡ r♦❧❡ t❤❛t ❡♥❣✐♥❡❡rs
❤❛✈❡ ♣❧❛②❡❞ ✐♥ t❤❡ t❡❝❤♥♦❧♦❣✐❝❛❧ r❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ t✇❡♥t✐❡t❤ ❝❡♥t✉r②✳
❙♦ ❤♦✇ ❞♦ ✇❡ ❤❛r♥❡ss ②♦✉r ❡♥t❤✉s✐❛s♠ ❢♦r t❤❡ ❣r❛♥❞ ❡♥t❡r♣r✐s❡ ♦❢ ❡♥✲
❣✐♥❡❡r✐♥❣❄

❍✐st♦r✐❝❛❧❧②✱ ✇❡ ❤❛✈❡ ❡♥r♦❧❧❡❞ ②♦✉ ✐♥ ❛ ❢r❡s❤♠❛♥ ❝✉rr✐❝✉❧✉♠

♦❢ ♠❛t❤❡♠❛t✐❝s✱ s❝✐❡♥❝❡✱ ❛♥❞ ❤✉♠❛♥✐t✐❡s✳
②♦✉ ✐♥ ❛♥ ❡♥❣✐♥❡❡r✐♥❣ ❝✉rr✐❝✉❧✉♠✳

■❢ ②♦✉ s✉❝❝❡❡❞❡❞✱ ✇❡ ❡♥r♦❧❧❡❞

❲❡ t❤❡♥ t❛✉❣❤t ②♦✉ t❤❡ ❞❡t❛✐❧s ♦❢

②♦✉r ♣r♦❢❡ss✐♦♥ ❛♥❞ ❡♥❝♦✉r❛❣❡❞ ②♦✉r ❢❛✐t❤ t❤❛t ✇❤❛t ②♦✉ ✇❡r❡ st✉❞②✐♥❣ ✐s

✇❤❛t ②♦✉ ♠✉st st✉❞② t♦ ❜❡ ❝r❡❛t✐✈❡ ❛♥❞ ♣r♦❞✉❝t✐✈❡ ❡♥❣✐♥❡❡rs✳ ❚❤❡ ❧♦♥❣❡r
②♦✉r ❢❛✐t❤ ❤❡❧❞✱ t❤❡ ♠♦r❡ ❧✐❦❡❧② ②♦✉ ✇❡r❡ t♦ ❝♦♠♣❧❡t❡ ②♦✉r st✉❞✐❡s✳ ❚❤✐s
s❡❡♠s ❧✐❦❡ ❛♥ ✐♠♣❡r✐♦✉s ❛♣♣r♦❛❝❤ t♦ ❡♥❣✐♥❡❡r✐♥❣ ❡❞✉❝❛t✐♦♥✱ ❡✈❡♥ t❤♦✉❣❤
♠❛t❤❡♠❛t✐❝s✱ ♣❤②s✐❝s✱ ❛♥❞ t❤❡ ❤✉♠❛♥✐t✐❡s ❛r❡ t❤❡ ❢♦✉♥❞❛t✐♦♥ ♦❢ ❡♥❣✐♥❡❡r✲
✐♥❣✱ ❛♥❞ ❞❡t❛✐❧s ❛r❡ ✇❤❛t ❢♦r♠ t❤❡ str✉❝t✉r❡ ♦❢ ❡♥❣✐♥❡❡r✐♥❣✳

■t s❡❡♠s t♦

✉s t❤❛t ❛ ❜❡tt❡r ✇❛② t♦ st✐♠✉❧❛t❡ ②♦✉r ❡♥t❤✉s✐❛s♠ ❛♥❞ ❡♥❝♦✉r❛❣❡ ②♦✉r
❢❛✐t❤ ✐s t♦ ✐♥tr♦❞✉❝❡ ②♦✉ ❡❛r❧② ✐♥ ②♦✉r st✉❞✐❡s t♦ ❡♥❣✐♥❡❡r✐♥❣ t❡❛❝❤❡rs ✇❤♦
✇✐❧❧ s❤❛r❡ t❤❡✐r ✐♥s✐❣❤ts ❛❜♦✉t s♦♠❡ ♦❢ t❤❡ ❢❛s❝✐♥❛t✐♥❣ ❛❞✈❛♥❝❡❞ t♦♣✐❝s
✐♥ ❡♥❣✐♥❡❡r✐♥❣✱ ✇❤✐❧❡ t❡❛❝❤✐♥❣ ②♦✉ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❞ ♣❤②s✐❝❛❧ ♣r✐♥✲
❝✐♣❧❡s ♦❢ ❡♥❣✐♥❡❡r✐♥❣✳

❇✉t ②♦✉ ♠✉st ♠❛t❝❤ t❤❡ t❡❛❝❤❡r✬s ❝♦♠♠✐t♠❡♥t

✇✐t❤ ②♦✉r ♦✇♥ ❝♦♠♠✐t♠❡♥t t♦ st✉❞②✳ ❚❤✐s ♠❡❛♥s t❤❛t ②♦✉ ♠✉st ❛tt❡♥❞
❧❡❝t✉r❡s✱ r❡❛❞ t❡①ts✱ ❛♥❞ ✇♦r❦ ♣r♦❜❧❡♠s✳

❨♦✉ ♠✉st ❜❡ ✐♥q✉✐s✐t✐✈❡ ❛♥❞

s❦❡♣t✐❝❛❧✳ ❆s❦ ②♦✉rs❡❧❢ ❤♦✇ ❛♥ ✐❞❡❛ ✐s ❧✐♠✐t❡❞ ✐♥ s❝♦♣❡ ❛♥❞ ❤♦✇ ✐t ♠✐❣❤t
❜❡ ❡①t❡♥❞❡❞ t♦ ❛♣♣❧② t♦ ❛ ✇✐❞❡r r❛♥❣❡ ♦❢ ♣r♦❜❧❡♠s✳
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

❋♦r✱ ❛❢t❡r ❛❧❧✱ ♦♥❡


✽

♦❢ t❤❡ ❣r❡❛t t❤❡♠❡s ♦❢ ❡♥❣✐♥❡❡r✐♥❣ ✐s t❤❛t ❛ ❢❡✇ ❢✉♥❞❛♠❡♥t❛❧ ✐❞❡❛s ❢r♦♠
♠❛t❤❡♠❛t✐❝s ❛♥❞ s❝✐❡♥❝❡✱ ❝♦✉♣❧❡❞ ✇✐t❤ ❛ ❢❡✇ ♣r✐♥❝✐♣❧❡s ♦❢ ❞❡s✐❣♥✱ ♠❛② ❜❡
❛♣♣❧✐❡❞ t♦ ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ❡♥❣✐♥❡❡r✐♥❣ ♣r♦❜❧❡♠s✳

●♦♦❞ ❧✉❝❦ ✇✐t❤ ②♦✉r

st✉❞✐❡s✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


❈❤❛♣t❡r ✶

❈♦♠♣❧❡① ◆✉♠❜❡rs
✶✳✶ ❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ■♥tr♦❞✉❝t✐♦♥
♥♦t❡✿ ❚❤✐s ♠♦❞✉❧❡ ✐s ♣❛rt ♦❢ t❤❡ ❝♦❧❧❡❝t✐♦♥✱

❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✳

✶

❆ ❋✐rst ❈♦✉rs❡ ✐♥

❚❤❡ ▲❛❚❡❳ s♦✉r❝❡ ✜❧❡s

❢♦r t❤✐s ❝♦❧❧❡❝t✐♦♥ ✇❡r❡ ❝r❡❛t❡❞ ✉s✐♥❣ ❛♥ ♦♣t✐❝❛❧ ❝❤❛r❛❝t❡r r❡❝♦❣✲
♥✐t✐♦♥ t❡❝❤♥♦❧♦❣②✱ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤✐s ♣r♦❝❡ss t❤❡r❡ ♠❛② ❜❡ ♠♦r❡
❡rr♦rs t❤❛♥ ✉s✉❛❧✳ P❧❡❛s❡ ❝♦♥t❛❝t ✉s ✐❢ ②♦✉ ❞✐s❝♦✈❡r ❛♥② ❡rr♦rs✳

✶✳✶✳✶ ◆♦t❡s t♦ ❚❡❛❝❤❡rs ❛♥❞ ❙t✉❞❡♥ts✿
❲❤❡♥ ✇❡ t❡❛❝❤ ❝♦♠♣❧❡① ♥✉♠❜❡rs t♦ ❜❡❣✐♥♥✐♥❣ ❡♥❣✐♥❡❡r✐♥❣ st✉❞❡♥ts✱ ✇❡
❡♥❝♦✉r❛❣❡ ❛

❣❡♦♠❡tr✐❝❛❧ ♣✐❝t✉r❡ s✉♣♣♦rt❡❞ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡✳

❊✈✲

❡r② ❛❧❣❡❜r❛✐❝ ♠❛♥✐♣✉❧❛t✐♦♥ ❝❛rr✐❡❞ ♦✉t ✐♥ ❛ ❧❡❝t✉r❡ ✐s ❛❝❝♦♠♣❛♥✐❡❞ ❜② ❛
❝❛r❡✲❢✉❧❧② ❞r❛✇♥ ♣✐❝t✉r❡ ✐♥ ♦r❞❡r t♦ ✜① t❤❡ ✐❞❡❛ t❤❛t ❣❡♦♠❡tr② ❛♥❞ ❛❧❣❡❜r❛
❣♦ ❤❛♥❞✲✐♥✲❣❧♦✈❡ t♦ ❝♦♠♣❧❡t❡ ♦✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❲❡
❛ss✐❣♥ ❡ss❡♥t✐❛❧❧② ❡✈❡r② ♣r♦❜❧❡♠ ❢♦r ❤♦♠❡✇♦r❦✳
❲❡ ✉s❡ t❤❡ ▼❆❚▲❆❇ ♣r♦❣r❛♠s ✐♥ t❤✐s ❝❤❛♣t❡r t♦ ✐❧❧✉str❛t❡ t❤❡ t❤❡♦r②
♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛♥❞ t♦ ❞❡✈❡❧♦♣ s❦✐❧❧ ✇✐t❤ t❤❡ ▼❆❚▲❆❇ ❧❛♥❣✉❛❣❡✳
❚❤❡ ♥✉♠❡r✐❝❛❧ ❡①♣❡r✐♠❡♥t ✭❙❡❝t✐♦♥ ✶✳✼✮ ✐♥tr♦❞✉❝❡s st✉❞❡♥ts t♦

t❤❡

❜❛s✐❝

q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ♦❢ ❡❧❡❝tr✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡r ❡♥❣✐♥❡❡r✐♥❣ ❛♥❞ s❤♦✇s ❤♦✇
t❤❡ r♦♦ts ♦❢ t❤✐s q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡
❡q✉❛t✐♦♥✳
✏❘❡♣r❡s❡♥t✐♥❣ ❈♦♠♣❧❡① ◆✉♠❜❡rs ✐♥ ❛ ❱❡❝t♦r ❙♣❛❝❡✱✑ ✭❙❡❝t✐♦♥ ✶✳✺✮ ✐s
❛ ❧✐tt❧❡ ❞❡♠❛♥❞✐♥❣ ❢♦r ❢r❡s❤♠❡♥ ❜✉t ❡❛s✐❧② ❛❝❝❡ss✐❜❧❡ t♦ s♦♣❤♦♠♦r❡s✳

■t

♠❛② ❜❡ ❝♦✈❡r❡❞ ❢♦r ❛❞❞✐t✐♦♥❛❧ ✐♥s✐❣❤t✱ s❦✐♣♣❡❞ ✇✐t❤♦✉t ❝♦♥s❡q✉❡♥❝❡✱ ♦r

✶ ❚❤✐s

❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✷✶✹✶✷✴✶✳✻✴❃✳
❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✾


❈❍❆P❚❊❘ ✶✳ ❈❖▼P▲❊❳ ◆❯▼❇❊❘❙

✶✵

❝♦✈❡r❡❞ ❛❢t❡r ❈❤❛♣t❡r ✹ ✭❙❡❝t✐♦♥ ✹✳✶✮✳ ✏❆♥ ❊❧❡❝tr✐❝ ❋✐❡❧❞ ❈♦♠♣✉t❛t✐♦♥✱✑
✭❙❡❝t✐♦♥ ✶✳✻✮ ✐s ✇❡❧❧ ❜❡②♦♥❞ ♠♦st ❢r❡s❤♠❡♥✱ ❛♥❞ ✐t ✐s ❞❡♠❛♥❞✐♥❣ ❢♦r s♦♣❤♦✲
♠♦r❡s✳

◆♦♥❡t❤❡❧❡ss✱ ❛♥ ❡①♣❡rt ✐♥ ❡❧❡❝tr♦♠❛❣♥❡t✐❝s ♠✐❣❤t ✇❛♥t t♦ ❝♦✈❡r

t❤❡ s❡❝t✐♦♥ ✧❆♥ ❡❧❡❝tr✐❝ ❋✐❡❧❞ ❈♦♠♣✉t❛t✐♦♥✧ ✭❙❡❝t✐♦♥ ✶✳✻✮ ❢♦r t❤❡ ✐♥s✐❣❤t
✐t ❜r✐♥❣s t♦ t❤❡ ✉s❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❢♦r r❡♣r❡s❡♥t✐♥❣ t✇♦✲❞✐♠❡♥s✐♦♥❛❧
r❡❛❧ q✉❛♥t✐t✐❡s✳

✶✳✶✳✷ ■♥tr♦❞✉❝t✐♦♥
■t ✐s ❤❛r❞ t♦ ♦✈❡r❡st✐♠❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❚❤❡② ✜rst ❛r♦s❡
✐♥ t❤❡ st✉❞② ♦❢ r♦♦ts ❢♦r q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳

❇✉t✱ ❛s ✇✐t❤ s♦ ♠❛♥②

♦t❤❡r ❣r❡❛t ❞✐s❝♦✈❡r✐❡s✱ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❤❛✈❡ ❢♦✉♥❞ ✇✐❞❡s♣r❡❛❞ ❛♣♣❧✐❝❛✲
t✐♦♥ ✇❡❧❧ ♦✉ts✐❞❡ t❤❡✐r ♦r✐❣✐♥❛❧ ❞♦♠❛✐♥ ♦❢ ❞✐s❝♦✈❡r②✳

❚❤❡② ❛r❡ ♥♦✇ ✉s❡❞

t❤r♦✉❣❤♦✉t ♠❛t❤❡♠❛t✐❝s✱ ❛♣♣❧✐❡❞ s❝✐❡♥❝❡✱ ❛♥❞ ❡♥❣✐♥❡❡r✐♥❣ t♦ r❡♣r❡s❡♥t
t❤❡ ❤❛r♠♦♥✐❝ ♥❛t✉r❡ ♦❢ ✈✐❜r❛t✐♥❣ s②st❡♠s ❛♥❞ ♦s❝✐❧❧❛t✐♥❣ ✜❡❧❞s✳

❋♦r ❡①✲

❛♠♣❧❡✱ ❝♦♠♣❧❡① ♥✉♠❜❡rs ♠❛② ❜❡ ✉s❡❞ t♦ st✉❞②

✐✳ tr❛✈❡❧✐♥❣ ✇❛✈❡s ♦♥ ❛ s❡❛ s✉r❢❛❝❡❀
✐✐✳ st❛♥❞✐♥❣ ✇❛✈❡s ♦♥ ❛ ✈✐♦❧✐♥ str✐♥❣❀
✐✐✐✳ t❤❡ ♣✉r❡ t♦♥❡ ♦❢ ❛ ❑✉r③✇❡✐❧ ♣✐❛♥♦❀
✐✈✳ t❤❡ ❛❝♦✉st✐❝ ✜❡❧❞ ✐♥ ❛ ❝♦♥❝❡rt ❤❛❧❧❀
✈✳ t❤❡ ❧✐❣❤t ♦❢ ❛ ❍❡✲◆❡ ❧❛s❡r❀
✈✐✳ t❤❡ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ ✜❡❧❞ ✐♥ ❛ ❧✐❣❤t s❤♦✇❀
✈✐✐✳ t❤❡ ✈✐❜r❛t✐♦♥s ✐♥ ❛ r♦❜♦t ❛r♠❀
✈✐✐✐✳ t❤❡ ♦s❝✐❧❧❛t✐♦♥s ♦❢ ❛ s✉s♣❡♥s✐♦♥ s②st❡♠❀
✐①✳ t❤❡ ❝❛rr✐❡r s✐❣♥❛❧ ✉s❡❞ t♦ tr❛♥s♠✐t ❆▼ ♦r ❋▼ r❛❞✐♦❀
①✳ t❤❡ ❝❛rr✐❡r s✐❣♥❛❧ ✉s❡❞ t♦ tr❛♥s♠✐t ❞✐❣✐t❛❧ ❞❛t❛ ♦✈❡r t❡❧❡♣❤♦♥❡ ❧✐♥❡s❀
❛♥❞
①✐✳ t❤❡ ✻✵ ❍③ s✐❣♥❛❧ ✉s❡❞ t♦ ❞❡❧✐✈❡r ♣♦✇❡r t♦ ❛ ❤♦♠❡✳

■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ❞❡✈❡❧♦♣ t❤❡ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr② ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs✳
■♥ ❈❤❛♣t❡r ✸ ✭❙❡❝t✐♦♥ ✸✳✶✮ ✇❡ ✇✐❧❧ s❤♦✇ ❤♦✇ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❛r❡ ✉s❡❞
t♦ ❜✉✐❧❞ ♣❤❛s♦r r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ♣♦✇❡r ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥ s✐❣♥❛❧s✳

✶✳✷ ❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ●❡♦♠❡tr② ♦❢ ❈♦♠♣❧❡①

◆✉♠❜❡rs
✷

♥♦t❡✿ ❚❤✐s ♠♦❞✉❧❡ ✐s ♣❛rt ♦❢ t❤❡ ❝♦❧❧❡❝t✐♦♥✱

❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✳
✷ ❚❤✐s

❆ ❋✐rst ❈♦✉rs❡ ✐♥

❚❤❡ ▲❛❚❡❳ s♦✉r❝❡ ✜❧❡s

❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✷✶✹✶✶✴✶✳✻✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


✶✶

❢♦r t❤✐s ❝♦❧❧❡❝t✐♦♥ ✇❡r❡ ❝r❡❛t❡❞ ✉s✐♥❣ ❛♥ ♦♣t✐❝❛❧ ❝❤❛r❛❝t❡r r❡❝♦❣✲
♥✐t✐♦♥ t❡❝❤♥♦❧♦❣②✱ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤✐s ♣r♦❝❡ss t❤❡r❡ ♠❛② ❜❡ ♠♦r❡
❡rr♦rs t❤❛♥ ✉s✉❛❧✳ P❧❡❛s❡ ❝♦♥t❛❝t ✉s ✐❢ ②♦✉ ❞✐s❝♦✈❡r ❛♥② ❡rr♦rs✳

❚❤❡ ♠♦st ❢✉♥❞❛♠❡♥t❛❧ ♥❡✇ ✐❞❡❛ ✐♥ t❤❡ st✉❞② ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs ✐s t❤❡
✏✐♠❛❣✐♥❛r② ♥✉♠❜❡r✑
r♦♦t ♦❢

j✳

❚❤✐s ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ sq✉❛r❡

−1✿

❚❤❡ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r

j

√

−1

✭✶✳✶✮

j 2 = −1.

✭✶✳✷✮

j=

✐s ✉s❡❞ t♦ ❜✉✐❧❞ ❝♦♠♣❧❡① ♥✉♠❜❡rs

x

❛♥❞

y

✐♥

t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿

z = x + jy.
❲❡ s❛② t❤❛t t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r

z

✭✶✳✸✮

❤❛s ✏r❡❛❧ ♣❛rt✑

x ❛♥❞ ✏✐♠❛❣✐♥❛r② ♣❛rt✑

y✿
z = Re [z] + jIm [z]

✭✶✳✹✮

Re [z] = x; Im [z] = y.

✭✶✳✺✮

x ✐s ❞❡♥♦t❡❞ ❜② r❡❛❧✭③✮✱ ❛♥❞ t❤❡ ✈❛r✐❛❜❧❡ y ✐s
✐♠❛❣✭③✮✳ ■♥ ❝♦♠♠✉♥✐❝❛t✐♦♥ t❤❡♦r②✱ x ✐s ❝❛❧❧❡❞ t❤❡ ✏✐♥✲♣❤❛s❡✑
❝♦♠♣♦♥❡♥t ♦❢ z ✱ ❛♥❞ y ✐s ❝❛❧❧❡❞ t❤❡ ✏q✉❛❞r❛t✉r❡✑ ❝♦♠♣♦♥❡♥t✳ ❲❡ ❝❛❧❧
z =x + jy t❤❡ ❈❛rt❡s✐❛♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ z ✱ ✇✐t❤ r❡❛❧ ❝♦♠♣♦♥❡♥t x ❛♥❞
✐♠❛❣✐♥❛r② ❝♦♠♣♦♥❡♥t y ✳ ❲❡ s❛② t❤❛t t❤❡ ❈❛rt❡s✐❛♥ ♣❛✐r (x, y)❝♦❞❡s t❤❡
❝♦♠♣❧❡① ♥✉♠❜❡r z ✳
❲❡ ♠❛② ♣❧♦t t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r z ♦♥ t❤❡ ♣❧❛♥❡ ❛s ✐♥ ❋✐❣✉r❡ ✶✳✶✳ ❲❡
■♥ ▼❆❚▲❆❇✱ t❤❡ ✈❛r✐❛❜❧❡

❞❡♥♦t❡❞ ❜②

❝❛❧❧ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s t❤❡ ✏r❡❛❧ ❛①✐s✑ ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s t❤❡ ✏✐♠❛❣✐♥❛r②
❛①✐s✳✑

❚❤❡ ♣❧❛♥❡ ✐s ❝❛❧❧❡❞ t❤❡ ✏❝♦♠♣❧❡① ♣❧❛♥❡✳✑

t❤❡ ❧✐♥❡ t♦ t❤❡ ♣♦✐♥t

z = x + jy

r=

x2 + y 2

θ = tan−1
❙❡❡ ❋✐❣✉r❡ ✶✳✶✳ ■♥ ▼❆❚▲❆❇✱
❜②

❛♥❣❧❡✭③✮✳

r

❚❤❡ r❛❞✐✉s ❛♥❞ ❛♥❣❧❡ ♦❢

❛r❡

✭✶✳✻✮

y
.
x

✐s ❞❡♥♦t❡❞ ❜②

✭✶✳✼✮

❛❜s✭③✮✱

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

❛♥❞

θ

✐s ❞❡♥♦t❡❞


❈❍❆P❚❊❘ ✶✳ ❈❖▼P▲❊❳ ◆❯▼❇❊❘❙

✶✷

❋✐❣✉r❡ ✶✳✶✿

◆✉♠❜❡r

❈❛rt❡s✐❛♥ ❛♥❞ P♦❧❛r ❘❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❈♦♠♣❧❡①

③

❚❤❡ ♦r✐❣✐♥❛❧ ❈❛rt❡s✐❛♥ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ r❛❞✐✉s
❛♥❞ ❛♥❣❧❡

θ

r

❛s ❢♦❧❧♦✇s✿

❚❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r

z
z

x = r❝♦sθ

✭✶✳✽✮

y = r sin θ.

✭✶✳✾✮

♠❛② t❤❡r❡❢♦r❡ ❜❡ ✇r✐tt❡♥ ❛s

=

x + jy

=

r❝♦sθ + jrs✐♥θ

✭✶✳✶✵✮

= r ( cosθ + j sin θ) .
❚❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r ❝♦s

θ + j s✐♥θ

✐s✱

✐ts❡❧❢✱

❛ ♥✉♠❜❡r t❤❛t ♠❛② ❜❡

r❡♣r❡s❡♥t❡❞ ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ❛♥❞ ❝♦❞❡❞ ✇✐t❤ t❤❡ ❈❛rt❡s✐❛♥ ♣❛✐r

(cosθ, s✐♥θ)✳ ❚❤✐s ✐s ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✉r❡ ✶✳✷✳ ❚❤❡
z = cosθ + j s✐♥θ ❛r❡ ✶ ❛♥❞ θ✳ ❈❛♥ ②♦✉ s❡❡

t❤❡ ♣♦✐♥t

r❛❞✐✉s ❛♥❞ ❛♥❣❧❡ t♦
✇❤②❄

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✶✸

❋✐❣✉r❡ ✶✳✷✿

❚❤❡ ❈♦♠♣❧❡① ◆✉♠❜❡r cos θ + jsin θ

θ + j s✐♥θ

❚❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r ❝♦s

✐s ♦❢ s✉❝❤ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rt❛♥❝❡

t♦ ♦✉r st✉❞② ♦❢ ❝♦♠♣❧❡① ♥✉♠❜❡rs t❤❛t ✇❡ ❣✐✈❡ ✐t t❤❡ s♣❡❝✐❛❧ s②♠❜♦❧

ejθ = cosθ + j s✐♥θ.
❆s ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✉r❡ ✶✳✷✱ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r
❛♥❣❧❡

θ✳

❲✐t❤ t❤❡ s②♠❜♦❧

ejθ ✱

✿

✭✶✳✶✶✮

e

jθ

❤❛s r❛❞✐✉s ✶ ❛♥❞

✇❡ ♠❛② ✇r✐t❡ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r

z = rejθ .
z = rejθ

ejθ

z

❛s

✭✶✳✶✷✮

♣♦❧❛r

r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✳ ❲❡
r∠θ❝♦❞❡s t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r z ✳ ■♥ t❤✐s ♣♦❧❛r
r❡♣r❡s❡♥t❛t✐♦♥✱ ✇❡ ❞❡✜♥❡ |z| = r t♦ ❜❡ t❤❡ ♠❛❣♥✐t✉❞❡ ♦❢ z ❛♥❞ arg (z) = θ
t♦ ❜❡ t❤❡ ❛♥❣❧❡✱ ♦r ♣❤❛s❡✱ ♦❢ z ✿
❲❡ ❝❛❧❧

❛

s❛② t❤❛t t❤❡ ♣♦❧❛r ♣❛✐r

|z| = r

✭✶✳✶✸✮

arg (z) = θ.

✭✶✳✶✹✮

❲✐t❤ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ♦❢ ♠❛❣♥✐t✉❞❡ ❛♥❞ ♣❤❛s❡✱ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ❝♦♠♣❧❡①
♥✉♠❜❡r

z

❛s

z = |z|ejarg(z) .

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✭✶✳✶✺✮


❈❍❆P❚❊❘ ✶✳ ❈❖▼P▲❊❳ ◆❯▼❇❊❘❙

✶✹

▲❡t✬s s✉♠♠❛r✐③❡ ♦✉r ✇❛②s ♦❢ ✇r✐t✐♥❣ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r

③

❛♥❞ r❡❝♦r❞

t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❣❡♦♠❡tr✐❝ ❝♦❞❡s✿

z

= x + jy

= rejθ

↓

↓

(x, y)

r∠θ

= |z|ej

arg(z)

.
✭✶✳✶✻✮

■♥ ✧❘♦♦ts ♦❢ ◗✉❛❞r❛t✐❝ ❊q✉❛t✐♦♥s✧ ✭❙❡❝t✐♦♥ ✶✳✹✮ ✇❡ s❤♦✇ t❤❛t t❤❡ ❞❡✜♥✐✲

ejθ = ❝♦sθ+j s✐♥θ ✐s ♠♦r❡ t❤❛♥ s②♠❜♦❧✐❝✳ ❲❡ s❤♦✇✱ ✐♥ ❢❛❝t✱ t❤❛t ejθ ✐s
x

❥✉st t❤❡ ❢❛♠✐❧✐❛r ❢✉♥❝t✐♦♥ e ❡✈❛❧✉❛t❡❞ ❛t t❤❡ ✐♠❛❣✐♥❛r② ❛r❣✉♠❡♥t x = jθ ✳
jθ
❲❡ ❝❛❧❧ e
❛ ✏❝♦♠♣❧❡① ❡①♣♦♥❡♥t✐❛❧✱✑ ♠❡❛♥✐♥❣ t❤❛t ✐t ✐s ❛♥ ❡①♣♦♥❡♥t✐❛❧
t✐♦♥

✇✐t❤ ❛♥ ✐♠❛❣✐♥❛r② ❛r❣✉♠❡♥t✳

❊①❡r❝✐s❡ ✶✳✷✳✶
Pr♦✈❡

2n

(j)

= (−1)

n

❛♥❞

2n+1

(j)

n

= (−1) j ✳

❊✈❛❧✉❛t❡

j3, j4, j5✳

❊①❡r❝✐s❡ ✶✳✷✳✷

ej[(π/2)+m2π] = j, ej[(3π/2)+m2π] = −j, ej(0+m2π) = 1✱ ❛♥❞
e
= −1✳ P❧♦t t❤❡s❡ ✐❞❡♥t✐t✐❡s ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✳
✭❆ss✉♠❡ m ✐s ❛♥ ✐♥t❡❣❡r✳✮
Pr♦✈❡

j(π+m2π)

❊①❡r❝✐s❡ ✶✳✷✳✸
❋✐♥❞ t❤❡ ♣♦❧❛r r❡♣r❡s❡♥t❛t✐♦♥

z = rejθ

❢♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣

❝♦♠♣❧❡① ♥✉♠❜❡rs✿
❛✳
❜✳
❝✳
❞✳

z
z
z
z

= 1 + j0❀
= 0 + j1❀
= 1 + j1❀
= −1 − j1✳

P❧♦t t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✳

❊①❡r❝✐s❡ ✶✳✷✳✹
❋✐♥❞ t❤❡ ❈❛rt❡s✐❛♥ r❡♣r❡s❡♥t❛t✐♦♥

z = x + jy

❢♦r ❡❛❝❤ ♦❢ t❤❡

❢♦❧❧♦✇✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡rs✿
❛✳
❜✳
❝✳
❞✳

z
z
z
z

√
= √2ejπ/2 ❀
= 2ejπ/4 ❀
= e√j3π/4 ❀

= 2ej3π/2 ✳

P❧♦t t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡✳

❊①❡r❝✐s❡ ✶✳✷✳✺
❚❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♦♠❡tr✐❝ ❝♦❞❡s r❡♣r❡s❡♥t ❝♦♠♣❧❡① ♥✉♠❜❡rs✳ ❉❡✲
❝♦❞❡ ❡❛❝❤ ❜② ✇r✐t✐♥❣ ❞♦✇♥ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♠♣❧❡① ♥✉♠❜❡r

③✿

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


✶✺

❛✳
❜✳
❝✳
❞✳

(0.7, −0.1) z =
(−1.0, 0.5) z =
1.6∠π/8 z =❄
0.4∠7π/8 z =❄

❄
❄

❊①❡r❝✐s❡ ✶✳✷✳✻

❙❤♦✇ t❤❛t

Im [jz] = Re [z]

❛♥❞

Re [−jz] = Im [z]✳

❉❡♠♦ ✶✳✶

✭▼❆❚▲❆❇✮✳ ❘✉♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ▼❆❚▲❆❇ ♣r♦❣r❛♠ ✐♥ ♦r❞❡r t♦
❝♦♠♣✉t❡ ❛♥❞ ♣❧♦t t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r

ejθ

❢♦r

θ = i2π/360, i =

1, 2, ..., 360✿

❥❂sqrt✭✲✶✮
♥❂✸✻✵
❢♦r ✐❂✶✿♥✱❝✐r❝❧❡✭✐✮❂❡①♣✭❥✯✷✯♣✐✯✐✴♥✮❀❡♥❞❀
❛①✐s✭✬sq✉❛r❡✬✮
♣❧♦t✭❝✐r❝❧❡✮
❘❡♣❧❛❝❡ t❤❡ ❡①♣❧✐❝✐t ❢♦r ❧♦♦♣ ♦❢ ❧✐♥❡ ✸ ❜② t❤❡ ✐♠♣❧✐❝✐t ❧♦♦♣

❝✐r❝❧❡❂❡①♣✭❥✯✷✯♣✐✯❬✶✿♥❪✴♥✮❀
t♦

s♣❡❡❞

t❤❛t

t❤❡

✉♣

t❤❡

❝❛❧❝✉❧❛t✐♦♥✳

❝♦♠♣❧❡①

♥✉♠❜❡r

2π/360, 2 (2π/360) , ...✱ t✉r♥s
❛♥❣❧❡ θ ❛♥❞ r❛❞✐✉s ✶✳ ❲❡ s❛②

❨♦✉

ejθ ✱

❝❛♥

s❡❡

❡✈❛❧✉❛t❡❞

❢r♦♠
❛t

❋✐❣✉r❡

✶✳✸

θ

=

❛♥❣❧❡s

♦✉t ❝♦♠♣❧❡① ♥✉♠❜❡rs t❤❛t ❧✐❡ ❛t
t❤❛t

ejθ

✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t

❧✐❡s ♦♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✳ ❲❡ ✇✐❧❧ ❤❛✈❡ ♠✉❝❤ ♠♦r❡ t♦ s❛② ❛❜♦✉t t❤❡
✉♥✐t ❝✐r❝❧❡ ✐♥ ❈❤❛♣t❡r ✷✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃


❈❍❆P❚❊❘ ✶✳ ❈❖▼P▲❊❳ ◆❯▼❇❊❘❙

✶✻

❋✐❣✉r❡ ✶✳✸✿

✶✳✶✮

❚❤❡ ❈♦♠♣❧❡① ◆✉♠❜❡rs ejθ ❢♦r 0 ≤ θ ≤ 2π ✭❉❡♠♦

✶✳✸ ❈♦♠♣❧❡① ◆✉♠❜❡rs✿ ❆❧❣❡❜r❛ ♦❢ ❈♦♠♣❧❡①
◆✉♠❜❡rs
✸

♥♦t❡✿ ❚❤✐s ♠♦❞✉❧❡ ✐s ♣❛rt ♦❢ t❤❡ ❝♦❧❧❡❝t✐♦♥✱

❊❧❡❝tr✐❝❛❧ ❛♥❞ ❈♦♠♣✉t❡r ❊♥❣✐♥❡❡r✐♥❣✳

❆ ❋✐rst ❈♦✉rs❡ ✐♥

❚❤❡ ▲❛❚❡❳ s♦✉r❝❡ ✜❧❡s

❢♦r t❤✐s ❝♦❧❧❡❝t✐♦♥ ✇❡r❡ ❝r❡❛t❡❞ ✉s✐♥❣ ❛♥ ♦♣t✐❝❛❧ ❝❤❛r❛❝t❡r r❡❝♦❣✲
♥✐t✐♦♥ t❡❝❤♥♦❧♦❣②✱ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤✐s ♣r♦❝❡ss t❤❡r❡ ♠❛② ❜❡ ♠♦r❡
❡rr♦rs t❤❛♥ ✉s✉❛❧✳ P❧❡❛s❡ ❝♦♥t❛❝t ✉s ✐❢ ②♦✉ ❞✐s❝♦✈❡r ❛♥② ❡rr♦rs✳

✸ ❚❤✐s

❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✷✶✹✵✽✴✶✳✻✴❃✳

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✶✼

❚❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❢♦r♠ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ✏✜❡❧❞✑ ♦♥ ✇❤✐❝❤ t❤❡ ✉s✉❛❧
♦♣❡r❛t✐♦♥s ♦❢ ❛❞❞✐t✐♦♥ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❛r❡ ❞❡✜♥❡❞✳ ❊❛❝❤ ♦❢ t❤❡s❡ ♦♣✲
❡r❛t✐♦♥s ❤❛s ❛ s✐♠♣❧❡ ❣❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥✳

✶✳✸✳✶ ❆❞❞✐t✐♦♥ ❛♥❞ ▼✉❧t✐♣❧✐❝❛t✐♦♥✳
❚❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs

z1

❛♥❞

z2

❛r❡

❛❞❞❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ r✉❧❡

z1 + z2

=

(x1 + jy1 ) + (x2 + jy2 )

=

(x1 + x2 ) + j (y1 + y2 ) .

❲❡ s❛② t❤❛t t❤❡ r❡❛❧ ♣❛rts ❛❞❞ ❛♥❞ t❤❡ ✐♠❛❣✐♥❛r② ♣❛rts ❛❞❞✳
tr❛t❡❞ ✐♥ ❋✐❣✉r❡ ✶✳✹✱ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡r
✏♣❛r❛❧❧❡❧♦❣r❛♠ r✉❧❡✱✑ ✇❤❡r❡✐♥
❢♦r♠❡❞ ❢r♦♠

z1

❛♥❞

z1 + z2

z1 + z2

✭✶✳✶✼✮

❆s ✐❧❧✉s✲

✐s ❝♦♠♣✉t❡❞ ❢r♦♠ ❛

❧✐❡s ♦♥ t❤❡ ♥♦❞❡ ♦❢ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠

z2 ✳

❊①❡r❝✐s❡ ✶✳✸✳✶

z1 = r1 ejθ1 ❛♥❞ z2 = r2 ejθ2 ✳ ❋✐♥❞ ❛ ♣♦❧❛r ❢♦r♠✉❧❛ z3 =r3 ejθ3
z3 = z1 + z2 t❤❛t ✐♥✈♦❧✈❡s ♦♥❧② t❤❡ ✈❛r✐❛❜❧❡s r1 , r2 , θ1 ✱ ❛♥❞
❚❤❡ ❢♦r♠✉❧❛ ❢♦r r3 ✐s t❤❡ ✏❧❛✇ ♦❢ ❝♦s✐♥❡s✳✑

▲❡t

❢♦r

θ2 ✳

❚❤❡ ♣r♦❞✉❝t ♦❢

③✶

❛♥❞

z1 z2

③✷

✐s

=

(x1 + jy1 ) (x2 + jy2 )

=

(x1 x2 − y1 y2 ) + j (y1 x2 + x1 y2 ) .

❆✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡ ❛t ❈♦♥♥❡①✐♦♥s
❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✵✻✽✺✴✶✳✷❃

✭✶✳✶✽✮