Linear maps in the plane do not preserve circles r rtt - - PowerPoint PPT Presentation

STRAIGHTENING THE SQUARE

❈♦♥❢♦r♠❛❧✴❛✣♥❡ ❣❡♦♠❡tr②✱ ✢❛t ❝♦♥♥❡❝t✐♦♥s ❛♥❞ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛

What if you live

in a flat world

where size is not well-defined?

❆r♥❛✉❞ ❈❤ér✐t❛t

❈◆❘❙✱ ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡

▼❛r✳ ✷✵✶✾

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶ ✴ ✸✸

Linear maps in the plane

do not preserve circles

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷ ✴ ✸✸

Beltrami derivative

• ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ ✷
• f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥

t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det(df ) > ✵✳ ❚❤❡♥ ✐s ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ ✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ ❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑ ✶✳ ❚❤❡ ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ✳ ✕ ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s ✶ ✕ ✵ ✐✛ ✐s ❛ s✐♠✐❧✐t✉❞❡

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

Beltrami derivative

• ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ ✷
• f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥

t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det(df ) > ✵✳ ❚❤❡♥ f ✐s holomorphic ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ conformal✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ ❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑ ✶✳ ❚❤❡ ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ✳ ✕ ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s ✶ ✕ ✵ ✐✛ ✐s ❛ s✐♠✐❧✐t✉❞❡

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

Beltrami derivative

• ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ ✷
• f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥

t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det(df ) > ✵✳ ❚❤❡♥ f ✐s holomorphic ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ conformal✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r df ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② df ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ K❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑> ✶✳ ❚❤❡ ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ✳ ✕ ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s ✶ ✕ ✵ ✐✛ ✐s ❛ s✐♠✐❧✐t✉❞❡

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

Beltrami derivative

• ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ ✷
• f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥

t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det(df ) > ✵✳ ❚❤❡♥ f ✐s holomorphic ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ conformal✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r df ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② df ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ K❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑> ✶✳ ❚❤❡ Beltrami derivative ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r Bf (z) t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② df ✳ ✕ Bf (z) ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s < ✶ ✕ Bf = ✵ ✐✛ df ✐s ❛ s✐♠✐❧✐t✉❞❡

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

Ellipse fields

❆♥ ellipse field ✐s t❤❡ ❞❛t❛ ♦❢ ✭✐♥✜♥✐t❡s✐♠❛❧✮ ❡❧❧✐♣s❡s ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ ❛ ❞♦♠❛✐♥✳ ❚❤❡✐r s✐③❡ ✐s ♥♦t r❡❧❡✈❛♥t✱ ♦♥❧② t❤❡✐r ❞✐r❡❝t✐♦♥ ❛♥❞ ✢❛t♥❡ss✱ ❡♥❝♦❞❡❞ ❜② ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r µ(z) ✇✐t❤ t❤❡ s❛♠❡ ❝♦♥✈❡♥t✐♦♥ ❛s ❢♦r Bf ✳

Straightening t❤❡ ❡❧❧✐♣s❡ ✜❡❧❞ ✐s s♦❧✈✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ Bf = µ✱

✐✳❡✳ ✜♥❞✐♥❣ ❛ ❞❡❢♦r♠❛t✐♦♥ f ♦❢ t❤❡ ♣❧❛♥❡ ✇❤♦s❡ ❞✐✛❡r❡♥t✐❛❧ s❡♥❞s ❛❧❧ t❤❡ ❡❧❧✐♣s❡s t♦ ❝✐r❝❧❡s✳ ❯s❡s✱ ❡①✐st❡♥❝❡✱ ❢♦r♠✉❧❛❄

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✹ ✴ ✸✸

Ellipse fields

❆♥ ellipse field ✐s t❤❡ ❞❛t❛ ♦❢ ✭✐♥✜♥✐t❡s✐♠❛❧✮ ❡❧❧✐♣s❡s ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ ❛ ❞♦♠❛✐♥✳ ❚❤❡✐r s✐③❡ ✐s ♥♦t r❡❧❡✈❛♥t✱ ♦♥❧② t❤❡✐r ❞✐r❡❝t✐♦♥ ❛♥❞ ✢❛t♥❡ss✱ ❡♥❝♦❞❡❞ ❜② ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r µ(z) ✇✐t❤ t❤❡ s❛♠❡ ❝♦♥✈❡♥t✐♦♥ ❛s ❢♦r Bf ✳

Straightening t❤❡ ❡❧❧✐♣s❡ ✜❡❧❞ ✐s s♦❧✈✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ Bf = µ✱

✐✳❡✳ ✜♥❞✐♥❣ ❛ ❞❡❢♦r♠❛t✐♦♥ f ♦❢ t❤❡ ♣❧❛♥❡ ✇❤♦s❡ ❞✐✛❡r❡♥t✐❛❧ s❡♥❞s ❛❧❧ t❤❡ ❡❧❧✐♣s❡s t♦ ❝✐r❝❧❡s✳ ❯s❡s✱ ❡①✐st❡♥❝❡✱ ❢♦r♠✉❧❛❄

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✹ ✴ ✸✸

The disk

❙❡tt✐♥❣✿ ❛♥ ❡❧❧✐♣s❡ ✜❡❧❞ ♦♥ t❤❡ ♣❧❛♥❡ t❤❛t ✐s ❝♦♥st❛♥t ✐♥ t❤❡ ✉♥✐t ❞✐s❦ ✭µ = a✮✱ ❛♥❞ ❝✐r❝❧❡s ♦✉ts✐❞❡ ✭µ = ✵✮✳ Pr♦❜❧❡♠✿ ✜♥❞ t❤❡ str❛✐❣❤t❡♥✐♥❣✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✺ ✴ ✸✸

The disk

an amazing coincidence

(✶ + a)x (✶ − a)y z + a z

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✻ ✴ ✸✸

The disk

an amazing coincidence

(✶ + a)x (✶ − a)y z + a z

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✻ ✴ ✸✸

The disk

an amazing coincidence

(✶ + a)x (✶ − a)y z + a z

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✻ ✴ ✸✸

The disk

an amazing coincidence (a conspiracy?)

❘❡❝❛❧❧ t❤❛t ✐❢ z = x + iy t❤❡♥ z = x − iy ❛♥❞ |z|✷ = x✷ + y✷ = zz✱ ❤❡♥❝❡

|z| = ✶ ⇐⇒ ✶ z = z.

■♥ ♣❛rt✐❝✉❧❛r ❢♦r ♦♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✿ ✶ ✶ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✼ ✴ ✸✸

The disk

an amazing coincidence (a conspiracy?)

❘❡❝❛❧❧ t❤❛t ✐❢ z = x + iy t❤❡♥ z = x − iy ❛♥❞ |z|✷ = x✷ + y✷ = zz✱ ❤❡♥❝❡

|z| = ✶ ⇐⇒ ✶ z = z.

■♥ ♣❛rt✐❝✉❧❛r ❢♦r z ♦♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✿ z + a z = z + az = x + iy + a(x − iy) = (✶ + a)x + i(✶ − a)y. ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✼ ✴ ✸✸

The disk

an amazing coincidence (a conspiracy?)

❘❡❝❛❧❧ t❤❛t ✐❢ z = x + iy t❤❡♥ z = x − iy ❛♥❞ |z|✷ = x✷ + y✷ = zz✱ ❤❡♥❝❡

|z| = ✶ ⇐⇒ ✶ z = z.

■♥ ♣❛rt✐❝✉❧❛r ❢♦r z ♦♥ t❤❡ ✉♥✐t ❝✐r❝❧❡✿ z + a z = z + az = x + iy + a(x − iy) = (✶ + a)x + i(✶ − a)y. ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✼ ✴ ✸✸

The square

❙❡tt✐♥❣✿ ❛♥ ❡❧❧✐♣s❡ ✜❡❧❞ ♦♥ t❤❡ ♣❧❛♥❡ t❤❛t ✐s ❝♦♥st❛♥t ✐♥ t❤❡ sq✉❛r❡ ✭µ = a✮✱ ❛♥❞ ❝✐r❝❧❡s ♦✉ts✐❞❡ ✭µ = ✵✮✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✽ ✴ ✸✸

The square

An anecdote

❆ mysterious drawing ♣✐♥♥❡❞ ♦♥ ❆✳ ❉♦✉❛❞②✬s ♦✣❝❡ ✇❛❧❧ ✐♥ t❤❡ ✶✾✾✵✬s✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✾ ✴ ✸✸

The square

An anecdote

❆ mysterious drawing ♣✐♥♥❡❞ ♦♥ ❆✳ ❉♦✉❛❞②✬s ♦✣❝❡ ✇❛❧❧ ✐♥ t❤❡ ✶✾✾✵✬s✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✾ ✴ ✸✸

The square

An anecdote

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✵ ✴ ✸✸

The square

By curiosity

❲❤❛t ❞♦❡s ✐t ❧♦♦❦ ❧✐❦❡ ♦♥ ❉♦✉❛❞②❄

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✶ ✴ ✸✸

The square

By curiosity

❲❤❛t ❞♦❡s ✐t ❧♦♦❦ ❧✐❦❡ ♦♥ ❉♦✉❛❞②❄

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✶ ✴ ✸✸

The square

a naive attempt

❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✷ ✴ ✸✸

The square

An obstruction to naiveness from potential theory

❇❡❝❛✉s❡ ❝♦♥❢♦r♠❛❧ ♠❛♣s ♠✉st ♣r❡s❡r✈❡ t❤❡ s♦❧✉t✐♦♥s ♦❢ ▲❛♣❧❛❝❡✬s ❡q✉❛t✐♦♥ ∆V = ✵✱ ❡♥❡r❣② ❝♦♥s✐❞❡r❛t✐♦♥s ✐♠♣❧② t❤❛t ❛ s✐❞❡ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ ❝❛♥♥♦t ❜❡ ♠❛♣♣❡❞ t♦ ❛ ❝✉r✈❡ ✇✐t❤ ❛ t♦♦ s♠❛❧❧ ❞✐❛♠❡t❡r✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✸ ✴ ✸✸

The square

Modified Laplacian approach

❚❤❡ s♦❧✉t✐♦♥ f ♦❢ t❤❡ ❇❡❧tr❛♠✐ ❡q✉❛t✐♦♥ ✐s ❤❛r♠♦♥✐❝ ❢♦r ❛ ♠♦❞✐✜❡❞ ▲❛♣❧❛❝✐❛♥✿

• ∆f = ✵.

❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✇❡❧❧✲st✉❞✐❡❞ s❝❤❡♠❡s t♦ s♦❧✈❡ t❤✐s ❦✐♥❞ ♦❢ ❡q✉❛t✐♦♥s ♥✉♠❡r✐❝❛❧❧②✳ ❚♦ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ♣✐❝t✉r❡s✱ ■ ✇♦r❦❡❞ ♦♥ ❛ ❣r✐❞✱ ✉s❡❞ ❛ ❞✐s❝r❡t❡ ♠♦❞✐✜❡❞ ❧❛♣❧❛❝✐❛♥✱ ❛♥❞ ❛♣♣r♦①✐♠❛t❡❞ ❛ s♦❧✉t✐♦♥ ✉s✐♥❣ t❤❡ Jacobi relaxation method✱ ❛♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ t❤❛t ❝♦♥✈❡r❣❡s r❛♣✐❞❧②✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✹ ✴ ✸✸

Numerically solving a modified Laplacian

K = ✷

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✺ ✴ ✸✸

Numerically solving a modified Laplacian

K = ✺

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✺ ✴ ✸✸

Numerically solving a modified Laplacian

K = ✶✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✺ ✴ ✸✸

Numerically solving a modified Laplacian

K = ✷✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✺ ✴ ✸✸

Limit as K → +∞

Guesses for the square

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✻ ✴ ✸✸

The square

Reformulation

❈❤❛rt ✶ ❈❤❛rt ✷ z + (i/✷) z − (i/✷) ✷z + ✶ ✷z − ✶

• ❧✉❡ t❤❡ t✇♦ ❝❤❛rts

❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✐♥✲ ❞✐❝❛t❡❞ ♠❛♣s ✶ ✶ ✶ ✶✴✷

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✼ ✴ ✸✸

The square

❚❤❡ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ ♦❢ t❤❡ ❢♦r♠ z → az + b✱ t❤✉s ❛r❡ ❤♦❧♦♠♦r♣❤✐❝✿ ✇❡ ❥✉st ❞❡✜♥❡❞ ❛ ❘✐❡♠❛♥♥ s✉r❢❛❝❡✳ ❇✉t ❜❡tt❡r✳ ✳ ✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ s✐♠✐❧✐t✉❞❡s✱ s♦ ✇❡ ✇♦r❦ ✇✐t❤ ❛ ♠♦r❡ r✐❣✐❞ ❝❛t❡❣♦r② ♦❢ ❣❡♦♠❡tr✐❝❛❧ ♦❜❥❡❝t✱ ✱ ✇✐t❤ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ❧✐❦❡✳ ✳ ✳ ✳ ✳ ✳ ❛ ❧♦❝❛❧❧② tr✐✈✐❛❧ ♣❛r❛❧❧❡❧ tr❛♥s♣♦rt✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✽ ✴ ✸✸

The square

❚❤❡ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ ♦❢ t❤❡ ❢♦r♠ z → az + b✱ t❤✉s ❛r❡ ❤♦❧♦♠♦r♣❤✐❝✿ ✇❡ ❥✉st ❞❡✜♥❡❞ ❛ ❘✐❡♠❛♥♥ s✉r❢❛❝❡✳ ❇✉t ❜❡tt❡r✳ ✳ ✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ s✐♠✐❧✐t✉❞❡s✱ s♦ ✇❡ ✇♦r❦ ✇✐t❤ ❛ ♠♦r❡ r✐❣✐❞ ❝❛t❡❣♦r② ♦❢ ❣❡♦♠❡tr✐❝❛❧ ♦❜❥❡❝t✱ ✱ ✇✐t❤ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ❧✐❦❡✳ ✳ ✳ ✳ ✳ ✳ ❛ ❧♦❝❛❧❧② tr✐✈✐❛❧ ♣❛r❛❧❧❡❧ tr❛♥s♣♦rt✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✽ ✴ ✸✸

The square

❚❤❡ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ ♦❢ t❤❡ ❢♦r♠ z → az + b✱ t❤✉s ❛r❡ ❤♦❧♦♠♦r♣❤✐❝✿ ✇❡ ❥✉st ❞❡✜♥❡❞ ❛ ❘✐❡♠❛♥♥ s✉r❢❛❝❡✳ ❇✉t ❜❡tt❡r✳ ✳ ✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ s✐♠✐❧✐t✉❞❡s✱ s♦ ✇❡ ✇♦r❦ ✇✐t❤ ❛ ♠♦r❡ r✐❣✐❞ ❝❛t❡❣♦r② ♦❢ ❣❡♦♠❡tr✐❝❛❧ ♦❜❥❡❝t✱ similarity surfaces✱ ✇✐t❤ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ❧✐❦❡✳ ✳ ✳ ✳ ✳ ✳ ❛ ❧♦❝❛❧❧② tr✐✈✐❛❧ ♣❛r❛❧❧❡❧ tr❛♥s♣♦rt✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✽ ✴ ✸✸

The square

❚❤❡ ❝❤❛♥❣❡s ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ ♦❢ t❤❡ ❢♦r♠ z → az + b✱ t❤✉s ❛r❡ ❤♦❧♦♠♦r♣❤✐❝✿ ✇❡ ❥✉st ❞❡✜♥❡❞ ❛ ❘✐❡♠❛♥♥ s✉r❢❛❝❡✳ ❇✉t ❜❡tt❡r✳ ✳ ✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ❛r❡ s✐♠✐❧✐t✉❞❡s✱ s♦ ✇❡ ✇♦r❦ ✇✐t❤ ❛ ♠♦r❡ r✐❣✐❞ ❝❛t❡❣♦r② ♦❢ ❣❡♦♠❡tr✐❝❛❧ ♦❜❥❡❝t✱ similarity surfaces✱ ✇✐t❤ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s ❧✐❦❡✳ ✳ ✳ ✳ ✳ ✳ ❛ ❧♦❝❛❧❧② tr✐✈✐❛❧ ♣❛r❛❧❧❡❧ tr❛♥s♣♦rt✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✽ ✴ ✸✸

The square

❍♦✇ ❞♦ ♦♥❡s ❧✐✈✐♥❣ t❤❡r❡ see t❤❡✐r ✇♦r❧❞❄ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶✾ ✴ ✸✸

Uniformization theorem

❚❤❡♦r❡♠✿ ✭P♦✐♥❝❛ré✱ ❑♦❡❜❡✮ ❆ ❘✐❡♠❛♥♥ s✉r❢❛❝❡ t❤❛t ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ ❛ s♣❤❡r❡ ✐s ♥❡❝❡ss❛r✐❧② ❝♦♥❢♦r♠❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡✳ ■♥ ♦✉r ❝❛s❡✱ ✇❡ ❝❛♥ ❝♦♠♣❧❡t❡ ♦✉r ❣❧✉✐♥❣ ❜② ❛❞❞✐♥❣ ✺ ♣♦✐♥ts✱ ♦♥❡ ❛t ✐♥✜♥✐t②✱ ❢♦✉r ❛t t❤❡ ❝♦r♥❡rs✱ ❛♥❞ ✺ ❘✐❡♠❛♥♥ ❝❤❛rts ♥❡❛r t❤❡s❡ ♣♦✐♥ts✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✵ ✴ ✸✸

Completing the Riemann surface

✶✳ ◆❡❛r ∞✱ t❤❡ ♠❛♣ z → ✶/z ❣✐✈❡s ❛ ❧♦❝❛❧ ❝❤❛rt ✭❡①❛❝t❧② ❧✐❦❡ t❤❡

Riemann sphere✮✳

✷✳ ◆❡❛r ❛ ❝♦r♥❡r✱ ✇❡ ❝❛♥ ❣❧✉❡ ♦♥❡ s✐❞❡ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ t♦ ♦♥❡ s✐❞❡ ♦❢ t❤❡ sq✉❛r❡ ❛♥❞ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧♦❝❛❧ ♣✐❝t✉r❡✿ ❛ s❧✐t ♣❧❛♥❡ ✇❤❡r❡ ♦♥❡ s✐❞❡ ♦❢ t❤❡ s❧✐t ✐s ❣❧✉❡❞ t♦ t❤❡ ♦t❤❡r s✐❞❡ ❜② ❛ ❤♦♠♦t❤❡t② ♦❢ r❛t✐♦ K✳ ❚❤❡♥ t❤❡ ♠❛♣ ✷ ✷ ✐s ❛ ❧♦❝❛❧ ❝❤❛rt✿ ✐♥ ♣❛rt✐❝✉❧❛r ✐t ❣❧✉❡s ❡❛❝❤ s✐❞❡ ♦❢ s❧✐t ❡①❛❝t❧② ❛❝❝♦r❞✐♥❣ t♦ t❤❡ r❡q✉✐r❡❞ ❤♦♠♦t❤❡t②✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✶ ✴ ✸✸

Completing the Riemann surface

✶✳ ◆❡❛r ∞✱ t❤❡ ♠❛♣ z → ✶/z ❣✐✈❡s ❛ ❧♦❝❛❧ ❝❤❛rt ✭❡①❛❝t❧② ❧✐❦❡ t❤❡

Riemann sphere✮✳

✷✳ ◆❡❛r ❛ ❝♦r♥❡r✱ ✇❡ ❝❛♥ ❣❧✉❡ ♦♥❡ s✐❞❡ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ t♦ ♦♥❡ s✐❞❡ ♦❢ t❤❡ sq✉❛r❡ ❛♥❞ ❛r❡ ❧❡❢t ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧♦❝❛❧ ♣✐❝t✉r❡✿ ❛ s❧✐t ♣❧❛♥❡ ✇❤❡r❡ ♦♥❡ s✐❞❡ ♦❢ t❤❡ s❧✐t ✐s ❣❧✉❡❞ t♦ t❤❡ ♦t❤❡r s✐❞❡ ❜② ❛ ❤♦♠♦t❤❡t② ♦❢ r❛t✐♦ K✳ ❚❤❡♥ t❤❡ ♠❛♣ z → zα,

α =

✷πi ✷πi ± logK ✐s ❛ ❧♦❝❛❧ ❝❤❛rt✿ ✐♥ ♣❛rt✐❝✉❧❛r ✐t ❣❧✉❡s ❡❛❝❤ s✐❞❡ ♦❢ s❧✐t ❡①❛❝t❧② ❛❝❝♦r❞✐♥❣ t♦ t❤❡ r❡q✉✐r❡❞ ❤♦♠♦t❤❡t②✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✶ ✴ ✸✸

A cultural remark

M.C. Escher’s lithography: Print Gallery (1956)

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✷ ✴ ✸✸

A solution via uniformization

❚❤❡ ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡ ♠✐♥✉s ♦♥❡ ♣♦✐♥t ✐s ❝♦♥❢♦r♠❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♣❧❛♥❡ ✭st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥✮✳

−→ t❤✐s ❛❧❧♦✇s t♦ ❝r❡❛t❡ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❇❡❧tr❛♠✐ ❡q✉❛t✐♦♥✿

❝♦♥❢♦r♠❛❧ ❣❧✉✐♥❣s

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✸ ✴ ✸✸

A solution via uniformization

❚❤❡ ❊✉❝❧✐❞❡❛♥ s♣❤❡r❡ ♠✐♥✉s ♦♥❡ ♣♦✐♥t ✐s ❝♦♥❢♦r♠❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♣❧❛♥❡ ✭st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥✮✳

−→ t❤✐s ❛❧❧♦✇s t♦ ❝r❡❛t❡ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❇❡❧tr❛♠✐ ❡q✉❛t✐♦♥✿

❝♦♥❢♦r♠❛❧ z → z (x,y) → (x, y K ) ❣❧✉✐♥❣s

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✸ ✴ ✸✸

Explicit uniformization?

❇✉t ✉s✉❛❧❧② ✜♥❞✐♥❣ t❤❡ ❡①♣❧✐❝✐t ✉♥✐❢♦r♠✐③❛t✐♦♥ ♦❢ ❛❜str❛❝t ❘✐❡♠❛♥♥ s✉r❢❛❝❡s ✐s ❛ ✈❡r② ❤❛r❞ ♣r♦❜❧❡♠✱ s♦ ✇❤❛t ❤❡❧♣s ✉s ❤❡r❡❄

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✹ ✴ ✸✸

❚❤❡ ❣❧♦❜❛❧ ❝❤❛rt \ {z✶,...,z✹} ✐s ❛ ❘✐❡♠❛♥♥ ❝❤❛rt ❜✉t ♥♦t ❛ s✐♠✲❝❤❛rt✳ ❚❤❡ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ❢r♦♠ t❤✐s ❝❤❛rt t♦ t❤❡ s✐♠✲❝❤❛rts ❛r❡

holomorphic ❢✉♥❝t✐♦♥s φ : U → ✇✐t❤ U ⊂ \ {z✶,...,z✹}✳ ❋♦r t✇♦ s✉❝❤

s✐♠✲❝❤❛rts✱ φ✶✱ φ✷✱ t❤❡♥ ♦♥ U✶ ∩ U✷ t❤❡② s❛t✐s❢② ✭❧♦❝❛❧❧②✮

φ✶ = aφ✷ + b

❢♦r s♦♠❡ ❝♦♥st❛♥ts a✱ b✳ ❍❡♥❝❡

φ′′

✷

φ′

✷

= φ′′

✶

φ′

✶

.

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✺ ✴ ✸✸

■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ❡①✐sts ❛ ❣❧♦❜❛❧ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥

η : \ {z✶,...,z✹} →

s✉❝❤ t❤❛t s✐♠✲❝❤❛rts ❛r❡ ❡①❛❝t❧② t❤❡ ✭❧♦❝❛❧✮ s♦❧✉t✐♦♥ φ ♦❢

φ′′ φ′ = η.

❋r♦♠ ✱ ♦♥❡ r❡tr✐❡✈❡s ❛s ❢♦❧❧♦✇s✿ ◆♦t❡✿ ✭❞✐✛❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ✈✐❡✇♣♦✐♥t✮ t❤❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❡①♣r❡ss✐♦♥✯ ♦❢ ❛ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ❧♦❝❛❧❧② ✢❛t ✳

✭✯✮ ❛✳❦✳❛✳ ❛ ❈❤r✐st♦✛❡❧ s②♠❜♦❧✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✻ ✴ ✸✸

■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ❡①✐sts ❛ ❣❧♦❜❛❧ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥

η : \ {z✶,...,z✹} →

s✉❝❤ t❤❛t s✐♠✲❝❤❛rts ❛r❡ ❡①❛❝t❧② t❤❡ ✭❧♦❝❛❧✮ s♦❧✉t✐♦♥ φ ♦❢

φ′′ φ′ = η.

❋r♦♠ η✱ ♦♥❡ r❡tr✐❡✈❡s φ ❛s ❢♦❧❧♦✇s✿

φ =

• exp
• η.

◆♦t❡✿ ✭❞✐✛❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ✈✐❡✇♣♦✐♥t✮ t❤❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ ❡①♣r❡ss✐♦♥✯ ♦❢ ❛ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ❧♦❝❛❧❧② ✢❛t ✳

✭✯✮ ❛✳❦✳❛✳ ❛ ❈❤r✐st♦✛❡❧ s②♠❜♦❧✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✻ ✴ ✸✸

■t ❢♦❧❧♦✇s t❤❛t t❤❡r❡ ❡①✐sts ❛ ❣❧♦❜❛❧ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥

η : \ {z✶,...,z✹} →

s✉❝❤ t❤❛t s✐♠✲❝❤❛rts ❛r❡ ❡①❛❝t❧② t❤❡ ✭❧♦❝❛❧✮ s♦❧✉t✐♦♥ φ ♦❢

φ′′ φ′ = η.

❋r♦♠ η✱ ♦♥❡ r❡tr✐❡✈❡s φ ❛s ❢♦❧❧♦✇s✿

φ =

• exp
• η.

◆♦t❡✿ ✭❞✐✛❡r❡♥t✐❛❧ ❣❡♦♠❡tr② ✈✐❡✇♣♦✐♥t✮ t❤❡ ❢✉♥❝t✐♦♥ η ✐s t❤❡ ❡①♣r❡ss✐♦♥✯ ♦❢ ❛ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ❧♦❝❛❧❧② ✢❛t connection✳

✭✯✮ ❛✳❦✳❛✳ ❛ ❈❤r✐st♦✛❡❧ s②♠❜♦❧✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✻ ✴ ✸✸

Analyzing η at the singularities

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡ ❢♦r t❤❡ ❝♦♥♥❡❝t✐♦♥✿ ✐❢ ♦♥❡ ❡①♣r❡ss❡s η ✐♥ t✇♦ ❘✐❡♠❛♥♥ ❝❤❛rts C✶ ❛♥❞ C✷ ✇✐t❤ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ψ ❜❡t✇❡❡♥ t❤❡♠✱ t❤❡♥ t❤❡ ❡①♣r❡ss✐♦♥s η✶ ❛♥❞ η✷ ✐♥ t❤❡ r❡s♣❡❝t✐✈❡ ❝❤❛rts ❛r❡ r❡❧❛t❡❞ ❜②✿

η✷ = ψ′ × η✶◦ψ + ψ′′ ψ′ .

✭✶✮ ✭■t ✐s ❛❧♠♦st ❧✐❦❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✮✳ ❋♦r t❤❡ s❧✐t ♣❧❛♥❡ ♠♦❞❡❧✱ r❡❝❛❧❧ t❤❡ ❣❧✉✐♥❣ ✇✐t❤

✷ ✷

✳ ❚❤❡♥

✶

❤❡♥❝❡

✶

✶✿ ✶

✷ ✶ ❇② ✭✶✮✱

✷ ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t

❛♥❞ ✐ts ♣♦❧❛r ♣❛rt ✐s

✷ ✶ ✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✼ ✴ ✸✸

Analyzing η at the singularities

❈❤❛♥❣❡ ♦❢ ✈❛r✐❛❜❧❡ ❢♦r t❤❡ ❝♦♥♥❡❝t✐♦♥✿ ✐❢ ♦♥❡ ❡①♣r❡ss❡s η ✐♥ t✇♦ ❘✐❡♠❛♥♥ ❝❤❛rts C✶ ❛♥❞ C✷ ✇✐t❤ ❝❤❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s ψ ❜❡t✇❡❡♥ t❤❡♠✱ t❤❡♥ t❤❡ ❡①♣r❡ss✐♦♥s η✶ ❛♥❞ η✷ ✐♥ t❤❡ r❡s♣❡❝t✐✈❡ ❝❤❛rts ❛r❡ r❡❧❛t❡❞ ❜②✿

η✷ = ψ′ × η✶◦ψ + ψ′′ ψ′ .

✭✶✮ ✭■t ✐s ❛❧♠♦st ❧✐❦❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ❢♦r♠✮✳ ❋♦r t❤❡ s❧✐t ♣❧❛♥❡ ♠♦❞❡❧✱ r❡❝❛❧❧ t❤❡ ❣❧✉✐♥❣ z → zα ✇✐t❤ α =

✷πi ✷πi±logK ✳

❚❤❡♥ φ = z✶/α ❤❡♥❝❡ φ′′/φ′ =

✶ α −✶

z ✿

η✶ = logK

✷πi · ✶ z . ❇② ✭✶✮✱ η✷ ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t zi ❛♥❞ ✐ts ♣♦❧❛r ♣❛rt ✐s logK

✷πi · ✶ z−zi ✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✼ ✴ ✸✸

Solution

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✿

• η ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t zk ✇✐t❤ r❡s✐❞✉❡ log(K)/✷πi✳
• η −→ ✵ ✇❤❡♥ z −→ ∞

❛♥❞ r❡❝❛❧❧ η ✐s ❤♦❧♦♠♦r♣❤✐❝ ♦♥ \ {z✶,...,z✹}✳ ❍❡♥❝❡✳ ✳ ✳ ✷ ✶

✶

✶

✷

✶

✸

✶

✹

◆♦✇ s♦❧✈✐♥❣ ❣✐✈❡s✿

✷ ✹ ✶ ✸

✷

❚❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ s♦✉❣❤t ❢♦r ✐s ❧♦❝❛❧❧② t❤❡ ♦❢ ✭❢♦r ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❝♦♥st❛♥ts ✱ ✮✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✽ ✴ ✸✸

Solution

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✿

• η ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t zk ✇✐t❤ r❡s✐❞✉❡ log(K)/✷πi✳
• η −→ ✵ ✇❤❡♥ z −→ ∞

❛♥❞ r❡❝❛❧❧ η ✐s ❤♦❧♦♠♦r♣❤✐❝ ♦♥ \ {z✶,...,z✹}✳ ❍❡♥❝❡✳ ✳ ✳

η = logK

✷πi ·

−✶ z − z✶ +

✶ z − z✷ +

−✶ z − z✸ +

✶ z − z✹

• ◆♦✇ s♦❧✈✐♥❣

❣✐✈❡s✿

✷ ✹ ✶ ✸

✷

❚❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ s♦✉❣❤t ❢♦r ✐s ❧♦❝❛❧❧② t❤❡ ♦❢ ✭❢♦r ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❝♦♥st❛♥ts ✱ ✮✳ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✽ ✴ ✸✸

Solution

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✿

• η ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t zk ✇✐t❤ r❡s✐❞✉❡ log(K)/✷πi✳
• η −→ ✵ ✇❤❡♥ z −→ ∞

❛♥❞ r❡❝❛❧❧ η ✐s ❤♦❧♦♠♦r♣❤✐❝ ♦♥ \ {z✶,...,z✹}✳ ❍❡♥❝❡✳ ✳ ✳

η = logK

✷πi ·

−✶ z − z✶ +

✶ z − z✷ +

−✶ z − z✸ +

✶ z − z✹

• ◆♦✇ s♦❧✈✐♥❣ φ′′/φ′ = η ❣✐✈❡s✿

φ = b + a z − z✷ z − z✹ · z − z✶ z − z✸ logK

✷πi

dz. ❚❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ s♦✉❣❤t ❢♦r ✐s ❧♦❝❛❧❧② t❤❡ ♦❢ ✭❢♦r ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❝♦♥st❛♥ts ✱ ✮✳

❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✽ ✴ ✸✸

Solution

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✿

• η ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t zk ✇✐t❤ r❡s✐❞✉❡ log(K)/✷πi✳
• η −→ ✵ ✇❤❡♥ z −→ ∞

❛♥❞ r❡❝❛❧❧ η ✐s ❤♦❧♦♠♦r♣❤✐❝ ♦♥ \ {z✶,...,z✹}✳ ❍❡♥❝❡✳ ✳ ✳

η = logK

✷πi ·

−✶ z − z✶ +

✶ z − z✷ +

−✶ z − z✸ +

✶ z − z✹

• ◆♦✇ s♦❧✈✐♥❣ φ′′/φ′ = η ❣✐✈❡s✿

φ = b + a z − z✷ z − z✹ · z − z✶ z − z✸ logK

✷πi

dz. ❚❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ s♦✉❣❤t ❢♦r ✐s ❧♦❝❛❧❧② t❤❡ inverse mapping ♦❢ φ ✭❢♦r ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❝♦♥st❛♥ts a✱ b✮✳

❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✽ ✴ ✸✸

Solution

❆s ❛ ❝♦♥s❡q✉❡♥❝❡✿

• η ❤❛s ❛ s✐♠♣❧❡ ♣♦❧❡ ❛t zk ✇✐t❤ r❡s✐❞✉❡ log(K)/✷πi✳
• η −→ ✵ ✇❤❡♥ z −→ ∞

❛♥❞ r❡❝❛❧❧ η ✐s ❤♦❧♦♠♦r♣❤✐❝ ♦♥ \ {z✶,...,z✹}✳ ❍❡♥❝❡✳ ✳ ✳

η = logK

✷πi ·

−✶ z − z✶ +

✶ z − z✷ +

−✶ z − z✸ +

✶ z − z✹

• ◆♦✇ s♦❧✈✐♥❣ φ′′/φ′ = η ❣✐✈❡s✿

φ = b + a z − z✷ z − z✹ · z − z✶ z − z✸ logK

✷πi

dz. ❚❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ s♦✉❣❤t ❢♦r ✐s ❧♦❝❛❧❧② t❤❡ inverse mapping ♦❢ φ ✭❢♦r ❛♣♣r♦♣r✐❛t❡ ❝❤♦✐❝❡s ♦❢ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❝♦♥st❛♥ts a✱ b✮✳

❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✽ ✴ ✸✸

The Schwarz-Christoffel formula

❚❤❡ ❢♦r♠✉❧❛ ✇❡ ❢♦✉♥❞ a + b

z − z✷ z − z✹ · z − z✶ z − z✸ logK

✷πi

dz ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛ t❤❛t ❣✐✈❡s ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ ❢r♦♠ t❤❡ ✉♣♣❡r ❤❛❧❢ ♣❧❛♥❡ t♦ ❛♥② ♣♦❧②❣♦♥ ✐♥ t❤❡ ♣❧❛♥❡✿ ❢♦r ❛♥ n✲❣♦♥ ✇✐t❤ ❛♥❣❧❡s αk ∈ (✵,✷π)✱ t❤❡r❡ ❡①✐sts r❡❛❧ ♥✉♠❜❡rs x✶✱✳ ✳ ✳ ✱xn s✉❝❤ t❤❛t f = a + b

• dz

(z − x✶)β✶ ···(z − xn)βn ✇✐t❤ βk = ✶ − αk

π ✳

❚❤❡ ❛r❡ ♠❛♣♣❡❞ t♦ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ♣♦❧②❣♦♥✳ ❚❤❡② ❝❛♥ ❜❡ ❤❛r❞ t♦ ❞❡t❡r♠✐♥❡✿ ❡❛❝❤ ❞❡♣❡♥❞s ♦♥ ❛❧❧ t❤❡ ❛♥❣❧❡s ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ ❛❧❧ s✐❞❡s ♦❢ t❤❡ ♣♦❧②❣♦♥✳ ❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ ✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✾ ✴ ✸✸

The Schwarz-Christoffel formula

❚❤❡ ❢♦r♠✉❧❛ ✇❡ ❢♦✉♥❞ a + b

z − z✷ z − z✹ · z − z✶ z − z✸ logK

✷πi

dz ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛ t❤❛t ❣✐✈❡s ❛♥ ❡①♣r❡ss✐♦♥ ♦❢ t❤❡ ❝♦♥❢♦r♠❛❧ ♠❛♣ ❢r♦♠ t❤❡ ✉♣♣❡r ❤❛❧❢ ♣❧❛♥❡ t♦ ❛♥② ♣♦❧②❣♦♥ ✐♥ t❤❡ ♣❧❛♥❡✿ ❢♦r ❛♥ n✲❣♦♥ ✇✐t❤ ❛♥❣❧❡s αk ∈ (✵,✷π)✱ t❤❡r❡ ❡①✐sts r❡❛❧ ♥✉♠❜❡rs x✶✱✳ ✳ ✳ ✱xn s✉❝❤ t❤❛t f = a + b

• dz

(z − x✶)β✶ ···(z − xn)βn ✇✐t❤ βk = ✶ − αk

π ✳

❚❤❡ xi ❛r❡ ♠❛♣♣❡❞ t♦ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ♣♦❧②❣♦♥✳ ❚❤❡② ❝❛♥ ❜❡ ❤❛r❞ t♦ ❞❡t❡r♠✐♥❡✿ ❡❛❝❤ ❞❡♣❡♥❞s ♦♥ ❛❧❧ t❤❡ ❛♥❣❧❡s ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ ❛❧❧ s✐❞❡s ♦❢ t❤❡ ♣♦❧②❣♦♥✳ ❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ parameter problem✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷✾ ✴ ✸✸

Parameter problem

❙✐♠✐❧❛r❧② ✐♥ ♦✉r q✉❡st✐♦♥ ✇❡ ❢❛❝❡ ❛ ♣❛r❛♠❡t❡r ♣r♦❜❧❡♠✿ ✜♥❞✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ z✶✱✳ ✳ ✳ ✱z✹✳ ❯s✐♥❣ t❤❡ s②♠♠❡tr✐❡s✱ t❤✐s r❡❞✉❝❡s t♦ ✜♥❞✐♥❣ t❤❡ s❤❛♣❡ r❛t✐♦ K ′ ♦❢ t❤❡ r❡❝t❛♥❣❧❡ z✶✱✳ ✳ ✳ ✱z✹ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ K✳ ✭◆♦t❡✿ K ′ = K✳✮ ❲❡ t❤✉s ❤❛✈❡ ❛ ♦♥❡ ✭r❡❛❧✮ ♣❛r❛♠❡t❡r ❡q✉❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ K ′✿ ✭❊✮

• [z✶,z✷] ω
• [z✹,z✶] ω

= iK. ✇✐t❤ ω = z−z✷

z−z✹ · z−z✶ z−z✸

logK

✷πi dz✳

■ r❡s♦rt❡❞ t♦ s♦❧✈❡ ✭❊✮ numerically ❢♦r ❡❛❝❤ ❡①♣❧✐❝✐t ✈❛❧✉❡ ♦❢ K ✭t❤✐s ✐s ♥♦t t♦♦ ❤❛r❞✮✳

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✵ ✴ ✸✸

K = ✷

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✺

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✺

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✺✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✷✵✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✵✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✹

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✻

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✾

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✷✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✺✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

K = ✶✵✺✵

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✶ ✴ ✸✸

The limit

❆s K −→ +∞ ✇❡ s❡❡ ❛ ❧✐♠✐t s❤❛♣❡ ❛♥❞ ❝❛♥ ♣r♦✈❡

ηK −→ η∞ = σ✵ (z − x✵)✷ − σ✵ (z + x✵)✷

❚❤✐s ❧✐♠✐t s❤❛♣❡ ❛❧s♦ ❤❛s ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ✐♥ t❡r♠s ♦❢ s✐♠✐❧❛r✐t② s✉r❢❛❝❡s✿ ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t ❈❧✐❝❦ ❤❡r❡ t♦ r✉♥ ❛♣♣❧❡t

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✷ ✴ ✸✸

The limit

❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸✸ ✴ ✸✸