Introduction to Riemann Surface

Introduction to Riemann Surface

Noh Seong Cheol

20106916

Physical Mathematics I Conference12 November 2010

Introduction to Riemann Surface

Contents

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Introduction

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Introduction

Question

Question

z()

4pi

2pi

0

2pi

4pi

f (z)

log z ...

How to make multi-valued functionto single-valued function?

Introduction to Riemann Surface

Introduction

Question

Question

z()

4pi

2pi

0

2pi

4pi

f (z)

log z ...

How to make multi-valued functionto single-valued function?

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

I Generalization of the complex plane consisting of more than onesheet.

I Once a Riemann surface is devised for a given function,

I the function is single-valued on the surface

I and the theory of single-valued functions applies there.

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

I Generalization of the complex plane consisting of more than onesheet.

I Once a Riemann surface is devised for a given function,

I the function is single-valued on the surface

I and the theory of single-valued functions applies there.

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

I Generalization of the complex plane consisting of more than onesheet.

I Once a Riemann surface is devised for a given function,

I the function is single-valued on the surface

I and the theory of single-valued functions applies there.

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

I Generalization of the complex plane consisting of more than onesheet.

I Once a Riemann surface is devised for a given function,

I the function is single-valued on the surface

I and the theory of single-valued functions applies there.

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

The main purpose is

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

The main purpose is

Multi-valued function Single-valued function

Introduction to Riemann Surface

Introduction

Definition of a Riemann surface

Definition of a Riemann surface

The main purpose is

Multi-valued function Single-valued function

Introduction to Riemann Surface

Construct Riemann Surface

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

z re i

log z = log r + i

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

z re i

log z = log r + i

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

log z = log r + i

log z is multi-valued function (n 1)

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut

cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut

cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut

cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

branch point

branch cut

cut

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

Im z

Re zO

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Im z

Re zO

R0

R1

R1

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

R0

R1

R1

R1 : 2pi < < 4pi

R0 : 0 < < 2pi

R1 : 2pi < < 0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

R0

R1

R1

R1 : 2pi < < 4pi

R0 : 0 < < 2pi

R1 : 2pi < < 0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

R0

R1

R1

R1 : 2pi < < 4pi

R0 : 0 < < 2pi

R1 : 2pi < < 0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

For Rn,

Rn : 2pin < < 2pi(n + 1)

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

For Rn,

Rn : 2pin < < 2pi(n + 1)

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

f (z) = log z

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

log z

Multi-valued function

Single-valued function

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

log z

Multi-valued function

Single-valued function

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = log z

log z

Multi-valued function

Single-valued function

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

z re i

z1/2 =re i/2 =

r [cos(/2) + i sin(/2)]

Period: 4pi

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

z re i

z1/2 =re i/2 =

r [cos(/2) + i sin(/2)]

Period: 4pi

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

z re i

z1/2 =re i/2 =

r [cos(/2) + i sin(/2)]

Period: 4pi

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

R0

R1

R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

R0

R1

R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

R0

R1

R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

R0

R1

R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/n

For z1n , (n: natural number)

z1n = nr cos n + i sin

n

,where period: 2npi apply same method!

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/n

For z1n , (n: natural number)

z1n = nr cos n + i sin

n

,where period: 2npi apply same method!

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/n

For z1n , (n: natural number)

z1n = nr cos n + i sin

n

,where period: 2npi

apply same method!

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = z1/2

f (z) = z1/n

For z1n , (n: natural number)

z1n = nr cos n + i sin

n

,where period: 2npi apply same method!

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

For f (z) = (z2 1)1/2

=r1r2e

i(1+2)/2,

{z 1 = r1e i1z + 1 = r2e

i2

1 1

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

For f (z) = (z2 1)1/2

=r1r2e

i(1+2)/2,{z 1 = r1e i1z + 1 = r2e

i2

1 1

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

For f (z) = (z2 1)1/2

=r1r2e

i(1+2)/2,{z 1 = r1e i1z + 1 = r2e

i2

1 1

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

For f (z) = (z2 1)1/2

=r1r2e

i(1+2)/2,{z 1 = r1e i1z + 1 = r2e

i2

1 1

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

For f (z) = (z2 1)1/2

=r1r2e

i(1+2)/2,{z 1 = r1e i1z + 1 = r2e

i2

1 1

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

f (z) =r1r2e

i(1+2)/2

1. closed curve encloses segmentP1P2

1 changes 0 to 2pi2 changes 0 to 2pi

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi 2piR : R0 R0 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

f (z) =r1r2e

i(1+2)/2

1. closed curve encloses segmentP1P2

1 changes 0 to 2pi2 changes 0 to 2pi

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi 2piR : R0 R0 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

f (z) =r1r2e

i(1+2)/2

1. closed curve encloses segmentP1P2

1 changes 0 to 2pi2 changes 0 to 2pi

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi 2piR : R0 R0 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

f (z) =r1r2e

i(1+2)/2

1. closed curve encloses segmentP1P2

1 changes 0 to 2pi2 changes 0 to 2pi

= (1 + 2)/2 changes 0 to 2pi

{ : 0 pi 2piR : R0 R0 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

f (z) =r1r2e

i(1+2)/2

1. closed curve encloses segmentP1P2

1 changes 0 to 2pi2 changes 0 to 2pi

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi 2piR : R0 R0 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

slit

f (z) =r1r2e

i(1+2)/2

2. closed curve encloses only P1(P2)

1(2) changes 0 to 4pi2(1) changes 0 to 0

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi(P1P2) 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

slit

f (z) =r1r2e

i(1+2)/2

2. closed curve encloses only P1(P2)

1(2) changes 0 to 4pi2(1) changes 0 to 0

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi(P1P2) 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

slit

f (z) =r1r2e

i(1+2)/2

2. closed curve encloses only P1(P2)

1(2) changes 0 to 4pi2(1) changes 0 to 0

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi(P1P2) 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2

slit

f (z) =r1r2e

i(1+2)/2

2. closed curve encloses only P1(P2)

1(2) changes 0 to 4pi2(1) changes 0 to 0

= (1 + 2)/2 changes 0 to 2pi

{ : 0 pi(P1P2) 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

P2(1, 0) P1(1, 0)

z

1

r1

2

r2slit

f (z) =r1r2e

i(1+2)/2

2. closed curve encloses only P1(P2)

1(2) changes 0 to 4pi2(1) changes 0 to 0

= (1 + 2)/2 changes 0 to 2pi{ : 0 pi(P1P2) 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = (z2 1)1/2

f (z) = (z2 1)1/2

The double-valued function f (z) = (z2 1)1/2 is considered as asingle-valued function of the points on the Riemann surface composed ofR0 and R1 just constructed.

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

For f (z) = [z(z2 1)]1/2

=rr1r2e

i(+1+2)/2,

z = rei

z 1 = r1e i1z + 1 = r2e

i2

1 1

r

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

For f (z) = [z(z2 1)]1/2

=rr1r2e

i(+1+2)/2, z = rei

z 1 = r1e i1z + 1 = r2e

i2

1 1

r

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

For f (z) = [z(z2 1)]1/2

=rr1r2e

i(+1+2)/2, z = rei

z 1 = r1e i1z + 1 = r2e

i2

1 1

r

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

For f (z) = [z(z2 1)]1/2

=rr1r2e

i(+1+2)/2, z = rei

z 1 = r1e i1z + 1 = r2e

i2

1 1

r

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

For f (z) = [z(z2 1)]1/2

=rr1r2e

i(+1+2)/2, z = rei

z 1 = r1e i1z + 1 = r2e

i2

1 1

r

z

1

r1

2

r2

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L1

f (z) =rr1r2e

i(+1+2)/2

1. closed curve encloses only P1

changes 0 to 4pi1 changes 0 to 02 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 2pi 4piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L1

f (z) =rr1r2e

i(+1+2)/2

1. closed curve encloses only P1

changes 0 to 4pi1 changes 0 to 02 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 2pi 4piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L1

f (z) =rr1r2e

i(+1+2)/2

1. closed curve encloses only P1

changes 0 to 4pi1 changes 0 to 02 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 2pi 4piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L1

f (z) =rr1r2e

i(+1+2)/2

1. closed curve encloses only P1

changes 0 to 4pi1 changes 0 to 02 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi

{ : 0 2pi 4piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L1

f (z) =rr1r2e

i(+1+2)/2

1. closed curve encloses only P1

changes 0 to 4pi1 changes 0 to 02 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 2pi 4piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

P3 P1P2

z

z

z

( + + )/2 > 2pi

L2

f (z) =rr1r2e

i(+1+2)/2

2. closed curve encloses only P2

changes 0 to 2pi1 changes 0 to 2pi2 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 ( + + )/2 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

P3 P1P2

z

z

z

( + + )/2 > 2pi

L2

f (z) =rr1r2e

i(+1+2)/2

2. closed curve encloses only P2

changes 0 to 2pi1 changes 0 to 2pi2 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 ( + + )/2 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

P3 P1P2

z

z

z

( + + )/2 > 2pi

L2

f (z) =rr1r2e

i(+1+2)/2

2. closed curve encloses only P2

changes 0 to 2pi1 changes 0 to 2pi2 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 ( + + )/2 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

P3 P1P2

z

z

z

( + + )/2 > 2pi

L2

f (z) =rr1r2e

i(+1+2)/2

2. closed curve encloses only P2

changes 0 to 2pi1 changes 0 to 2pi2 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi

{ : 0 ( + + )/2 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

P3 P1P2

z

z

z

( + + )/2 > 2pi

L2

f (z) =rr1r2e

i(+1+2)/2

2. closed curve encloses only P2

changes 0 to 2pi1 changes 0 to 2pi2 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi

{ : 0 ( + + )/2 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

P3 P1P2

z

z

z

( + + )/2 > 2pi

L2

f (z) =rr1r2e

i(+1+2)/2

2. closed curve encloses only P2

changes 0 to 2pi1 changes 0 to 2pi2 changes 0 to 0

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 ( + + )/2 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L2

f (z) =rr1r2e

i(+1+2)/2

3. closed curve encloses only P3

changes 0 to 01 changes 0 to 02 changes 0 to 4pi

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 pi 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L2

f (z) =rr1r2e

i(+1+2)/2

3. closed curve encloses only P3

changes 0 to 01 changes 0 to 02 changes 0 to 4pi

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 pi 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L2

f (z) =rr1r2e

i(+1+2)/2

3. closed curve encloses only P3

changes 0 to 01 changes 0 to 02 changes 0 to 4pi

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 pi 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L2

f (z) =rr1r2e

i(+1+2)/2

3. closed curve encloses only P3

changes 0 to 01 changes 0 to 02 changes 0 to 4pi

= (+ 1 + 2)/2 changes 0 to2pi

{ : 0 pi 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

P3 P1P2

z

r1r2r

12

L2

f (z) =rr1r2e

i(+1+2)/2

3. closed curve encloses only P3

changes 0 to 01 changes 0 to 02 changes 0 to 4pi

= (+ 1 + 2)/2 changes 0 to2pi{ : 0 pi 2piR : R0 R1 R0

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

f (z) = [z(z2 1)]1/2

The double-valued function f (z) = [z(z2 1)]1/2 is considered as asingle-valued function of the points on the Riemann surface composed ofR0 and R1 just constructed.

Introduction to Riemann Surface

Construct Riemann Surface

f (z) = [z(z2 1)]1/2

Examples

z1/3 z1/4 arcsin z

Introduction to Riemann Surface

Conclusion

Introduction to Riemann Surface

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef (z) = log zf (z) = z1/2

f (z) = (z2 1)1/2f (z) = [z(z2 1)]1/2

Conclusion

Introduction to Riemann Surface

Conclusion

Conclusion

I Riemann surfaces can be thought of as deformed versions of thecomplex plane.

I Every Riemann surface is a two-dimensional real analytic surface,but it contains more complex structure.

I Riemann surfaces make given multiple-valued functions tosingle-valued functions.

Introduction to Riemann Surface

Conclusion

Conclusion

I Riemann surfaces can be thought of as deformed versions of thecomplex plane.

I Every Riemann surface is a two-dimensional real analytic surface,but it contains more complex structure.

I Riemann surfaces make given multiple-valued functions tosingle-valued functions.

Introduction to Riemann Surface

Conclusion

Conclusion

I Riemann surfaces can be thought of as deformed versions of thecomplex plane.

I Every Riemann surface is a two-dimensional real analytic surface,but it contains more complex structure.

I Riemann surfaces make given multiple-valued functions tosingle-valued functions.

Introduction to Riemann Surface

Thank you for your attention

THANK YOU FOR YOUR ATTENTION!

IntroductionQuestionDefinition of a Riemann surface

Construct Riemann Surfacef(z)=logzf(z)=z1/2f(z)=(z2-1)1/2f(z)=[z(z2-1)]1/2

Conclusion