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A powerpoint presentation by Noh Seong Cheol at the physical mathematics conference 2010 constructing some elementary Riemann surfaces.
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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