Review of complex numbers 1=Re -R₂e z=Re 2122 R Re₂) Caz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z=Rea-bi, which is the reflection of z across the real axis. Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8=2/3 Note that Iz zz Re Re = R2e0 = R2 z= √√zz √√a²+ b² = R. Statement: Consider the complex function f(2) = √z, defined initially on C\(-0,0], where the branch cut is taken along the negative real axis. 1. Analytic Continuation: ⚫ Construct all possible analytic continuations of f(z) around the origin by traversing paths in C\{0}. Describe the resulting Riemann surface of √. 2. Monodromy Representation: Define the monodromy group associated with the function and the covering map: C\(-0,0] C\{0}). Show that the monodromy group is isomorphic to Z/2Z. 3. Branch Points and Cuts: Identify all branch points of f(x) and discuss how different choices of branch cuts affect the analytic continuation of f(). 4. Extension to Multi-Valued Functions: Generalize the analysis to the function f(z) = 21/n for an integer n > 1. Determine the structure of the monodromy group and describe the corresponding Riemann surface. Requirements for Solution: ⚫ Apply the concept of analytic continuation along different paths. . Understand and describe Riemann surfaces for multi-valued functions.
Review of complex numbers 1=Re -R₂e z=Re 2122 R Re₂) Caz-Re(+2x/8) The complex conjugate of z = Rei=a+bi is z=Rea-bi, which is the reflection of z across the real axis. Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8=2/3 Note that Iz zz Re Re = R2e0 = R2 z= √√zz √√a²+ b² = R. Statement: Consider the complex function f(2) = √z, defined initially on C\(-0,0], where the branch cut is taken along the negative real axis. 1. Analytic Continuation: ⚫ Construct all possible analytic continuations of f(z) around the origin by traversing paths in C\{0}. Describe the resulting Riemann surface of √. 2. Monodromy Representation: Define the monodromy group associated with the function and the covering map: C\(-0,0] C\{0}). Show that the monodromy group is isomorphic to Z/2Z. 3. Branch Points and Cuts: Identify all branch points of f(x) and discuss how different choices of branch cuts affect the analytic continuation of f(). 4. Extension to Multi-Valued Functions: Generalize the analysis to the function f(z) = 21/n for an integer n > 1. Determine the structure of the monodromy group and describe the corresponding Riemann surface. Requirements for Solution: ⚫ Apply the concept of analytic continuation along different paths. . Understand and describe Riemann surfaces for multi-valued functions.
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.3: The Complex Plane
Problem 2ECP
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![Review of complex numbers
1=Re
-R₂e
z=Re
2122 R Re₂)
Caz-Re(+2x/8)
The complex conjugate of z = Rei=a+bi is
z=Rea-bi,
which is the reflection of z across the real axis.
Do not use AI, I need real solution, attach required graph and code wherever needed.
For reference I have attached the image, but if you need any reference then check out the book by
Churchill only.
8=2/3
Note that
Iz zz Re Re = R2e0 = R2
z= √√zz √√a²+ b² = R.
Statement:
Consider the complex function f(2) = √z, defined initially on C\(-0,0], where the branch cut
is taken along the negative real axis.
1. Analytic Continuation:
⚫ Construct all possible analytic continuations of f(z) around the origin by traversing paths
in C\{0}. Describe the resulting Riemann surface of √.
2. Monodromy Representation:
Define the monodromy group associated with the function and the covering map:
C\(-0,0] C\{0}). Show that the monodromy group is isomorphic to Z/2Z.
3. Branch Points and Cuts:
Identify all branch points of f(x) and discuss how different choices of branch cuts affect
the analytic continuation of f().
4. Extension to Multi-Valued Functions:
Generalize the analysis to the function f(z) = 21/n for an integer n > 1. Determine the
structure of the monodromy group and describe the corresponding Riemann surface.
Requirements for Solution:
⚫ Apply the concept of analytic continuation along different paths.
.
Understand and describe Riemann surfaces for multi-valued functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46aab64c-4858-4efc-94a9-de505c329d9b%2F8a0a82ca-6df3-4aab-8dda-34e0408229da%2F38wf56_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Review of complex numbers
1=Re
-R₂e
z=Re
2122 R Re₂)
Caz-Re(+2x/8)
The complex conjugate of z = Rei=a+bi is
z=Rea-bi,
which is the reflection of z across the real axis.
Do not use AI, I need real solution, attach required graph and code wherever needed.
For reference I have attached the image, but if you need any reference then check out the book by
Churchill only.
8=2/3
Note that
Iz zz Re Re = R2e0 = R2
z= √√zz √√a²+ b² = R.
Statement:
Consider the complex function f(2) = √z, defined initially on C\(-0,0], where the branch cut
is taken along the negative real axis.
1. Analytic Continuation:
⚫ Construct all possible analytic continuations of f(z) around the origin by traversing paths
in C\{0}. Describe the resulting Riemann surface of √.
2. Monodromy Representation:
Define the monodromy group associated with the function and the covering map:
C\(-0,0] C\{0}). Show that the monodromy group is isomorphic to Z/2Z.
3. Branch Points and Cuts:
Identify all branch points of f(x) and discuss how different choices of branch cuts affect
the analytic continuation of f().
4. Extension to Multi-Valued Functions:
Generalize the analysis to the function f(z) = 21/n for an integer n > 1. Determine the
structure of the monodromy group and describe the corresponding Riemann surface.
Requirements for Solution:
⚫ Apply the concept of analytic continuation along different paths.
.
Understand and describe Riemann surfaces for multi-valued functions.
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