Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN: 9781337278461
Author: Ron Larson
Publisher: Cengage Learning
Bartleby Related Questions Icon

Related questions

Question
Review of complex numbers
1=R₁e
z=Re
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.
Ca-e2/8
22=
R₂e
2122=R₁ Re(+₂)
01+02
R
Saz-Re40+2x/8)
The complex conjugate of z = Rei = a + bi is
z= Rea-bi.
which is the reflection of z across the real axis.
Note that
Iz²=z z=Re Re-i = R2e0 = R²
=>
Consider a meromorphic function f : C+C (the Riemann sphere), where C = CU{0}.
1. Weierstrass Factorization Theorem:
State and prove the Weierstrass Factorization Theorem. Then, use it to express an entire
function f(x) with prescribed zeros {a} (without accumulation points in C) as a product
involving its zeros.
2. Picard's Theorem:
Prove the Great Picard Theorem: If an analytic function f has an essential singularity at a
point zo, then in any neighborhood of zu, f takes on every complex value, with at most a
single exception, infinitely often.
3. Nevanlinna Theory Fundamentals:
Introduce the Nevanlinna characteristic function T(r, f) for a meromorphic function f.
Prove the First Fundamental Theorem of Nevanlinna Theory, which relates T(r. f), the
counting function N(r,a), and the proximity function m(r, a) for a value a € C.
4. Second Fundamental Theorem and Defect Relations:
State the Second Fundamental Theorem of Nevanlinna Theory. Use it to derive defect
relations, demonstrating that for a non-constant meromorphic function, the sum of defects
for all distinct values in C is at most 2.
5. Applications to Value Distribution:
Apply Nevanlinna Theory to prove that any non-constant entire function f omitting more
than one value in Ĉ must be constant, thereby providing an alternative proof of the Little
Picard Theorem.
expand button
Transcribed Image Text:Review of complex numbers 1=R₁e z=Re 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. Ca-e2/8 22= R₂e 2122=R₁ Re(+₂) 01+02 R Saz-Re40+2x/8) The complex conjugate of z = Rei = a + bi is z= Rea-bi. which is the reflection of z across the real axis. Note that Iz²=z z=Re Re-i = R2e0 = R² => Consider a meromorphic function f : C+C (the Riemann sphere), where C = CU{0}. 1. Weierstrass Factorization Theorem: State and prove the Weierstrass Factorization Theorem. Then, use it to express an entire function f(x) with prescribed zeros {a} (without accumulation points in C) as a product involving its zeros. 2. Picard's Theorem: Prove the Great Picard Theorem: If an analytic function f has an essential singularity at a point zo, then in any neighborhood of zu, f takes on every complex value, with at most a single exception, infinitely often. 3. Nevanlinna Theory Fundamentals: Introduce the Nevanlinna characteristic function T(r, f) for a meromorphic function f. Prove the First Fundamental Theorem of Nevanlinna Theory, which relates T(r. f), the counting function N(r,a), and the proximity function m(r, a) for a value a € C. 4. Second Fundamental Theorem and Defect Relations: State the Second Fundamental Theorem of Nevanlinna Theory. Use it to derive defect relations, demonstrating that for a non-constant meromorphic function, the sum of defects for all distinct values in C is at most 2. 5. Applications to Value Distribution: Apply Nevanlinna Theory to prove that any non-constant entire function f omitting more than one value in Ĉ must be constant, thereby providing an alternative proof of the Little Picard Theorem.
Expert Solution
Check Mark
Knowledge Booster
Background pattern image
Similar questions
Recommended textbooks for you
Text book image
Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781337278461
Author:Ron Larson
Publisher:Cengage Learning
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Text book image
Intermediate Algebra
Algebra
ISBN:9781285195728
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Cengage Learning