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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.

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.

Trigonometry (MindTap Course List)
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=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.
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.
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