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Review of complex numbers 01+02 1=R₁e Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. CB-2/8 R Caz-Re10+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 Iz zz Re Re = R² e = R2 z=√√zz √√a2+ b² = R. Problem 1: Application of the Maximum Modulus Principle and Liouville's Theorem Statement: Let f : C→ C be an entire function (holomorphic on the entire complex plane). 1. Bounded Entire Functions: ⚫ Theorem (Liouville): Prove that if f is entire and bounded (ie., there exists a constant M>0 such that f(=) 0 and integer n >0. Prove that f must be a polynomial of degree at most n 3. Entire Functions with Specific Growth Rates: Let f be an entire function such that f() < Cealal for some constants C, a > 0. Discuss the possible forms off and relate your findings to the classification of entire functions based on their order. 4. Application of the Maximum Modulus Principle: etf: D-C be ⚫ Let DCC be a bounded domain with a smooth boundary, and let f: holomorphic on D and continuous on D. Prove that the maximum of f(z) on Dis attained on the boundary ǝD. Requirements for Solution: ⚫ Utilize Liouville's Theorem and understand its implications for entire functions.

Review of complex numbers 01+02 1=R₁e Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. CB-2/8 R Caz-Re10+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 Iz zz Re Re = R² e = R2 z=√√zz √√a2+ b² = R. Problem 1: Application of the Maximum Modulus Principle and Liouville's Theorem Statement: Let f : C→ C be an entire function (holomorphic on the entire complex plane). 1. Bounded Entire Functions: ⚫ Theorem (Liouville): Prove that if f is entire and bounded (ie., there exists a constant M>0 such that f(=) 0 and integer n >0. Prove that f must be a polynomial of degree at most n 3. Entire Functions with Specific Growth Rates: Let f be an entire function such that f() < Cealal for some constants C, a > 0. Discuss the possible forms off and relate your findings to the classification of entire functions based on their order. 4. Application of the Maximum Modulus Principle: etf: D-C be ⚫ Let DCC be a bounded domain with a smooth boundary, and let f: holomorphic on D and continuous on D. Prove that the maximum of f(z) on Dis attained on the boundary ǝD. Requirements for Solution: ⚫ Utilize Liouville's Theorem and understand its implications for entire functions.

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 01+02 1=R₁e Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. CB-2/8 R Caz-Re10+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 Iz zz Re Re = R² e = R2 z=√√zz √√a2+ b² = R. Problem 1: Application of the Maximum Modulus Principle and Liouville's Theorem Statement: Let f : C→ C be an entire function (holomorphic on the entire complex plane). 1. Bounded Entire Functions: ⚫ Theorem (Liouville): Prove that if f is entire and bounded (ie., there exists a constant M>0 such that f(=) <M for all C), then f is constant. 2. Polynomial Growth: Suppose f is entire and satisfies |f(z)| <Al" + B for some constants A, B > 0 and integer n >0. Prove that f must be a polynomial of degree at most n 3. Entire Functions with Specific Growth Rates: Let f be an entire function such that f() < Cealal for some constants C, a > 0. Discuss the possible forms off and relate your findings to the classification of entire functions based on their order. 4. Application of the Maximum Modulus Principle: etf: D-C be ⚫ Let DCC be a bounded domain with a smooth boundary, and let f: holomorphic on D and continuous on D. Prove that the maximum of f(z) on Dis attained on the boundary ǝD. Requirements for Solution: ⚫ Utilize Liouville's Theorem and understand its implications for entire functions.
Transcribed Image Text:Review of complex numbers 01+02 1=R₁e Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. CB-2/8 R Caz-Re10+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 Iz zz Re Re = R² e = R2 z=√√zz √√a2+ b² = R. Problem 1: Application of the Maximum Modulus Principle and Liouville's Theorem Statement: Let f : C→ C be an entire function (holomorphic on the entire complex plane). 1. Bounded Entire Functions: ⚫ Theorem (Liouville): Prove that if f is entire and bounded (ie., there exists a constant M>0 such that f(=) <M for all C), then f is constant. 2. Polynomial Growth: Suppose f is entire and satisfies |f(z)| <Al" + B for some constants A, B > 0 and integer n >0. Prove that f must be a polynomial of degree at most n 3. Entire Functions with Specific Growth Rates: Let f be an entire function such that f() < Cealal for some constants C, a > 0. Discuss the possible forms off and relate your findings to the classification of entire functions based on their order. 4. Application of the Maximum Modulus Principle: etf: D-C be ⚫ Let DCC be a bounded domain with a smooth boundary, and let f: holomorphic on D and continuous on D. Prove that the maximum of f(z) on Dis attained on the boundary ǝD. Requirements for Solution: ⚫ Utilize Liouville's Theorem and understand its implications for entire functions.
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