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Review of complex numbers 1=R₁e 22-R₂e 2122=R₁ Re(+82). z=Re 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. 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. Note that Iz zz Re Re = R²e = R² z=√√zz=√√a²+ b² = R. Consider a harmonic function u: D→ R. where DCC is an open connected domain. 1. Harmonic Conjugates: a. Prove that if u is harmonic on D and simply connected, then there exists a harmonic conjugate such that fu+ie is holomorphic on D. ⚫b. Provide an explicit construction of given u in a specific example, such as u(x, y) = 22-y². 2. Maximum Principle for Harmonic Functions: ⚫a. State and prove the Maximum Principle for harmonic functions, asserting that if u attains its maximum (or minimum) value in the interior of D, then u is constant. b. Apply the Maximum Principle to show that a non-constant harmonic function cannot attain its maximum value inside a bounded domain. 3. Dirichlet Problem: ⚫a. Formulate the Dirichlet problem for finding a harmonic function on a bounded domain D with prescribed continuous boundary values. ⚫b. Prove the existence and uniqueness of the solution to the Dirichlet problem using the method of harmonic conjugates or Perron's method. 4. Green's Function: ⚫a. Define Green's function G(z, C) for the domain D with a pole at C € D. ⚫ b. Derive the representation of a harmonic function u on D in terms of its boundary values and Green's function.

Review of complex numbers 1=R₁e 22-R₂e 2122=R₁ Re(+82). z=Re 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. 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. Note that Iz zz Re Re = R²e = R² z=√√zz=√√a²+ b² = R. Consider a harmonic function u: D→ R. where DCC is an open connected domain. 1. Harmonic Conjugates: a. Prove that if u is harmonic on D and simply connected, then there exists a harmonic conjugate such that fu+ie is holomorphic on D. ⚫b. Provide an explicit construction of given u in a specific example, such as u(x, y) = 22-y². 2. Maximum Principle for Harmonic Functions: ⚫a. State and prove the Maximum Principle for harmonic functions, asserting that if u attains its maximum (or minimum) value in the interior of D, then u is constant. b. Apply the Maximum Principle to show that a non-constant harmonic function cannot attain its maximum value inside a bounded domain. 3. Dirichlet Problem: ⚫a. Formulate the Dirichlet problem for finding a harmonic function on a bounded domain D with prescribed continuous boundary values. ⚫b. Prove the existence and uniqueness of the solution to the Dirichlet problem using the method of harmonic conjugates or Perron's method. 4. Green's Function: ⚫a. Define Green's function G(z, C) for the domain D with a pole at C € D. ⚫ b. Derive the representation of a harmonic function u on D in terms of its boundary values and Green's function.

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 22-R₂e 2122=R₁ Re(+82). z=Re 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. 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. Note that Iz zz Re Re = R²e = R² z=√√zz=√√a²+ b² = R. Consider a harmonic function u: D→ R. where DCC is an open connected domain. 1. Harmonic Conjugates: a. Prove that if u is harmonic on D and simply connected, then there exists a harmonic conjugate such that fu+ie is holomorphic on D. ⚫b. Provide an explicit construction of given u in a specific example, such as u(x, y) = 22-y². 2. Maximum Principle for Harmonic Functions: ⚫a. State and prove the Maximum Principle for harmonic functions, asserting that if u attains its maximum (or minimum) value in the interior of D, then u is constant. b. Apply the Maximum Principle to show that a non-constant harmonic function cannot attain its maximum value inside a bounded domain. 3. Dirichlet Problem: ⚫a. Formulate the Dirichlet problem for finding a harmonic function on a bounded domain D with prescribed continuous boundary values. ⚫b. Prove the existence and uniqueness of the solution to the Dirichlet problem using the method of harmonic conjugates or Perron's method. 4. Green's Function: ⚫a. Define Green's function G(z, C) for the domain D with a pole at C € D. ⚫ b. Derive the representation of a harmonic function u on D in terms of its boundary values and Green's function.
Transcribed Image Text:Review of complex numbers 1=R₁e 22-R₂e 2122=R₁ Re(+82). z=Re 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. 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. Note that Iz zz Re Re = R²e = R² z=√√zz=√√a²+ b² = R. Consider a harmonic function u: D→ R. where DCC is an open connected domain. 1. Harmonic Conjugates: a. Prove that if u is harmonic on D and simply connected, then there exists a harmonic conjugate such that fu+ie is holomorphic on D. ⚫b. Provide an explicit construction of given u in a specific example, such as u(x, y) = 22-y². 2. Maximum Principle for Harmonic Functions: ⚫a. State and prove the Maximum Principle for harmonic functions, asserting that if u attains its maximum (or minimum) value in the interior of D, then u is constant. b. Apply the Maximum Principle to show that a non-constant harmonic function cannot attain its maximum value inside a bounded domain. 3. Dirichlet Problem: ⚫a. Formulate the Dirichlet problem for finding a harmonic function on a bounded domain D with prescribed continuous boundary values. ⚫b. Prove the existence and uniqueness of the solution to the Dirichlet problem using the method of harmonic conjugates or Perron's method. 4. Green's Function: ⚫a. Define Green's function G(z, C) for the domain D with a pole at C € D. ⚫ b. Derive the representation of a harmonic function u on D in terms of its boundary values and Green's function.
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